A Second-Order Nonlocal Approximation for Surface Poisson Model with Dirichlet Boundary
AA SECOND-ORDER NONLOCAL APPROXIMATION FOR SURFACEPOISSON MODEL WITH DIRICHLET BOUNDARY ∗ ZUOQIANG SHI † AND
YAJIE ZHANG ‡ Abstract.
Partial differential equations on manifolds have been widely studied and plays a crucial role inmany subjects. In our previous work, a class of integral equations was introduced to approximatethe Poisson problems on manifolds with Dirichlet and Neumann type boundary conditions. In thispaper, we restrict our domain into a compact, two dimensional manifold(surface) embedded in highdimensional Euclid space with Dirichlet boundary. Under such special case, a class of more accuratenonlocal models are set up to approximate the Poisson model. One innovation of our model is that,the normal derivative on the boundary is regarded as a variable so that the second order normalderivative can be explicitly expressed by such variable and the curvature of the boundary. Ourconcentration is on the well-posedness analysis of the weak formulation corresponding to the integralmodel and the study of convergence to its PDE counterpart. The main result of our work is that, suchsurface nonlocal model converges to the standard Poisson problem in a rate of O ( δ ) in H norm,where δ is the parameter that denotes the range of support for the kernel of the integral operators.Such convergence rate is currently optimal among all integral models according to the literature.Two numerical experiments are included to illustrate our convergence analysis on the other side. Key words.
Manifold Poisson equation, Dirichlet boundary, integral approximation, well-posedness, second order convergence.
AMS subject classifications.
1. Introduction.
Partial differential equations on manifolds have never failedto attract researchers in the past decades. Its application includes material science[9] [16] , fluid flow [18] [20], and biology physics [6] [17] [27]. In addition, many recentwork on machine learning [7] [11] [23] [25] [31] and image processing [10] [19] [21] [22][28] [30] [37] relates their model with manifold PDEs. One frequently used manifoldmodel is the Poisson equation. Apart from its broad application, such equation ismathematically interesting by itself since it reveals much information of the manifold.Among all the recent study on the numerical analysis of the Poisson models, one ∗ This work of YZ and ZS were supported by NSFC grant 11671005. † Yau Mathematical Sciences Center, Tsinghua University, Beijing, China, 100084.
Email:[email protected] ‡ Department of Mathematical Sciences, Tsinghua University, Beijing, China, 100084.
Email:[email protected] a r X i v : . [ m a t h . NA ] J a n pproach is its integral approximation. The main advantage for integral model isthat the use of explicit spatial differential operators is avoided, hence new numericalschemes can be explored. In spite of the large quantity of articles on the integralapproximation of Poisson model on Euclid spaces, less work have been done on itsextension from Euclid spaces to the manifolds. Due to the demand of numericalmethods on manifold PDEs, it is necessary to introduce one integral model thatapproximate the manifold Poisson equation with high accuracy, hence able to solve itby proper numerical scheme.In this paper, we mainly concentrate on seeking for an integral approximation ofthe following Poisson equation with Dirichlet boundary condition − ∆ M u ( x ) = f ( x ) x ∈ M ; u ( x ) = 0 x ∈ ∂ M . (1.1)Here M is a compact, smooth two dimensional manifold(surface) embedded in R d ,with ∂ M a smooth one dimensional curve with bounded curvature. f is an L functionon M . ∆ M = div ∇ M is the Laplace-Beltrami operator. Now to define ∆ M , weintroduce the definition of manifold divergence and manifold gradient ∇ M : let Φ : Ω ⊂R k → M ⊂ R d be a local parametrization of M and θ ∈ M . For any differentiablefunction f : M → R , we define the gradient on the manifold ∇ M f (Φ( θ )) = m (cid:88) i,j =1 g ij ( θ ) ∂ Φ ∂θ i ( θ ) ∂f (Φ( θ )) ∂θ j ( θ ) , (1.2)and for vector field F : M → T x M on M , where T M is the tangent space of M at x ∈ M , the divergence is defined as div ( F ) = 1 √ det G d (cid:88) k =1 m (cid:88) i,j =1 ∂∂θ i ( √ det Gg ij F k (Φ( θ )) ∂ Φ k ∂θ j ) , (1.3)where ( g ij ) i,j =1 ,...,k = G − , det G is the determinant of matrix G and G ( θ ) =( g ij ) i,j =1 , ,...,k is the first fundamental form which is defined by g ij ( θ ) = d (cid:88) k =1 ∂ Φ k ∂θ i ( θ ) ∂ Φ k ∂θ j ( θ ) , i, j = 1 , ..., m. (1.4)and ( F ( x ) , ..., F d ( x )) t is the representation of F in the embedding coordinates.Now the operator ∆ M has been defined. According to the literature, for anyinteger m ≥ f ∈ H m ( M ), there exists a unique solution u ∈ H m +2 ( M ) that olves the problem (1.1). In this paper, to make the exact solution u sufficientlyregular, we always assume f ∈ H ( M ) so that u ∈ H ( M ).For the simpler case with Euclid domain M = Ω ⊂ R , a widely-known integralapproximation of Poisson equation ∆ u = f is: L δ u ( x ) = 1 δ (cid:90) Ω ( u ( x ) − u ( y )) R δ ( x , y ) dµ y = f ( x ) , x ∈ Ω , (1.5)with appropriate boundary conditions in a layer adjacent to the boundary. The nor-malized interaction kernel function R δ ( x , · ) is supported in a small neighborhood of x with radius 2 δ and is given some additional assumptions. Such model usually ap-pears in the model of peridynamics [3] [8] [13] [29] [34] [36], where the singularityof materials can be effectively evaluated. According to the literature, the solutionof (1.5) converges to the solution of Poisson equation in a rate of O ( δ ) as δ → O ( δ ) convergence rate under Dirich-let/Neumann/Robin type boundaries due to its low accuracy near the boundary, see[4] [5] [12] [14] for the Neumann boundary case and [1] [2] [15] [26] [39] for other typeof boundaries. Specially, in one dimensional [35] and two dimensional [38] cases withNeumann boundary , one can obtain O ( δ ) convergence by doing several modificationsof (1.5) on the boundary layer.In our previous work [32] [33], the model (1.5) was extended to smooth manifoldwith Neumann and Dirichlet boundary condition, where the following approximationof Laplace-Beltrami operator(LBO) is used: − (cid:90) M ∆ M u ( y ) ¯ R δ ( x , y ) dµ y ≈ (cid:90) M δ R δ ( x , y )( u ( x ) − u ( y )) dµ y − (cid:90) ∂ M ¯ R δ ( x , y ) ∂u∂ n ( y ) dτ y , (1.6)where M can be any smooth k -dimensional manifold embedded in R d , n ( y ) is theoutward normal vector of M at y and ∂u∂ n = ∇ M u · n , dµ y and dτ y are the volumeform of M and ∂ M . The kernel functions R δ ( x , y ) and ¯ R δ ( x , y ) are defined as R δ ( x , y ) = C δ R (cid:0) | x − y | δ (cid:1) , ¯ R δ ( x , y ) = C δ ¯ R (cid:0) | x − y | δ (cid:1) , (1.7)where R ∈ C ( R + ) is a positive function that is integrable over [0 , ∞ ), and ¯ R ( r ) = (cid:82) + ∞ r R ( s ) ds . C δ = πδ ) k/ is the normalization factor. According to our previouscalculation, the approximation (1.6) has an error of O ( δ ) in the interior region and O (1) on the boundary layer with width 2 δ . Based on (1.6), we constructed two integralmodels that converges to the manifold Poisson equation linearly on δ under Neumannand Dirichlet boundary. imilar to the cases with Euclid domain, one natural question to consider is theexistence of new integral model with second order convergence to the manifold Poissonequation. To this end, we need a more accurate approximation of LBO than (1.6). Infact, further analysis indicates that the error of (1.6) in the interior region is actually O ( δ ) rather than O ( δ ). However, in order to reduce the error on the boundarylayer, we need the information of the second order normal derivative of u along theboundary ∂ M . Fortunately, such information can be revealed in the case of Dirichletboundary, where the 2nd order normal derivative of u can be expressed by a functionof ∂u∂ n and ∆ M u . This function can be explicitly written when the dimension of M is2. Using such fact, one integral equation can be obtained by adding a correspondingterm to the right hand side of (1.6), which gives more accuracy. Next, to interpretthe Dirichlet boundary condition, we write the second equation as a relation between ∂u∂ n and the value of u on the boundary layer. Such equation has to be constructedin the basis of the first equation so that the cross term can be properly eliminated inthe regularity analysis. We then set up our integral model by combining these twoequations.The main results of our paper is the well-posedness of such model, its second orderconvergence to its local counterpart, and the numerical simulation of our model bypoint cloud method(PIM) that illustrates such convergence rate. Our analytic resultscan be easily generalized into the case with non-homogeneous Dirichlet boundarycondition. The numerical methods PIM have been introduced in our previous paper[24], where the crucial part is to discretize the integral operators on the point cloudthat samples the manifold.The paper is organized as follows: we derive the nonlocal model that approx-imates the Poisson model in section 2. In section 3, the truncation error of theintegral equations is controlled. Next, we describe the properties of the bilinear formcorresponding to the integral equations in section 4. In section 5, we prove that theweak formulation of the system is well-posed. The convergence of our model to thePoisson model is presented in section 6. In section 7, we simulate our model by pointcloud method to realize such convergence rate. Finally, discussion and conclusion isincluded in section 8.
2. Nonlocal Model.
In this section, let us introduce the derivation of our non-local Poisson model. As we mentioned, we approximate the Laplace-Beltrami operator n [32] by the following integral equation − (cid:90) M ∆ M u ( y ) ¯ R δ ( x , y ) dµ y ≈ (cid:90) M δ R δ ( x , y )( u ( x ) − u ( y )) dµ y − (cid:90) ∂ M ¯ R δ ( x , y ) ∂u∂ n ( y ) dτ y , (2.1)Here kernel function R plays a crucial role in the integral model and is related withmost of the calculation. A better choice of R results in much less work on the erroranalysis. To this end, we make the following assumptions on the kernel R ( r ):1. Smoothness: d dr R ( r ) is bounded, i.e., for any r ≥ (cid:12)(cid:12) d dr R ( r ) (cid:12)(cid:12) ≤ C ;2. Nonnegativity: R ( r ) > r ≥ R ( r ) = 0 for any r > ∃ δ > R ( r ) ≥ δ > ≤ r ≤ / R not blow up near the origin,while the fourth condition assures a non-degenerate interval for both R and ¯ R . Afrequently used kernel function R is the following: R ( r ) = (1 + cos πr ) , ≤ r ≤ , , r > . (2.2)The control of the error terms of (2.1) were given in the theorem 4 of [32] under theabove assumption. Our work [32] gives a nonlocal Poisson model with homogeneousNeumann boundary by removing the last term of (2.1) and replacing − ∆ M u by f :1 δ (cid:90) M ( u δ ( x ) − u δ ( y )) R δ ( x , y ) dµ y = (cid:90) M f ( y ) ¯ R δ ( x , y ) dµ y , ∀ x ∈ M . (2.3)In this paper, we aim to construct a nonlocal Poisson model with homogeneousDirichlet boundary. The term that involves the normal derivative of u in (2.1) willno longer vanish in our case. Nevertheless, based on the approximation (2.1), we canstill construct a nonlocal Poisson model with Dirichlet boundary by approximatingthe boundary condition as a relation between u and ∂u∂ n . Same as the model (2.3)with Neumann boundary, such model gives at most first order convergence to its localcounterpart. To make an innovation, instead of applying (2.1) directly into a nonlocalmodel, let us introduce a new approximation by adding a higher order term to theRHS of (2.1): (cid:90) M ∆ M u ( y ) ¯ R δ ( x , y ) dµ y ≈ δ (cid:90) M ( u ( x ) − u ( y )) R δ ( x , y ) dµ y − (cid:90) ∂ M ∂u∂ n ( y ) ¯ R δ ( x , y ) dτ y − (cid:90) ∂ M (( x − y ) · n ( y )) ∂ u∂ n ( y ) ¯ R δ ( x , y ) dτ y . (2.4)Here ∂ u∂ n = d (cid:80) i =1 d (cid:80) j =1 n i n j ∇ i M ∇ j M u and n i , n j is the i, j th component of n . To simplifythe notation, we always write as ∂ u∂ n = n i n j ∇ i ∇ j u or u nn . By a simple observation,we see the term we added is supported on the boundary layer with width 2 δ . In ourcoming analysis, we will show such approximation gives more accuracy than (2.1) onthis layer. For the interior part away from the layer, the estimate (2.4) remains thesame as (2.1). However, a standard symmetric analysis on the higher order termsindicates that the interior error can be further controlled.The main difficulty to apply such approximation into a nonlocal model is thenumber of unknown functions. There are three functions to solve: u , ∂u∂ n and ∂ u∂ n ,but we can only have up to two equations to describe their relation. In fact, as theshape of the boundary ∂ M is given and u ≡ u on ∂ M . Specially, in the twodimensional case as we are studying, we can explicitly write down the equality: ∂ u∂ τ ( y ) = κ ( y )( n ( y ) · n b ( y )) ∂u∂ n ( y ) , ∀ y ∈ ∂ M , (2.5)where τ ( y ) represents the unit tangent vector of ∂ M at y , κ ( y ) denotes the curvatureof ∂ M at y , and n b ( y ) denotes the unit principal normal vector of ∂ M at y . Inaddition, we can derive the following important equality∆ M u ( y ) = ∂ u∂ n ( y ) + ∂ u∂ τ ( y ) . ∀ y ∈ ∂ M . (2.6)The above two equation gives ∂ u∂ n ( y ) = ∆ M u ( y ) − κ ( y )( n ( y ) · n b ( y )) ∂u∂ n ( y ) , ∀ y ∈ ∂ M , (2.7)thereafter, we return to (2.4) to discover − (cid:90) M ∆ M u ( y ) ¯ R δ ( x , y ) dµ y ≈ δ (cid:90) M ( u ( x ) − u ( y )) R δ ( x , y ) dµ y − (cid:90) ∂ M ∂u∂ n ( y ) ¯ R δ ( x , y ) dτ y − (cid:90) ∂ M (( x − y ) · n ( y ))(∆ M u ( y ) − κ ( y )( n ( y ) · n b ( y )) ∂u∂ n ( y )) ¯ R δ ( x , y ) dτ y . (2.8) he proof of the equality (2.7) will be introduced in the lemma 3.3 in the comingsection. We see the term u nn has been expressed by u n and other functions. A closerapproximation to the equation ∆ M u = f can be obtained after substituting ∆ M u with f in (2.8). Now another important question is that how to retain the coercivityof our nonlocal model. We aim to construct a second equation based on the Dirichletboundary condition. Since the coefficient of u n in (2.8) is known, we impose thesame coefficient on u in the second equation so that the cross terms can be properlyeliminated in the coming analysis. The coefficient of u n in the second equation canthen be calculated using Taylor expansion and the Dirichlet boundary condition. Adetailed calculation will be introduced in the next section.As a consequence, we can eventually set up our model as follows: L δ u δ ( x ) − G δ v δ ( x ) = P δ f ( x ) , x ∈ M , D δ u δ ( x ) + ˜ R δ ( x ) v δ ( x ) = Q δ f ( x ) , x ∈ ∂ M . (2.9)where the operators are defined as L δ u δ ( x ) = 1 δ (cid:90) M ( u δ ( x ) − u δ ( y )) R δ ( x , y ) dµ y , (2.10) G δ v δ ( x ) = (cid:90) ∂ M v δ ( y ) (2 − ( x − y ) · p ( y ) n ( y )) ¯ R δ ( x , y ) dτ y , (2.11) D δ u δ ( x ) = (cid:90) M u δ ( y ) (2 + ( x − y ) · p ( x ) n ( x )) ¯ R δ ( x , y ) dµ y , (2.12)˜ R δ ( x ) = 4 δ (cid:90) ∂ M = R δ ( x , y ) dτ y + (cid:90) M p ( x ) (( x − y ) · n ( x )) ¯ R δ ( x , y ) dµ y , (2.13) P δ f ( x ) = (cid:90) M f ( y ) ¯ R δ ( x , y ) dµ y − (cid:90) ∂ M (( x − y ) · n ( y )) f ( y ) ¯ R δ ( x , y ) dτ y , (2.14) Q δ f ( x ) = − δ (cid:90) M f ( y ) = R δ ( x , y ) dµ y ; (2.15)and the function p ( x ) = κ ( x )( n ( x ) · n b ( x )) . (2.16)Our goal in this paper is to present that such model assures a unique solution( u δ , v δ ) and it approaches ( u, ∂u∂ n ) with second order accuracy on δ . In other words,our main results are the following two theorems. Theorem 2.1 (
Well-Posedness ). . For each fixed δ > and f ∈ H ( M ) , there exists a unique solution u δ ∈ L ( M ) , v δ ∈ L ( ∂ M ) to the integral model (2.9) , with the following estimate (cid:107) u δ (cid:107) L ( M ) + δ (cid:107) v δ (cid:107) L ( ∂ M ) ≤ C (cid:107) f (cid:107) H ( M ) . (2.17)
2. In addition, we have u δ ∈ H ( M ) as well, with (cid:107) u δ (cid:107) H ( M ) ≤ C (cid:107) f (cid:107) H ( M ) . (2.18) Here the constant C in the above inequalities are independent on δ . Theorem 2.2 (
Quadratic Convergence Rate ). Let u be the solution to the Pois-son model (1.1) , and ( u δ , v δ ) be the solution to the integral model (2.9) , then we havethe following estimate (cid:107) u − u δ (cid:107) H ( M ) + δ / (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂ n − v δ (cid:13)(cid:13)(cid:13)(cid:13) L ( ∂ M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) , (2.19) where the constant is independent to δ . This two theorem indicates that the solution of (2.9) exist uniquely and u δ ap-proaches u in a rate of O ( δ ) in H norm, while v δ converges to ∂u∂ n in a rate of O ( δ / )in L norm. In the following two sections, the error control and the coercivity of (2.9)will be given. The proof of theorem 2.1 and 2.2 will be given separately in the section5 and 6.
3. Analysis of the Truncation Errors.
In the previous section, we describedhow our integral model (2.9) was formulated. In order to present the convergence rateclaimed in the theorem 2.2, one crucial part is to control the truncation error betweenour model and the original Poisson equation (1.1). Let u be the exact solution of(1.1) and ∂u∂ n be its normal derivative at ∂ M , we denote the truncation errors as r in ( x ) = L δ u ( x ) − G δ ∂u∂ n ( x ) − P δ f ( x ) , x ∈ M , (3.1) r bd ( x ) = D δ u ( x ) + ˜ R δ ( x ) ∂u∂ n ( x ) − Q δ f ( x ) , x ∈ ∂ M . (3.2)The RHS of (3.1) and (3.2) equals 0 after replacing ( u, ∂u∂ n ) with ( u δ , v δ ). To makethis approximation more accurate, we want the terms r in and r bd to be sufficientlysmall. Next, we are ready to state the main theorem of this section. Theorem 3.1.
Let u ∈ H ( M ) solves the system (1.1) , v ( x ) = ∂u∂ n ( x ) for x ∈ ∂ M , and r in , r bd are the functions given in the equation (3.1) and (3.2) , then we an decompose r in into r in = r it + r bl , where r it is supported in the whole domain M ,with the following bound δ (cid:107) r it (cid:107) L ( M ) + (cid:107)∇ r it (cid:107) L ( M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) ; (3.3) and r bl is supported in the layer adjacent to the boundary ∂ M with width δ : supp ( r bl ) ⊂ { x (cid:12)(cid:12) x ∈ M , dist ( x , ∂ M ) ≤ δ } , (3.4) and satisfy the following two estimates δ (cid:107) r bl (cid:107) L ( M ) + (cid:107)∇ r bl (cid:107) L ( M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) ; (3.5) (cid:90) M r bl ( x ) f ( x ) dµ x ≤ Cδ (cid:107) u (cid:107) H ( M ) ( (cid:107) f (cid:107) H ( M ) + (cid:13)(cid:13) ¯ f (cid:13)(cid:13) H ( M ) + (cid:13)(cid:13)(cid:13)(cid:13) = f (cid:13)(cid:13)(cid:13)(cid:13) H ( M ) ) , ∀ f ∈ H ( M ) , (3.6) where the notations ¯ f ( x ) = 1¯ ω δ ( x ) (cid:90) M f ( y ) ¯ R δ ( x , y ) dµ y , = f ( x ) = 1 = ω δ ( x ) (cid:90) M f ( x ) = R δ ( x , y ) dµ y , (3.7) represents the weighted average of f in B δ ( x ) with respect to ¯ R and = R , and ¯ ω δ ( x ) = (cid:90) M ¯ R δ ( x , y ) dµ y , = ω δ ( x ) = (cid:90) M = R δ ( x , y ) dµ y , ∀ x ∈ M . In addition, we have the following estimate for r bd : (cid:107) r bd (cid:107) L ( ∂ M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) . (3.8)This theorem gives a complete control on the truncation errors r in and r bd , andis a crucial step for presenting the convergence results claimed in the theorem 2.2.In a simple word, this theorem indicates that r in is O ( δ ) in the interior region,and is O ( δ ) in the boundary layer with width 2 δ , while it satisfies some additionalsymmetric properties on such layer. Since r in and r bd is defined on M , we need a localparametrization of the surface M to write down their explicit formulation. Next, tobetter illustrate the control on these two terms, we are ready to introduce a speciallocal parametrization of M .We let δ be sufficiently small so that it is less than 0 . M and ∂ M . For the definition of the reach of the manifold, one can refer to the first art of section 5 of [32]. Under such assumption, for each x ∈ M , its neighborhood B δ x is holomorphic to the Euclid space R . Since M is compact, there exists a 2 δ -net, N δ = { q i ∈ M , i = 1 , ..., N } , such that M ⊂ N (cid:91) i =1 B δ q i , (3.9)and there exists a partition of M , {O i , i = 1 , ..., N } , such that O i ∩ O j = ∅ , ∀ i (cid:54) = j and M = N (cid:91) i =1 O i , O i ⊂ B δ q i , i = 1 , , ..., N. (3.10) Lemma 3.2.
There exist a parametrization φ i : Ω i ⊂ R → U i ⊂ M , i =1 , , ..., N , such that1. (Convexity) B δ q i ∩ M ⊂ U i and Ω i is convex,2. (Smoothness) φ i ∈ C (Ω i ) ,3. (Local small deformation) For any points θ , θ ∈ Ω i . | θ − θ | ≤ (cid:107) φ i ( θ ) − φ i ( θ ) (cid:107) ≤ | θ − θ | . (3.11)This lemma is a corollary of the proposition 1 of [32]. Such lemma gives a localparametrization of M . It indicates that for each x ∈ M , there is a unique index J ( x ) ∈ { , , ..., N } such that x ∈ O J ( x ) , while B δ x ∈ U J ( x ) . For any y ∈ B δ x , wedefine the function ξ ( x , y ) = φ − J ( x ) ( y ) − φ − J ( x ) ( x ) , (3.12)and the auxiliary function η ( x , y ) = < η , η , ..., η d > , where η j ( x , y ) = ξ i ( x , y ) ∂ i φ jJ ( x ) ( y ) , i = 1 , . (3.13)Here η = η ( x , y ) is an auxiliary function that approximates the vector ( y − x ) inthe tangential plane T M ( x ). To simplify our notations, we always write α to denote φ − J ( x ) ( x ), and β to denote φ − J ( x ) ( y ), with ξ = β − α .Now let us return to the theorem 3.1 regarding r in . In fact, by a simple observa-tion, r in ( x ) = (cid:90) M ∆ M u ( y ) ¯ R δ ( x , y ) dµ y + 1 δ (cid:90) M ( u ( x ) − u ( y )) R δ ( x , y ) dµ y − (cid:90) ∂ M ∂u∂ n ( y ) ¯ R δ ( x , y ) dτ y − (cid:90) ∂ M (( x − y ) · n ( y ))(∆ M u ( y ) − κ ( y )( n ( y ) · n b ( y )) ∂u∂ n ( y )) ¯ R δ ( x , y ) dτ y . (3.14) ere the first line of r in in (3.14) is exactly the function r in the page 10 of [32]. Inother words, r in = r − r (cid:48) , where r (cid:48) ( x ) = (cid:90) ∂ M (( x − y ) · n ( y ))(∆ M u ( y ) − κ ( y )( n ( y ) · n b ( y )) ∂u∂ n ( y )) ¯ R δ ( x , y ) dτ y . (3.15)By a similar way as we decomposed r in [32], we split r in into four terms r in = r + r + r − r , where r ( x ) = 1 δ (cid:90) M ( u ( x ) − u ( y ) − ( x − y ) ·∇ u ( y ) − η i η j ( ∇ i ∇ j u ( y ))) R δ ( x , y ) dµ y , (3.16) r ( x ) = 12 δ (cid:90) M η i η j ( ∇ i ∇ j u ( y )) R δ ( x , y ) dµ y − (cid:90) M η i ( ∇ i ∇ j u ( y )) ∇ j ¯ R δ ( x , y ) dµ y , (3.17) r ( x ) = (cid:90) M η i ( ∇ i ∇ j u ( y )) ∇ j ¯ R δ ( x , y ) dµ y + (cid:90) M div ( η i ( ∇ i ∇ u ( y )) ¯ R δ ( x , y ) dµ y − r (cid:48) , (3.18) r ( x ) = (cid:90) M div ( η i ( ∇ i ∇ u ( y )) ¯ R δ ( x , y ) dµ y + (cid:90) M ∆ M u ( y ) ¯ R δ ( x , y ) dµ y . (3.19)Here the term r i , i = 1 , , r i in [32], and the term r is theold r subtracted by r (cid:48) . In order to give a complete control on the error term r in ,we need to study each term r i , i = 1 , , , r wasalready proved to satisfy the control (3.3). while a weaker estimate was presentedfor r , r . Our purpose here is to decompose r and r into an interior term and aboundary layer term via integration by parts, thus to seek for a stronger estimate.For the term r , we obtain from integration by parts that r ( x ) = (cid:90) ∂ M n j η i ( ∇ i ∇ j u ( y )) ¯ R δ ( x , y ) dτ y − r (cid:48) ( x ) , (3.20)where n j denotes the j th component of n ( x ). since the vector η ( x , y ) lies on thetangent plane of M at y , We can decompose r into r ( x ) = (cid:90) ∂ M ( η · n ) n j n i ( ∇ i ∇ j u ( y )) ¯ R δ ( x , y ) dτ y + (cid:90) ∂ M ( η · τ ) n j τ i ( ∇ i ∇ j u ( y )) ¯ R δ ( x , y ) dτ y − r (cid:48) ( x )= (cid:0) (cid:90) ∂ M (( x − y ) · n ( y )) u nn ( y ) ¯ R δ ( x , y ) dτ y − r (cid:48) ( x ) (cid:1) + (cid:90) ∂ M ( η · τ ) u n τ ( y ) ¯ R δ ( x , y ) dτ y − (cid:90) ∂ M n ( y ) · (( x − y ) − η ( x , y )) u nn ( y ) ¯ R δ ( x , y ) dτ y , (3.21) here τ ( y ) represents the unit tangent vector of ∂ M at y , and τ i denotes the i th component of τ ( y ). Like r and r , the last two terms are of higher order and willbe controlled later using integration by parts. For the first term, we aim to show it iszero by proving the following lemma: Lemma 3.3.
For each x ∈ ∂ M , we have the following equality (equation (2.7) )hold: n j n i ( ∇ i ∇ j u ( x )) = ∆ M u ( x ) − κ ( x )( n ( x ) · n b ( x )) ∂u∂ n ( x ) , (3.22)The above lemma indicates that the new r is a higher order term compared tothe old r in [32]. This is the crucial reason why the error can be reduced after addingthe second order term in (2.4).The next lemma can be applied to several terms that composes r , r and r .It gives an orientation on the control of the higher order terms supported on theboundary layer. Lemma 3.4.
Let f : M → R , g : M → R m . If f ∈ H ( M ) and g ∈ [ H ( M )] m , then we have (cid:90) ∂ M (cid:90) M f ( x ) g ( y ) · ( x − y − η ( x , y )) R δ ( x , y ) dµ x dτ y ≤ Cδ ( (cid:107) f (cid:107) H ( M ) + (cid:13)(cid:13) ¯ f (cid:13)(cid:13) H ( M ) ) (cid:107) g (cid:107) H ( M ) , (3.23) where ¯ f is the function defined in (3.7) , and η is defined in (3.13) . Next, we are ready to prove the main theorem of this section.
Proof . [Proof of Theorem 3.1] Since r in = r + r + r − r , the idea to control r in is to decompose r i , i = 1 , , , r bd , wecan decompose it into four terms by going through the derivation of (3.2), and thencontrol each of them using proper techniques.1. Let us first study r , recall r ( x ) = 1 δ (cid:90) M ( u ( x ) − u ( y ) − ( x − y ) ·∇ u ( y ) − η i η j ( ∇ i ∇ j u ( y ))) R δ ( x , y ) dµ y , (3.24)In the paper [32], this term was decomposed into three lower order terms usingCauchy-Leibniz formula, and the lower order terms have been shown to be O ( δ ). In this paper, we decompose these 3 terms again using Cauchy-Leibniz ormula and get three O ( δ ) terms and three O ( δ ) terms. After integrationby parts on the parametrized domain Ω for several times, each of the three O ( δ ) terms can be split into two parts, while one satisfies (3.3) and anothersatisfies (3.5) and (3.6). we move the control of r into appendix due to itscomplicated calculation.2. For the term r , The equation (27) of [32] directly gives (cid:90) M | r ( x ) | dµ x ≤ Cδ (cid:107) u (cid:107) H ( M ) , (3.25) (cid:90) M |∇ r ( x ) | dµ x ≤ Cδ (cid:107) u (cid:107) H ( M ) . (3.26)Hence it satisfies the control (3.3).3. Next, let us analyze r . The equation (3.21) and the lemma 3.3 gives r ( x ) = (cid:90) ∂ M ( η · τ ) u n τ ( y ) ¯ R δ ( x , y ) dτ y − (cid:90) ∂ M n ( y ) · (( x − y ) − η ( x , y )) u nn ( y ) ¯ R δ ( x , y ) dτ y , (3.27)For the second term of (3.27), we directly apply the lemma 3.4 to obtain that (cid:90) M f ( x ) (cid:90) ∂ M n ( y ) · (( x − y ) − η ( x , y )) u nn ( y ) ¯ R δ ( x , y ) dτ y ≤ C ( (cid:107) f (cid:107) H ( M ) + (cid:13)(cid:13)(cid:13)(cid:13) = f (cid:13)(cid:13)(cid:13)(cid:13) H ( M ) ) (cid:107) u (cid:107) H ( M ) , ∀ f ∈ H ( M ) , (3.28)where the term = f is defined in (3.7). For the first term of (3.27), we candecompose it as (cid:90) ∂ M ( η · τ ) u n τ ( y ) ¯ R δ ( x , y ) dτ y = 2 δ (cid:90) ∂ M τ ( y ) u n τ ( y ) · ∇ y = R δ ( x , y ) dτ y +2 δ (cid:90) ∂ M τ ( y ) u n τ ( y ) · ( 12 δ η ( x , y ) ¯ R δ ( x , y ) − ∇ y = R δ ( x , y )) dτ y , (3.29)and we can calculate (cid:90) ∂ M τ ( y ) u n τ ( y ) · ∇ y = R δ ( x , y ) dτ y = (cid:90) γ dds = R δ ( x , γ ( s )) u τ n ( γ ( s )) ds = − (cid:90) γ = R δ ( x , γ ( s )) dds u τ n ( γ ( s )) ds = − (cid:90) ∂ M = R δ ( x , y ) u ττ n ( y ) dµ y , (3.30) here γ is the equation of the curve ∂ M with | γ (cid:48) | ≡ ∇ j y = R δ ( x , y ) − η j δ ¯ R δ ( x , y )= 12 δ ξ i (cid:48) ξ j (cid:48) ∂ m (cid:48) φ i g m (cid:48) n (cid:48) ∂ n (cid:48) φ j ( (cid:90) (cid:90) s∂ j (cid:48) ∂ i (cid:48) φ j ( α + τ sξ ) dτ ds ) ¯ R δ ( x , y )= 14 δ ξ i (cid:48) ξ j (cid:48) ∂ m (cid:48) φ i g m (cid:48) n (cid:48) ∂ n (cid:48) φ j ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) ¯ R δ ( x , y )+ 12 δ ξ i (cid:48) ξ j (cid:48) ξ k (cid:48) ∂ m (cid:48) φ i g m (cid:48) n (cid:48) ∂ n (cid:48) φ j ( (cid:90) (cid:90) (cid:90) s τ ∂ j (cid:48) ∂ i (cid:48) ∂ k (cid:48) φ j ( α + τ stξ ) dtdτ ds ) ¯ R δ ( x , y ) , (3.31)which indicates that we can follow the control of d in the equation (9.42)(9.43) (9.44) in the appendix to obtain a same bound for r .4. for r , we have from (33) of [32] that r ( x ) = (cid:90) M ξ l (cid:112) det G ( y ) ∂ i (cid:48) (cid:0) √ det Gg i (cid:48) j (cid:48) ( ∂ j (cid:48) φ j )( ∂ l φ i )( ∇ i ∇ j u ( y )) (cid:1) ¯ R δ ( x , y ) dµ y . (3.32)We apply (9.10) to discover r ( x ) = (cid:90) M (cid:112) det G ( y ) ∂ i (cid:48) (cid:0)(cid:112) det G ( y ) g i (cid:48) j (cid:48) ( ∂ j (cid:48) φ j )( ∂ l φ i )( ∇ i ∇ j u ( y )) (cid:1) ξ l ¯ R δ ( x , y ) dµ y =2 δ (cid:90) M (cid:112) det G ( y ) ∂ i (cid:48) (cid:0)(cid:112) det G ( y ) g i (cid:48) j (cid:48) ( ∂ j (cid:48) φ j )( ∂ l φ i )( ∇ i ∇ j u ( y )) (cid:1) g k (cid:48) l ( y ) ∂ y k (cid:48) = R δ ( x , y ) dµ y + r =2 δ (cid:90) Ω ∂ i (cid:48) (cid:0)(cid:112) det G ( y ) g i (cid:48) j (cid:48) ( ∂ j (cid:48) φ j )( ∂ l φ i )( ∇ i ∇ j u ( y )) (cid:1) g k (cid:48) l ( y ) ∂ y k (cid:48) = R δ ( x , y ) dA y + r = − δ (cid:90) Ω ∂ k (cid:48) (cid:16) ∂ i (cid:48) (cid:0)(cid:112) det G ( y ) g i (cid:48) j (cid:48) ( ∂ j (cid:48) φ j )( ∂ l φ i )( ∇ i ∇ j u ( y )) (cid:1) g k (cid:48) l ( y ) (cid:17) = R δ ( x , y ) dA y + 2 δ (cid:90) ∂ Ω ∂ i (cid:48) (cid:0)(cid:112) det G ( y ) g i (cid:48) j (cid:48) ( ∂ j (cid:48) φ j )( ∂ l φ i )( ∇ i ∇ j u ( y )) (cid:1) g k (cid:48) l ( y ) n k (cid:48) M ( y ) = R δ ( x , y ) dA y + r (3.33)where the control for the first term is similar to the control of d in theappendix, and the control for the second term is similar to (9.15), while thethird term r = (cid:90) M (cid:112) det G ( y ) ∂ i (cid:48) (cid:0)(cid:112) det G ( y ) g i (cid:48) j (cid:48) ( ∂ j (cid:48) φ j )( ∂ l φ i )( ∇ i ∇ j u ( y )) (cid:1) d l dµ y (3.34)is obviously a lower order term. Here d is the error function defined as(9.35) in the appendix. . So far, we have finished the control of all the terms that composes r in . Inconclusion, they can be classified into two groups r it and r bl , where r it satisfies(3.3) and r bl has support on the boundary layer and satisfies (3.5) (3.6).Finally, what left in the theorem is the control of the term r bd . Recall r bd ( x ) =2 (cid:90) M u ( y ) ¯ R δ ( x , y ) dµ y + 4 δ ∂u∂ n ( x ) (cid:90) ∂ M = R δ ( x , y ) dτ y + 2 δ (cid:90) M f ( y ) = R δ ( x , y ) dµ y + p ( x ) (cid:0) (cid:90) M u ( y ) ( x − y ) · n ( x ) ¯ R δ ( x , y ) dµ y + ∂u∂ n ( x ) (cid:90) M (( x − y ) · n ( x )) ¯ R δ ( x , y ) dµ y (cid:1) , (3.35)we then split r bd into r bd = 2 r + 2 r + 2 δ ¯ r in − pr , where r = δ (cid:90) ∂ M (( x − y ) · n ( y ))( f ( y ) + p ( y ) ∂u∂ n ( y )) = R δ ( x , y ) dτ y , (3.36) r = 2 δ ( (cid:90) ∂ M ∂u∂ n ( y ) = R δ ( x , y ) dτ y − (cid:90) ∂ M ∂u∂ n ( x ) = R δ ( x , y ) dτ y ) , (3.37) r = (cid:90) M (cid:0) u ( y ) − u ( x ) + ( x − y ) · ∇ u ( x ) (cid:1) (( x − y ) · n ( x )) ¯ R δ ( x , y ) dµ y , (3.38)¯ r in = (cid:90) M f ( y ) = R δ ( x , y ) dµ y − δ (cid:90) M ( u ( x ) − u ( y )) ¯ R δ ( x , y ) dµ y + 2 (cid:90) ∂ M ∂u∂ n ( y ) = R δ ( x , y ) dµ y + (cid:90) ∂ M (( x − y ) · n ( y ))( − f ( y ) − p ( y ) ∂u∂ n ( y )) = R δ ( x , y ) dµ y . (3.39)Here the term u ( x ) ≡ r bd . In fact,since the term ( x − y ) · n ( x ) is O ( δ ) for x , y ∈ ∂ M with | x − y | < δ due tothe smoothness of ∂ M , we have the following bound for r : (cid:107) r (cid:107) L ( ∂ M ) = (cid:90) ∂ M ( δ (cid:90) ∂ M (( x − y ) · n ( y )) ( f ( y ) + p ( y ) ∂u∂ n ( y )) = R δ ( x , y ) dτ y ) dτ x ≤ Cδ (cid:90) ∂ M ( (cid:90) ∂ M δ |∇ M u ( y ) | = R δ ( x , y ) dτ y ) dτ x ≤ Cδ (cid:90) ∂ M ( (cid:90) ∂ M |∇ M u ( y ) | R δ ( x , y ) dτ y ) ( (cid:90) ∂ M = R δ ( x , y ) dτ y ) dτ x ≤ Cδ (cid:90) ∂ M |∇ M u ( y ) | dτ y = Cδ (cid:107) u (cid:107) H ( ∂ M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) . (3.40) or the next term r , the trick we need here is to apply the approximation(2.1). Since ∂ M is a one dimensional closed curve, we use (2.1) in one di-mensional case on the smooth curve ∂ M to obtain1 δ (cid:90) ∂ M ( ∂u∂ n ( y ) − ∂u∂ n ( x )) = R δ ( x , y ) dτ y = (cid:90) ∂ M ∂ ∂ τ ∂u∂ n ( y ) ≡ R δ ( x , y ) dτ y + r , (3.41)and r satisfy the property (3.5) as well. Here the function ≡ R δ ( x , y ) = C δ ≡ R (cid:0) | x − y | δ (cid:1) , and ≡ R ( r ) = (cid:82) + ∞ r = R ( s ) ds . This gives (cid:107) r (cid:107) L ( ∂ M ) ≤ δ (cid:90) ∂ M ( ∂ ∂ τ ∂u∂ n ( y ) ≡ R δ ( x , y ) dτ y ) dτ x + 4 δ (cid:107) r (cid:107) L ( ∂ M ) ≤ δ (cid:90) ∂ M ( |∇ M u ( y ) | ≡ R δ ( x , y ) dτ y ) dτ x + 4 δ (cid:13)(cid:13) ∇ M u (cid:13)(cid:13) L ( ∂ M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) , (3.42)where last inequality is the same as r . Next, for the term r , we have | r | ≤ Cδ (cid:90) M | u ( y ) − u ( x ) − ( y − x ) · ∇ u ( x ) | ¯ R δ ( x , y ) dµ y ≤ Cδ (cid:90) M | u ( y ) − u ( x ) − ( y − x ) · ∇ u ( x ) − η i η j ∇ i ∇ j u ( x ) | ¯ R δ ( x , y ) dµ y + Cδ (cid:90) M | η i η j ∇ i ∇ j u ( x ) | ¯ R δ ( x , y ) dµ y ≤ Cδ δ (cid:90) M | u ( y ) − u ( x ) − ( y − x ) · ∇ u ( x ) − η i η j ∇ i ∇ j u ( x ) | ¯ R δ ( x , y ) dµ y + Cδ |∇ u ( x ) | . (3.43)For the second term, it is clear that its boundary L norm is bounded by δ (cid:107) u (cid:107) H ( M ) . What left is the control of the first term. We denote it as ¯ r .Recall in (3.24) we have controlled the term r ( x ) = 1 δ (cid:90) M ( u ( y ) − u ( x ) − ( y − x ) · ∇ u ( x ) − η i η j ∇ i ∇ j u ( x )) R δ ( x , y ) dµ y , (3.44)in the sense that (cid:107) r (cid:107) H ( M ) ≤ Cδ − (cid:107) u (cid:107) H ( M ) . (3.45)By replacing R with the kernel ¯ R , we have the same argument hold for ¯ r as r : (cid:107) ¯ r (cid:107) H ( M ) ≤ Cδ δ − (cid:107) u (cid:107) H ( M ) = Cδ (cid:107) u (cid:107) H ( M ) , (3.46) hich gives (cid:107) ¯ r (cid:107) L ( ∂ M ) ≤ C (cid:107) ¯ r (cid:107) H ( M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) , (3.47)therefore (cid:107) r (cid:107) L ( ∂ M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) . (3.48)Finally, what left is the term ¯ r in . By a simple observation, we see ¯ r in isexactly the term r in after replacing the kernel functions R δ in (3.1) by ¯ R δ in(3.39). recall our estimate on r in in the first part of theorem: (cid:107) r in (cid:107) L ( ∂ M ) ≤ (cid:107) r it (cid:107) L ( ∂ M ) + (cid:107) r bl (cid:107) L ( ∂ M ) ≤ C ( (cid:107) r it (cid:107) H ( M ) + (cid:107) r bl (cid:107) H ( M ) ) ≤ Cδ (cid:107) u (cid:107) H ( M ) , our previous calculation indicates that the same bound holds for ¯ r in : (cid:107) ¯ r in (cid:107) L ( ∂ M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) . Finally, we sum up all the above estimates to conclude (cid:107) r bd (cid:107) L ( ∂ M ) ≤ (cid:107) r (cid:107) L ( ∂ M ) + (cid:107) r (cid:107) L ( ∂ M ) + (cid:107) p r (cid:107) L ( ∂ M ) + 2 δ (cid:107) ¯ r in (cid:107) L ( ∂ M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) . (3.49)Hence we have completed the proof of theorem 3.1.
4. Stability Analysis of Nonlocal Model.
Now we have completed the anal-ysis on the truncation error of our integral model (2.9). In this section, our goal isto evaluate the stability of the integral model (2.9). In other words, we aim to findthe connection between the functions ( m δ , n δ ) and ( p δ , q δ ), where m δ , p δ ∈ L ( M ), n δ , q δ ∈ L ( ∂ M ) that satisfies L δ m δ ( x ) − G δ n δ ( x ) = p δ ( x ) , x ∈ M , D δ m δ ( x ) + ˜ R δ ( x ) n δ ( x ) = q δ ( x ) , x ∈ ∂ M , (4.1)To begin with, let us construct the corresponding bilinear form of (2.9). For any w δ ∈ L ( M ), s δ ∈ L ( ∂ M ), we define the following bilinear function: B δ [ m δ , n δ ; w δ , s δ ] = (cid:90) M w δ ( x )( L δ m δ ( x ) − G δ n δ ( x )) dµ x + (cid:90) ∂ M s δ ( x )( D δ m δ ( x ) + ˜ R δ ( x ) n δ ( x )) dτ x = (cid:90) M w δ ( x ) L δ m δ ( x ) dµ x − (cid:90) M w δ ( x ) G δ n δ ( x ) dµ x + (cid:90) ∂ M s δ ( x ) D δ m δ ( x ) dτ x + (cid:90) ∂ M s δ ( x ) n δ ( x ) ˜ R δ ( x ) dτ x , (4.2) nd the system (4.1) implies B δ [ m δ , n δ ; w δ , s δ ] = (cid:90) M w δ ( x ) p δ ( x ) dµ x + (cid:90) ∂ M s δ ( x ) q δ ( x ) dτ x , ∀ w δ ∈ L ( M ) , s δ ∈ L ( ∂ M ) . (4.3)Since w δ and s δ are arbitrary L functions, we can easily deduce that the weakformulation (4.3) is equivalent to the integral model (4.1). In this section, we mainlyconcentrate on presenting the properties of the bilinear function (4.2). Lemma 4.1 (
Non-Negativity of the Bilinear Form ). we have B δ [ m δ , n δ ; m δ , n δ ] = 12 δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ x dµ y + (cid:90) ∂ M n δ ( x ) ˜ R δ ( x ) dτ x . (4.4) Proof . [Proof of Lemma 4.1]We calculate each term of the bilinear form in (4.2) after substituting ( w δ , s δ ) by( m δ , n δ ): (cid:90) M m δ ( x ) L δ m δ ( x ) dµ x = 1 δ (cid:90) M m δ ( x ) (cid:90) M ( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ y dµ x = 1 δ (cid:90) M m δ ( y ) (cid:90) M ( m δ ( y ) − m δ ( x )) R δ ( x , y ) dµ x dµ y = 12 δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y ))( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ x dµ y = 12 δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ x dµ y , (4.5) (cid:90) M m δ ( x ) G δ n δ ( x ) dµ x = (cid:90) M m δ ( x ) (cid:90) ∂ M n δ ( y ) (2 − κ ( y ) ( x − y ) · n ( y )) ¯ R δ ( x , y ) dτ y dµ x = (cid:90) M m δ ( y ) (cid:90) ∂ M n δ ( x ) (2 + κ ( x ) ( x − y ) · n ( x )) ¯ R δ ( x , y ) dτ x dµ y = (cid:90) ∂ M n δ ( x ) (cid:90) M m δ ( y ) (2 + κ ( x ) ( x − y ) · n ( x )) ¯ R δ ( x , y ) dµ y dτ x = (cid:90) ∂ M n δ ( x ) D δ m δ ( x ) dτ x , (4.6)the above equation (4.5) and (4.6) gives B δ [ m δ , n δ ; m δ , n δ ] = 12 δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ x dµ y + (cid:90) ∂ M n δ ( x ) ˜ R δ ( x ) dτ x , (4.7) here the cross terms are eliminated by each other. Lemma 4.2 (
Regularity ). For any functions m δ , p δ ∈ L ( M ) , and n δ , q δ ∈ L ( ∂ M ) that satisfies the system of equations (4.1) ,1. there exists a constant C independent to δ such that B δ [ m δ , n δ ; m δ , n δ ] + 1 δ (cid:107) q δ (cid:107) L ( ∂ M ) ≥ C ( (cid:107) m δ (cid:107) L ( M ) + δ (cid:107) n δ (cid:107) L ( ∂ M ) ); (4.8)
2. If in addition, p δ satisfies the following conditions(a) (cid:107)∇ p δ (cid:107) L ( M ) + 1 δ (cid:107) p δ (cid:107) L ( M ) ≤ F ( δ ) (cid:107) p (cid:107) H β ( M ) , (4.9) (b) (cid:90) M p δ ( x ) f ( x ) dµ x ≤ G ( δ )( (cid:107) f (cid:107) H ( M ) + (cid:13)(cid:13) ¯ f (cid:13)(cid:13) H ( M ) + (cid:13)(cid:13)(cid:13)(cid:13) = f (cid:13)(cid:13)(cid:13)(cid:13) H ( M ) ) (cid:107) p (cid:107) H β ( M ) , (4.10) for all function f ∈ H ( M ) and some function p ∈ H β ( M ) and someconstant F ( δ ) , G ( δ ) depend on δ , with the notations ¯ f ( x ) = 1¯ ω δ ( x ) (cid:90) M f ( y ) ¯ R δ ( x , y ) dµ y , = f ( x ) = 1 = ω δ ( x ) (cid:90) M f ( x ) = R δ ( x , y ) dµ y , (4.11) and ¯ ω δ ( x ) = (cid:90) M ¯ R δ ( x , y ) dµ y , = ω δ ( x ) = (cid:90) M = R δ ( x , y ) dµ y , ∀ x ∈ M , then we will have m δ ∈ H ( M ) , with the estimate (cid:107) m δ (cid:107) H ( M ) + δ (cid:107) n δ (cid:107) L ( ∂ M ) ≤ C (cid:0) ( G ( δ )+ δ F ( δ )) (cid:107) p (cid:107) H β ( M ) + 1 δ (cid:107) q δ (cid:107) L ( ∂ M ) (cid:1) . (4.12)Before we present the proof of the lemma 4.2, let us denote the following weightedaverage functions of m δ :¯ m δ ( x ) = 1¯ ω δ ( x ) (cid:90) M m δ ( y ) ¯ R δ ( x , y ) dµ y , ˆ m δ ( x ) = 1 ω δ ( x ) (cid:90) M m δ ( y ) R δ ( x , y ) dµ y , ∀ x ∈ M , (4.13)where in addition ω δ ( x ) = (cid:82) M R δ ( x , y ) dµ y . The following 2 estimates will be appliedin the proof of lemma 4.2:12 δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ x dµ y ≥ C (cid:107)∇ ˆ m δ (cid:107) L ( M ) , (4.14) δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ x dµ y ≥ C (cid:107)∇ ¯ m δ (cid:107) L ( M ) , (4.15)where the constant C is independent to δ and m δ .The first inequality is from the theorem 7 of [32]. For the second inequality, weapply the lemma 3 of [32]:12 δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R ( (cid:107) x − y (cid:107) δ ) dµ x dµ y ≥ C δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R ( (cid:107) x − y (cid:107) δ ) dµ x dµ y , (4.16)hence12 δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ x dµ y = C δ δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R ( (cid:107) x − y (cid:107) δ ) dµ x dµ y ≥ C C δ δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R ( (cid:107) x − y (cid:107) δ ) dµ x dµ y ≥ C C δ δ (cid:90) | y − x |≤ δ (cid:90) M ( m δ ( x ) − m δ ( y )) R ( (cid:107) x − y (cid:107) δ ) dµ x dµ y ≥ C δ δ C δ δ (cid:90) | y − x |≤ δ (cid:90) M ( m δ ( x ) − m δ ( y )) dµ x dµ y ≥ C δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) ¯ R δ ( x , y ) dµ x dµ y ≥ C (cid:107)∇ ¯ m δ (cid:107) L ( M ) , (4.17)where the last inequality is a direct corollary of (4.14). Proof . [Proof of Lemma 4.2]1. By the lemma 4.1 we have B δ [ m δ , n δ ; m δ , n δ ] = 12 δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ x dµ y + (cid:90) ∂ M n δ ( x ) ˜ R δ ( x ) dτ x . (4.18)Therefore, if we can present the following inequalities(a) C δ ≥ ˜ R δ ≥ C δ, (b) (cid:90) ∂ M n δ ( x ) ˜ R δ ( x ) dτ x ≥ C (cid:107) ¯ m δ (cid:107) L ( ∂ M ) − δ (cid:107) q δ (cid:107) L ( ∂ M ) − δ (cid:107) m δ (cid:107) L ( M ) , (c) (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ x dµ y ≥ C (cid:107) m δ − ¯ m δ (cid:107) L ( M ) , d) 12 δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ x dµ y ≥ C (cid:107)∇ ¯ m δ (cid:107) L ( M ) , (e) (cid:107)∇ ¯ m δ (cid:107) L ( M ) + (cid:107) ¯ m δ (cid:107) L ( ∂ M ) ≥ C (cid:107) ¯ m δ (cid:107) L ( M ) , Then the first inequality implies (cid:90) ∂ M n δ ( x ) ˜ R δ ( x ) dτ x ≥ Cδ (cid:107) n δ (cid:107) L ( ∂ M ) , (4.19)and the direct sum of the last 4 inequalities illustrate12 δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ x dµ y + (cid:90) ∂ M n δ ( x ) ˜ R δ ( x ) dτ x + 12 δ (cid:107) q δ (cid:107) L ( ∂ M ) ≥ C (cid:107) m δ (cid:107) L ( M ) , (4.20)we will then conclude (4.8). Now let us prove these estimates in order.(a) For small δ and each x ∈ ∂ M , our assumption on the kernel function¯ R δ gives˜ R δ ( x ) =4 δ (cid:90) ∂ M = R δ ( x , y ) dτ y + (cid:90) M p ( x ) (( x − y ) · n ( x )) ¯ R δ ( x , y ) dµ y ≥ δ (cid:90) ∂ M = R δ ( x , y ) dτ y ≥ C δ, (4.21)on the other hand,˜ R δ ( x ) = 4 δ (cid:90) ∂ M = R δ ( x , y ) dτ y + (cid:90) M p ( x ) (( x − y ) · n ( x )) ¯ R δ ( x , y ) dµ y ≤ δ (cid:90) ∂ M = R δ ( x , y ) dτ y + δ p ( x ) (cid:90) M ¯ R δ ( x , y ) dµ y ≤ C δ. (4.22)Hence we have both upper and lower bounds for ˜ R δ .(b) We apply the inequality (cid:107) a (cid:107) L ( M ) + (cid:107) b − a (cid:107) L ( M ) ≥ C (cid:107) b (cid:107) L ( M ) to de- uce (cid:90) ∂ M n δ ( x ) ˜ R δ ( x ) dτ x + 12 δ (cid:107) q δ (cid:107) L ( ∂ M ) = (cid:90) ∂ M R δ ( x ) ( q δ ( x ) − D δ m δ ( x )) dτ x + 12 δ (cid:107) q δ (cid:107) L ( ∂ M ) ≥ Cδ (cid:90) ∂ M ( q δ ( x ) − D δ m δ ( x )) dτ x + 12 δ (cid:90) ∂ M q δ ( x ) dτ x ≥ Cδ (cid:90) ∂ M ( D δ m δ ( x )) dτ x ≥ C (cid:90) ∂ M ( (cid:90) M m δ ( y ) (2 + p ( x ) ( x − y ) · n ( x )) ¯ R δ ( x , y ) dµ y ) dτ x . (4.23)In addition, we estimate the term (cid:90) ∂ M ( (cid:90) M m δ ( y ) p ( x ) ( x − y ) · n ( x ) ¯ R δ ( x , y ) dµ y ) dτ x ≤ δ (cid:90) ∂ M ( (cid:90) M | m δ ( y ) | p ( x ) ¯ R δ ( x , y ) dµ y ) dτ x ≤ δ (cid:90) ∂ M κ ( x ) ( (cid:90) M | m δ ( y ) | ¯ R δ ( x , y ) dµ y ) ( (cid:90) M ¯ R δ ( x , y ) dµ y ) dτ x ≤ δ (cid:90) M ( (cid:90) ∂ M p ( x ) ¯ R δ ( x , y ) dτ x ) | m δ ( y ) | dµ y ≤ Cδ (cid:107) m δ (cid:107) L ( M ) . (4.24)Using again the inequality (cid:107) a (cid:107) L ( M ) + (cid:107) b − a (cid:107) L ( M ) ≥ C (cid:107) b (cid:107) L ( M ) andthe control (4.24), we discover C (cid:90) ∂ M ( (cid:90) M m δ ( y ) (2 + p ( x ) ( x − y ) · n ( x )) ¯ R δ ( x , y ) dµ y ) dτ x + δ (cid:107) m δ (cid:107) L ( M ) ≥ C (cid:90) ∂ M ( (cid:90) M m δ ( y ) ¯ R δ ( x , y ) dµ y ) dτ x ≥ C (cid:90) ∂ M = m δ ( x ) dτ x = C (cid:107) ¯ m δ (cid:107) L ( ∂ M ) . (4.25)Hence we combine (4.23) and (4.25) to conclude (cid:90) ∂ M n δ ( x ) ˜ R δ ( x ) dτ x + 12 δ (cid:107) q δ (cid:107) L ( ∂ M ) + δ (cid:107) m δ (cid:107) L ( M ) ≥ C (cid:107) ¯ m δ (cid:107) L ( ∂ M ) . (4.26) c) We can calculate (cid:107) ¯ m δ − m δ (cid:107) L ( M ) = (cid:90) M (cid:0) (cid:90) M ω δ ( x ) ( m δ ( x ) − m δ ( y )) ¯ R δ ( x , y ) dµ y (cid:1) dµ x ≤ C (cid:90) M (cid:0) (cid:90) M ( m δ ( x ) − m δ ( y )) ¯ R δ ( x , y ) dµ y (cid:1) dµ x ≤ C (cid:90) M ( (cid:90) M ¯ R δ ( x , y ) dµ y ) ( (cid:90) M ¯ R δ ( x , y )( m δ ( x ) − m δ ( y )) dµ y ) dµ x ≤ C (cid:90) M (cid:90) M ¯ R δ ( x , y )( m δ ( x ) − m δ ( y )) dµ y dµ x ≤ C (cid:90) M (cid:90) M R δ ( x , y )( m δ ( x ) − m δ ( y )) dµ y dµ x ≤ Cδ B δ [ m δ , n δ ; m δ , n δ ] . (4.27)(d) This is exactly the equation (4.15).(e) This is the manifold version of Poincare inequality for = m δ ∈ H ( M ).2. As usual, we split the proof into the following steps(a) (cid:107)∇ ˆ m δ (cid:107) L ( M ) ≤ C δ (cid:90) M (cid:90) M ( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ x dµ y , (4.28)(b) (cid:107)∇ ( m δ ( x ) − ˆ m δ ( x )) (cid:107) L ( M ) ≤ C ( δ F ( δ ) (cid:107) p (cid:107) H β ( M ) + δ (cid:107) n δ (cid:107) L ( ∂ M ) ) , (4.29)(c) (cid:107) m δ (cid:107) L ( M ) + δ (cid:107) n δ (cid:107) L ( ∂ M ) ≤ C ( B δ [ m δ , n δ ; m δ , n δ ] + 1 δ (cid:107) q δ (cid:107) L ( ∂ M ) ) , (4.30)(d) C B δ [ m δ , n δ ; m δ , n δ ] ≤ (cid:107) m δ (cid:107) H ( M ) + δ (cid:107) n δ (cid:107) L ( ∂ M ) + C ( G ( δ ) (cid:107) p (cid:107) H β ( M ) + 1 δ (cid:107) q δ (cid:107) L ( ∂ M ) ) , (4.31)where the first 3 inequalities imply (cid:107) m δ (cid:107) H ( M ) + δ (cid:107) n δ (cid:107) L ( ∂ M ) ≤ C ( B δ [ m δ , n δ ; m δ , n δ ]+ 1 δ (cid:107) q δ (cid:107) L ( ∂ M ) + δ F ( δ ) (cid:107) p (cid:107) H β ( M ) ) , (4.32)together with the 4 th inequality, we will then deduce (4.12). Now let us provethem in order a) This is exactly the inequality (4.14).(b) This inequality is derived from the equation L δ m δ ( x ) −G δ n δ ( x ) = p δ ( x ),or in other words,1 δ (cid:90) M ( m δ ( x ) − m δ ( y )) R δ ( x , y ) dµ y − (cid:90) ∂ M n δ ( y ) (2 − p ( y ) ( x − y ) · n ( y )) ¯ R δ ( x , y ) dτ y = p δ ( x ) . (4.33)Recall the definition of ¯ m δ , we have1 δ ω δ ( x )( m δ ( x ) − ˆ m δ ( x )) − (cid:90) ∂ M n δ ( y ) (2 − p ( y ) ( x − y ) · n ( y )) ¯ R δ ( x , y ) dτ y = p δ ( x ) , (4.34)this is m δ ( x ) − ˆ m δ ( x ) = δ (cid:90) ∂ M ω δ ( x ) n δ ( y ) (2 − p ( y ) ( x − y ) · n ( y )) ¯ R δ ( x , y ) dτ y + δ p δ ( x ) ω δ ( x ) . (4.35)Hence we obtain (cid:107)∇ ( m δ − ˆ m δ ) (cid:107) L ( M ) ≤ δ (cid:13)(cid:13)(cid:13)(cid:13) ∇ p δ ( x ) ω δ ( x ) (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) + δ (cid:13)(cid:13)(cid:13)(cid:13) ∇ (cid:90) ∂ M ω δ ( x ) n δ ( y ) (2 − p ( y ) ( x − y ) · n ( y )) ¯ R δ ( x , y ) dτ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) . (4.36)The first term of (4.36) can be controlled by (cid:13)(cid:13)(cid:13)(cid:13) ∇ p δ ( x ) ω δ ( x ) (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) = (cid:13)(cid:13)(cid:13)(cid:13) ω δ ( x ) ∇ p δ ( x ) − p δ ( x ) ∇ ω δ ( x ) ω δ ( x ) (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) ω δ ( x ) ∇ p δ ( x ) (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) + 2 (cid:13)(cid:13)(cid:13)(cid:13) p δ ( x ) ∇ ω δ ( x ) ω δ ( x ) (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) ≤ C ( (cid:107)∇ p δ ( x ) (cid:107) L x ( M ) + 1 δ (cid:107) p δ ( x ) (cid:107) L x ( M ) ) ≤ C F ( δ ) (cid:107) p (cid:107) H β ( M ) , (4.37)where the second inequality results from the fact that C ≤ ω δ ( x ) ≤ C and |∇ ω δ ( x ) | = | (cid:90) M ∇ x M R δ ( x , y ) dµ y | = | (cid:90) M ∇ y R δ ( x , y ) dµ y | = | (cid:90) ∂ M R δ ( x , y ) n ( y ) dτ y | ≤ (cid:90) ∂ M R δ ( x , y ) dτ y ≤ C δ , ∀ x ∈ M . (4.38)The control on second term of (4.36) is more complicated in calculation. imilar to (4.37), we have (cid:13)(cid:13)(cid:13)(cid:13) ∇ (cid:90) ∂ M ω δ ( x ) n δ ( y ) (2 − p ( y ) ( x − y ) · n ( y )) ¯ R δ ( x , y ) dτ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) ≤ C ( (cid:13)(cid:13)(cid:13)(cid:13) ∇ (cid:90) ∂ M n δ ( y ) (2 − p ( y ) ( x − y ) · n ( y )) ¯ R δ ( x , y ) dτ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) + 1 δ (cid:13)(cid:13)(cid:13)(cid:13) (cid:90) ∂ M n δ ( y ) (2 − p ( y ) ( x − y ) · n ( y )) ¯ R δ ( x , y ) dτ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) ) ≤ C ( (cid:13)(cid:13)(cid:13)(cid:13) (cid:90) ∂ M n δ ( y ) (2 − p ( y ) ( x − y ) · n ( y )) ∇ x M ¯ R δ ( x , y ) dτ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) + (cid:13)(cid:13)(cid:13)(cid:13) (cid:90) ∂ M n δ ( y ) p ( y ) n ( y ) ¯ R δ ( x , y ) dτ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) + 1 δ (cid:13)(cid:13)(cid:13)(cid:13) (cid:90) ∂ M n δ ( y ) (2 − p ( y ) ( x − y ) · n ( y )) ¯ R δ ( x , y ) dτ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) ) ≤ C ( (cid:13)(cid:13)(cid:13)(cid:13) (cid:90) ∂ M | n δ ( y ) | δ | x − y | R δ ( x , y ) dτ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) + (cid:13)(cid:13)(cid:13)(cid:13) (cid:90) ∂ M | n δ ( y ) | ¯ R δ ( x , y ) dτ y (cid:13)(cid:13)(cid:13)(cid:13) L ( M ) + 1 δ (cid:13)(cid:13)(cid:13)(cid:13) (cid:90) ∂ M | n δ ( y ) | ¯ R δ ( x , y ) dτ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) ) ≤ Cδ (cid:13)(cid:13)(cid:13)(cid:13) (cid:90) ∂ M | n δ ( y ) | R δ ( x , y ) dτ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) ≤ Cδ ( (cid:90) M ( (cid:90) ∂ M n δ ( y ) R δ ( x , y ) dτ y )( (cid:90) ∂ M R δ ( x , y ) dτ y ) dµ x ) ≤ Cδ ( (cid:90) ∂ M (cid:90) M δ n δ ( y ) R δ ( x , y ) dµ x dτ y ) ≤ Cδ − (cid:107) n δ (cid:107) L ( ∂ M ) . (4.39)We therefore conclude (4.36), (4.37) and (4.39) to discover (cid:107)∇ ( m δ ( x ) − ˆ m δ ( x )) (cid:107) L ( M ) ≤ C ( δ F ( δ ) (cid:107) p (cid:107) H β ( M ) + δ (cid:107) n δ (cid:107) L ( ∂ M ) ) , (4.40)(c) This is exactly the first part of the lemma.(d) On the other hand, the bilinear form of the system (4.1) gives us2 C B δ [ m δ , n δ ; m δ , n δ ] = 2 C (cid:90) M m δ ( x ) p δ ( x ) dµ x + 2 C (cid:90) ∂ M n δ ( x ) q δ ( x ) dτ x ≤ C G ( δ )( (cid:107) m δ (cid:107) H ( M ) + (cid:107) ¯ m δ (cid:107) H ( M ) + (cid:13)(cid:13)(cid:13) = m δ (cid:13)(cid:13)(cid:13) H ( M ) ) (cid:107) p (cid:107) H β ( M ) + 2 C (cid:107) n δ (cid:107) L ( ∂ M ) (cid:107) q δ (cid:107) L ( ∂ M ) . (4.41)Similar as the equation (4.39), we follow the calculation of (4.27) to ventually obtain (cid:107) ¯ m δ − m δ (cid:107) H ( M ) = (cid:90) M (cid:0) (cid:90) M ω δ ( x ) ( m δ ( x ) − m δ ( y )) ¯ R δ ( x , y ) dµ y (cid:1) dµ x + (cid:90) M (cid:0) (cid:90) M ∇ x M ω δ ( x ) ( m δ ( x ) − m δ ( y )) ¯ R δ ( x , y ) dµ y (cid:1) dµ x ≤ C ( δ B δ [ m δ , n δ ; m δ , n δ ] + B δ [ m δ , n δ ; m δ , n δ ]) ≤ CB δ [ m δ , n δ ; m δ , n δ ] . (4.42)By substituting ¯ R δ by = R in (4.42), we can obtain the following propertyfor = m δ : (cid:13)(cid:13)(cid:13) = m δ − m δ (cid:13)(cid:13)(cid:13) H ( M ) ≤ CB δ [ m δ , n δ ; m δ , n δ ] . (4.43)This indicates2 C G ( δ )( (cid:107) m δ (cid:107) H ( M ) + (cid:107) ¯ m δ (cid:107) H ( M ) + (cid:13)(cid:13)(cid:13) = m δ (cid:13)(cid:13)(cid:13) H ( M ) ) (cid:107) p (cid:107) H β ( M ) ≤ C G ( δ )(3 (cid:107) m δ (cid:107) H ( M ) + C B δ [ m δ , n δ ; m δ , n δ ]) (cid:107) p (cid:107) H β ( M ) ≤ (cid:107) m δ (cid:107) H ( M ) + C B δ [ m δ , n δ ; m δ , n δ ] + (18 C + CC ) G ( δ ) (cid:107) p (cid:107) H β ( M ) , (4.44)On the other hand, we have2 C (cid:107) n δ (cid:107) L ( ∂ M ) (cid:107) q δ (cid:107) L ( ∂ M ) ≤ δ (cid:107) n δ (cid:107) L ( ∂ M ) + 2 C δ (cid:107) q δ (cid:107) L ( ∂ M ) , (4.45)We then combine the equations (4.41) (4.44) (4.45) to obtain C B δ [ m δ , n δ ; m δ , n δ ] ≤ (cid:107) m δ (cid:107) H ( M ) + δ (cid:107) n δ (cid:107) L ( ∂ M ) + (18 C + CC ) G ( δ ) (cid:107) p (cid:107) H β ( M ) + 2 C δ (cid:107) q δ (cid:107) L ( ∂ M ) . (4.46)Hence we have completed our proof.
5. Well-Posedness of Integral Model.
The main purpose of this section is toprove theorem 2.1. Based on the lemma 4.2, let us return to the integral model (2.9)to establish a well-posedness argument. According to the Lax-Milgram theorem, onemodel is well-posed if its corresponding bilinear form satisfy certain properties. Fromthe first part of lemma 4.2, we see the bilinear form B δ of (2.9) satisfies B δ [ m δ , n δ ; m δ , n δ ] + 1 δ (cid:107) q δ (cid:107) L ( ∂ M ) ≥ C ( (cid:107) m δ (cid:107) L ( M ) + δ (cid:107) n δ (cid:107) L ( ∂ M ) ) , (5.1) here q δ = D δ m δ + ˜ R δ ( x ) n δ . Here the term δ (cid:107) q δ (cid:107) L ( ∂ M ) is not removable, otherwisewe can easily find a counterexample. Nevertheless, we can show that the system (2.9)assures a unique solution. The main idea is to write the first equation of (2.9) asan equation of u δ by eliminating v δ through the second equation of (2.9), and thenprove the weak formulation of such equation satisfies the conditions of Lax-Milgramtheorem. Proof . [Proof of Theorem 2.1]1. Recall the second equation of our model (2.9): D δ u δ ( x ) + ˜ R δ ( x ) v δ ( x ) = Q δ f ( x ) , x ∈ ∂ M , (5.2)this gives v δ ( x ) = Q δ f ( x )˜ R δ ( x ) − D δ u δ ( x )˜ R δ ( x ) , x ∈ ∂ M , (5.3)and we apply it to the first equation of (2.9) to discover L δ u δ ( x ) + ( G δ D δ u δ ˜ R δ )( x ) = P δ f ( x ) + G δ ( Q δ f ( x )˜ R δ ( x ) ) , x ∈ M . (5.4)Our purpose here is to show there exists a unique solution u δ ∈ L ( M ) tothe equation (5.4), and thus v δ ( x ) exists and is unique as well. To achievethis goal, we have to prove the following 3 estimates(a) (cid:90) M u δ ( x )( L δ u δ ( x ) + ( G δ D δ u δ ˜ R δ )( x )) dµ x ≥ C (cid:107) u δ (cid:107) L ( M ) , (b) (cid:90) M w δ ( x )( L δ u δ ( x )+( G δ D δ u δ ˜ R δ )( x )) dµ x ≤ C δ (cid:107) u δ (cid:107) L ( M ) (cid:107) w δ (cid:107) L ( M ) , ∀ w δ ∈ L ( M ) , (c) (cid:90) M w δ ( x ) P δ f ( x ) dµ x + (cid:90) M w δ ( x ) G δ ( Q δ f ( x )˜ R δ ( x ) ) dµ x ≤ C (cid:107) f (cid:107) H ( M ) (cid:107) w δ (cid:107) L ( M ) , ∀ w δ ∈ L ( M ) . Then the existence and uniqueness of solution will be given directly by theLax-Milgram theorem. Now let us prove them in order.(a) We denote ˜ v δ ( x ) = D δ u δ ( x )˜ R δ ( x ) , x ∈ ∂ M . (5.5) rom the proof of the lemma 4.1, we know that (cid:90) M G δ ˜ v δ ( x ) u δ ( x ) dµ x = (cid:90) ∂ M ˜ v δ ( x ) D δ u δ ( x ) dτ x , hence (cid:90) M ( L δ u δ ( x ) + ( G δ D δ u δ ˜ R δ )( x )) u δ ( x ) dµ x = (cid:90) M ( L δ u δ ( x ) + G δ ˜ v δ ( x )) u δ ( x ) dµ x = (cid:90) M ( L δ u δ ( x )) u δ ( x ) dµ x + (cid:90) ∂ M ˜ v δ ( x ) D δ u δ ( x ) dτ x = (cid:90) M ( L δ u δ ( x )) u δ ( x ) dµ x + (cid:90) ∂ M ˜ R δ ( x )˜ v δ ( x ) dτ x = B δ [ u δ , ˜ v δ ; u δ , ˜ v δ ];(5.6)and we apply the part 1 of the lemma 4.2, B δ [ u δ , ˜ v δ ; u δ , ˜ v δ ] ≥ C (cid:107) u δ (cid:107) L ( M ) . (5.7)(b) For any u δ , w δ ∈ L ( M ) we can calculate (cid:90) M ( L δ u δ ( x ) + ( G δ D δ u δ ˜ R δ )( x )) w δ ( x ) dµ x = (cid:90) M (cid:90) M w δ ( x )( u δ ( x ) − u δ ( y )) R δ ( x , y ) dµ y dµ x + (cid:90) M (cid:90) ∂ M (cid:90) M u ( s ) (2 − κ ( y ) ( y − s ) · n ( y )) ¯ R δ ( y , s ) d s (2 + κ ( y ) ( x − y ) · n ( y )) ¯ R δ ( x , y ) dτ y w δ ( x ) dµ x , (5.8)here (cid:12)(cid:12) (cid:90) M (cid:90) M w δ ( x )( u δ ( x ) − u δ ( y )) R δ ( x , y ) dµ y dµ x (cid:12)(cid:12) ≤ C δ ( (cid:90) M (cid:90) M | w δ ( x ) u δ ( x ) | dµ y dµ x + (cid:90) M (cid:90) M | w δ ( x ) u δ ( y ) | dµ y dµ x ) ≤ C δ ( (cid:107) w δ (cid:107) L ( M ) (cid:107) u δ (cid:107) L ( M ) + (cid:107) w δ (cid:107) L ( M ) (cid:107) u δ (cid:107) L ( M ) ) ≤ C δ (cid:107) w δ (cid:107) L ( M ) (cid:107) u δ (cid:107) L ( M ) ;(5.9)and (cid:12)(cid:12) (cid:90) M (cid:90) ∂ M (cid:90) M u δ ( s ) (2 + κ ( y ) ( y − s ) · n ( y )) ¯ R δ ( y , s ) d s (2 − κ ( y ) ( x − y ) · n ( y )) ¯ R δ ( x , y ) dτ y w δ ( x ) dµ x (cid:12)(cid:12) ≤ C δ (cid:90) M (cid:90) ∂ M (cid:90) M | u δ ( s ) w δ ( x ) | dµ s dτ y dµ x ≤ C δ (cid:107) w δ (cid:107) L ( M ) (cid:107) u δ (cid:107) L ( M ) ≤ C δ (cid:107) w δ (cid:107) L ( M ) (cid:107) u δ (cid:107) L ( M ) . (5.10) he above 2 inequalities implies that (cid:90) M ( L δ u δ ( x ) + ( G δ D δ u δ ˜ R δ )( x )) w δ ( x ) dµ x ≤ C δ (cid:107) w δ (cid:107) L ( M ) (cid:107) u δ (cid:107) L ( M ) , (5.11)where C δ is a constant depend on δ and independent on u δ and w δ .(c) Now our next step is to control the right hand side of (5.4). Recall (cid:90) M w δ ( x ) P δ f ( x ) dµ x = (cid:90) M w δ ( x ) (cid:90) M f ( y ) ¯ R δ ( x , y ) dµ y dµ x − (cid:90) M w δ ( x ) (cid:90) ∂ M (( x − y ) · n ( y )) f ( y ) ¯ R δ ( x , y ) dτ y dµ x , (5.12)and we can calculate (cid:90) M w δ ( x ) (cid:90) M f ( y ) ¯ R δ ( x , y ) dµ y dµ x ≤ (cid:104) (cid:90) M w δ ( x ) dµ x (cid:90) M ( (cid:90) M f ( y ) ¯ R δ ( x , y ) dµ y ) dµ x (cid:105) ≤ (cid:104) (cid:90) M w δ ( x ) dµ x (cid:90) M ( (cid:90) M f ( y ) ¯ R δ ( x , y ) dµ y (cid:90) M ¯ R δ ( x , y ) dµ y ) dµ x (cid:105) ≤ (cid:104) (cid:90) M w δ ( x ) dµ x (cid:90) M (cid:90) M f ( y ) ¯ R δ ( x , y ) dµ y dµ x (cid:105) ≤ (cid:104) (cid:90) M w δ ( x ) dµ x (cid:90) M f ( y ) dµ y (cid:105) ≤ (cid:107) f (cid:107) L ( M ) (cid:107) w δ (cid:107) L ( M ) ≤ (cid:107) f (cid:107) H ( M ) (cid:107) w δ (cid:107) L ( M ) , (5.13) (cid:90) M w δ ( x ) (cid:90) ∂ M (( x − y ) · n ( y )) f ( y ) ¯ R δ ( x , y ) dτ y dµ x ≤ δ (cid:90) M | w δ ( x ) | (cid:90) ∂ M | f ( y ) | ¯ R δ ( x , y ) dτ y dµ x ≤ δ (cid:104) (cid:90) M w δ ( x ) dµ x (cid:90) M ( (cid:90) ∂ M | f ( y ) | ¯ R δ ( x , y ) dτ y ) dµ x (cid:105) ≤ δ (cid:104) (cid:90) M w δ ( x ) dµ x (cid:90) M ( (cid:90) ∂ M f ( y ) ¯ R δ ( x , y ) dτ y (cid:90) ∂ M ¯ R δ ( x , y ) dµ y ) dµ x (cid:105) ≤ δ (cid:104) (cid:90) M w δ ( x ) dµ x (cid:90) M (cid:90) ∂ M f ( y ) ¯ R δ ( x , y ) dτ y dµ x (cid:105) ≤ δ (cid:104) (cid:90) M w δ ( x ) dµ x (cid:90) ∂ M f ( y ) dτ y (cid:105) ≤ δ (cid:107) f (cid:107) L ( ∂ M ) (cid:107) w δ (cid:107) L ( M ) ≤ (cid:107) f (cid:107) H ( M ) (cid:107) w δ (cid:107) L ( M ) ; (5.14) n addition, we have (cid:90) M w δ ( x ) G δ ( Q δ f ( x )˜ R δ ( x ) ) dµ x = (cid:90) M w δ ( x ) (cid:90) ∂ M Q δ f ( y )˜ R δ ( y ) (2 − p ( y )( x − y ) · n ( y )) ¯ R δ ( x , y ) dτ y dµ x ≤ (cid:90) M | w δ ( x ) | (cid:90) ∂ M C δ δ | f ( y ) | R δ ( x , y ) dτ y dµ x ≤ Cδ (cid:90) ∂ M | f ( y ) | (cid:90) M | w δ ( x ) | ¯ R δ ( x , y ) dµ x dτ y ≤ Cδ (cid:104) (cid:90) ∂ M f ( y ) dτ y (cid:90) ∂ M ( (cid:90) M w δ ( x ) ¯ R δ ( x , y ) dµ x ) ( (cid:90) M ¯ R δ ( x , y ) dµ x ) dτ y (cid:105) ≤ Cδ (cid:104) (cid:90) ∂ M f ( y ) dτ y (cid:90) M δ w δ ( x ) dµ x (cid:105) ≤ Cδ (cid:107) f (cid:107) L ( ∂ M ) (cid:107) w δ (cid:107) L ( M ) ≤ C (cid:107) f (cid:107) H ( M ) (cid:107) w δ (cid:107) L ( M ) , (5.15)The above three inequalities reveals (cid:90) M w δ ( x ) P δ f ( x ) dµ x + (cid:90) M w δ ( x ) G δ ( Q δ f ( x )˜ R δ ( x ) ) dµ x ≤ C (cid:107) f (cid:107) H ( M ) (cid:107) w δ (cid:107) L ( M ) . (5.16)Hence we have completed all the 3 estimates, and the existence and uniquenessof solution ( u δ , v δ ) is then given by the Lax-Milgram theorem, with (cid:107) u δ (cid:107) L ( M ) + δ (cid:107) v δ (cid:107) L ( ∂ M ) ≤ C (cid:107) f (cid:107) H ( M ) . (5.17)2. Now we apply a weaker argument of Lemma 4.2(i) to the model (2.9): if wecan show(a) (cid:107)∇ M ( P δ f ) (cid:107) L ( M ) + 1 δ (cid:107)P δ f (cid:107) L ( M ) ≤ Cδ (cid:107) f (cid:107) H ( M ) , (5.18)and(b) (cid:90) M P δ f ( x ) f ( x ) dµ x ≤ C (cid:107) f (cid:107) H ( M ) (cid:107) f (cid:107) H ( M ) , ∀ f ∈ H ( M );(5.19)then the lemma 4.2 will give us (cid:107) u δ (cid:107) H ( M ) + δ (cid:107) v δ (cid:107) L ( ∂ M ) ≤ C ( (cid:107) f (cid:107) H ( M ) + 1 δ (cid:107)Q δ f (cid:107) L ( ∂ M ) + δ (cid:107) f (cid:107) H ( M ) ) , (5.20) onsequently, (cid:107) u δ (cid:107) H ( M ) ≤ C (cid:107) f (cid:107) H ( M ) . (5.21)In fact, the estimate (2b) has been shown in (5.16) in the part 1 as (cid:90) M f ( x ) P δ f ( x ) dµ x ≤ C (cid:107) f (cid:107) H ( M ) (cid:107) f (cid:107) L ( M ) , (5.22)so what remains to present is (2a). Recall P δ f ( x ) = (cid:90) M f ( y ) ¯ R δ ( x , y ) dµ y + (cid:90) ∂ M (( x − y ) · n ( y )) f ( y ) ¯ R δ ( x , y ) dµ y , (5.23)hence (cid:107)∇ M ( P δ f ) (cid:107) L ( M ) + 1 δ (cid:107)P δ f (cid:107) L ( M ) ≤ δ (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) M f ( y ) ¯ R δ ( x , y ) dµ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) + (cid:13)(cid:13)(cid:13)(cid:13) ∇ x M (cid:90) M f ( y ) ¯ R δ ( x , y ) dµ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) + 1 δ (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) ∂ M (( x − y ) · n ( y )) f ( y ) ¯ R δ ( x , y ) dµ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) + (cid:13)(cid:13)(cid:13)(cid:13) ∇ x M (cid:90) ∂ M (( x − y ) · n ( y )) f ( y ) ¯ R δ ( x , y ) dµ y (cid:13)(cid:13)(cid:13)(cid:13) L x ( M ) . (5.24)The control for the above 4 terms are exactly the same as the control for theequations (5.13) (5.14) (5.15). As a consequence, (cid:107)∇ M ( P δ f ) (cid:107) L ( M ) + 1 δ (cid:107)P δ f (cid:107) L ( M ) ≤ Cδ (cid:107) f (cid:107) L ( M ) + Cδ (cid:107) f (cid:107) L ( ∂ M ) ≤ Cδ (cid:107) f (cid:107) H ( M ) . (5.25)Therefore we proved the condition (2a). Together with the condition (2b)shown in (5.22), we can eventually conclude (cid:107) u δ (cid:107) H ( M ) ≤ C (cid:107) f (cid:107) H ( M ) .
6. Vanishing Nonlocality.
Our goal in this section is to prove theorem 2.2.So far we have shown that our integral model (2.9) gives a unique solution ( u δ , v δ ),and the original Poisson model (1.1) gives a regular solution u ∈ H ( M ). Thereafter,we are ready to establish the convergence argument u δ → u and v δ → ∂u∂ n in properspaces.Let us denote the error function e δ = u − u δ on M , and e n δ = ∂u∂ n − v δ on ∂ M . e then compare the equation (2.9) with the truncation errors (3.1) (3.2) to discover L δ e δ ( x ) − G δ e n δ ( x ) = r in , x ∈ M , D δ e δ ( x ) + ˜ R δ ( x ) e n δ ( x ) = r bd , x ∈ ∂ M , (6.1)and r in = r it + r bl . In the theorem 3.1, we have completely controlled the truncationerrors r it , r bl , r bd . Next, we will apply the second part of lemma 4.2 to control theerror e δ and e n δ as well. Proof . [Proof of Theorem 2.2] Our purpose is to prove (cid:107) e δ (cid:107) H ( M ) + δ / (cid:107) e n δ (cid:107) L ( ∂ M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) , (6.2)Since the system of equations (6.1) describes the relation between ( e δ , e n δ ) and ( r it + r bl , r bd ), we apply the second part of the lemma 4.2: if we can present1. 1 δ (cid:107) r in (cid:107) L ( M ) + (cid:107)∇ r in (cid:107) L ( M ) + 1 δ (cid:107) r bl (cid:107) L ( M ) + (cid:107)∇ r bl (cid:107) L ( M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) , (6.3)2. (cid:90) M r in ( x ) f ( x ) dµ x + (cid:90) M r bl ( x ) f ( x ) dµ x ≤ Cδ (cid:107) u (cid:107) H ( M ) ( (cid:107) f (cid:107) H ( M ) + (cid:13)(cid:13) ¯ f (cid:13)(cid:13) H ( M ) + (cid:13)(cid:13)(cid:13)(cid:13) = f (cid:13)(cid:13)(cid:13)(cid:13) H ( M ) ) , ∀ f ∈ H ( M ) , (6.4)3. (cid:107) r bd (cid:107) L ( ∂ M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) , (6.5)then we will have the estimate (cid:107) e δ (cid:107) H ( M ) + δ (cid:107) e n δ (cid:107) L ( ∂ M ) ≤ δ ( Cδ (cid:107) u (cid:107) H ( M ) )+ 1 δ ( Cδ (cid:107) u (cid:107) H ( M ) )+ Cδ (cid:107) u (cid:107) H ( M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) . (6.6)In fact, we have already completed most of the above estimates (6.3) (6.4) and(6.5) in the theorem 3.1, what left is to present (cid:90) M r in ( x ) f ( x ) dµ x ≤ Cδ (cid:107) u (cid:107) H ( M ) (cid:107) f (cid:107) H ( M ) , ∀ f ∈ H ( M ) , (6.7)In fact, from the theorem 3.1 we have (cid:107) r in (cid:107) L ( M ) ≤ δ (cid:107) u (cid:107) H ( M ) , the inequality (6.7)is directly obtained from Cauchy’s inequality. Hence we have completed our proof. . Discretization of Model. The analysis in the previous sections indicatesthat our integral model (2.9) approximates the Poisson model (1.1) in the quadraticrate. So far our results are all on the continuous setting. Nevertheless, a natural think-ing is to numerically implement such integral model with proper numerical method,where the operators can be approximated by certain discretization technique. As wementioned in the beginning, a corresponding numerical method named point integralmethod(PIM) can be applied to discretize our integral model. The main idea is tosample the manifold and its boundary with a set of sample points, which is usuallycalled point cloud. Given a proper density of points, one can approximate the integralof a function by adding up the value of the function at each sample point multipliedby its volume weight. The calculation of the volume weights involves the use of K -nearest neighbors to construct local mesh around each points. For our model (2.9),We can easily discretize each term of it since differential operators are nonexistent.It will result in a linear system and provide an approximation of the solution to theoriginal Poisson equation.Now assuming we are given the set of points { p i } ni =1 ⊂ M , { q k } mk =1 ⊂ ∂ M . Inaddition, we are given the area weight A i for p i ∈ M , and the length weight L k for q k ∈ ∂ M . Then according to the description of PIM method, we can discretize themodel (2.9) into the following linear system: n (cid:80) j =1 L ijδ ( u i − u j ) − m (cid:80) k =1 G ikδ v k = f i δ i = 1 , , ...n. n (cid:80) j =1 D ljδ u j + ˜ R lδ v l = f l δ l = 1 , , ..., m. (7.1)where the discretized coefficients are given as follows L ijδ = 1 δ R δ ( p i , p j ) A j , (7.2) G ikδ = (2 − κ ( q k ))(( p i − q k ) · n k ) ¯ R δ ( p i , q k ) L k , (7.3) f i δ = n (cid:88) j =1 f ( p j ) ¯ R δ ( p i , p j ) A j − m (cid:88) k =1 ( p i − q k ) · n k f ( q k ) ¯ R δ ( p i , q k ) L k , (7.4) D ljδ = (2 + κ ( q l )( q l − p j ) · n l ) ¯ R δ ( q l , p j ) A j , (7.5)˜ R lδ = 4 δ m (cid:88) k =1 = R δ ( q l , p k ) L k + n (cid:88) j =1 κ ( q l )(( q l − p j ) · n l ) ¯ R δ ( q l , p j ) A j , (7.6) l δ = − δ n (cid:88) j =1 f ( p j ) = R δ ( q l , p j ) A j . (7.7)The system (7.1) gives a system of linear equations on the unknown values { u i } i =1 ,...,n , { v k } k =1 , ,...,m . The stiff matrix of the system is symmetric positive definite by mul-tiplying a positive diagonal matrix. According to the algorithm of PIM, the exactsolution u to the Poisson equation (1.1) at the point p i can be approximated by u i ,while its normal derivative u n at q k can be approximated by v k . To evaluate the ac-curacy of such method, we use the following two terms to record the L error betweenthe numerical solution and the exact solution: e = interior L error = (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) n (cid:80) j =1 ( u j − u ( p j )) A jn (cid:80) j =1 u ( p j ) A j , (7.8) e b = boundary L error = (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) m (cid:80) k =1 ( v k − ∂u∂ n ( q k )) L km (cid:80) k =1 ∂u∂ n ( q k ) L k . (7.9)Now let us study an example. In such example, we let the manifold M be thehemisphere x + y + z = 1 , z ≥ . (7.10)By a simple observation, its boundary is the unit circle x + y = 1 , z = 0. To comparethe exact solution with our numerical solution, we let u ( x, y, z ) = z so that u ≡ ∂ M , and by calculation f ( x, y, z ) = ∆ M u = − z .In our experiment, we always let δ = ( n ) , where n denotes the number ofinterior points in the point cloud. and hence h = (cid:113) n represents the average distancebetween each adjacent points on the point cloud. To make our simulation simpler,all the points p i , q k are randomly chosen by Matlab. After solving the linear system(7.1), we record the error terms on the following diagram and graph: nterior points boundary points δ e rate e b rate512 64 0.250 0.0158 N/A 0.0862 N/A1250 100 0.200 0.0099 2.0950 0.0353 4.00102592 144 0.167 0.0078 1.3076 0.0122 5.82734802 196 0.143 0.0056 2.1496 0.0089 2.04608192 256 0.125 0.0040 2.5198 0.0071 1.692213122 324 0.111 0.0033 1.6333 0.0067 0.492320000 400 0.100 0.0026 2.2628 0.0050 2.777829282 484 0.091 0.0020 2.7527 0.0045 1.1054 Fig. 7.1 . Diagram: Convergence of PIM
Fig. 7.2 . blue line: log e ; red line: y = 2 x − . ig. 7.3 . blue line: log e b ; red line: y = 1 . x − . . We now extend our probleminto the case where u is no longer zero along the boundary but equals to some smoothfunction g : − ∆ u ( x ) = f ( x ) x ∈ M ; u ( x ) = g ( x ) x ∈ ∂ M . (7.11)In fact, by going through the calculation in the section 3, we see two additionalboundary terms should be added to our integral model in such non-homogeneous case,to eventually conclude the following equations: L δ u δ ( x ) − G δ v δ ( x ) = P δ f ( x ) + S δ g ( x ) , x ∈ M , D δ u δ ( x ) + ˜ R δ ( x ) v δ ( x ) = Q δ f ( x ) + ˜ P δ ( x ) g ( x ) , x ∈ ∂ M . (7.12)where the operator S δ g ( x ) = − (cid:90) ∂ M (( x − y ) · n ( y )) d g ( γ ( s )) ds ( y ) ¯ R δ ( x , y ) dµ y , (7.13)and the function ˜ P δ ( x ) = (cid:90) M (2 + p ( x ) ( x − y ) · n ( x )) ¯ R δ ( x , y ) dµ y , (7.14) ere γ : s → ∂ M is a vector-valued function that describes the boundary curve ∂ M with | γ (cid:48) ( s ) | ≡ n (cid:80) j =1 L ijδ ( u i − u j ) − m (cid:80) k =1 G ikδ v k = f i δ + g i δ i = 1 , , ...n. n (cid:80) j =1 D ljδ u j + ˜ R lδ v l = f l δ + g l δ l = 1 , , ..., m. (7.15)here g i δ = − m (cid:88) k =1 (( p i − q k ) · n k ) d g ( γ ( s )) ds ( q k ) ¯ R δ ( p i , q k ) L k , (7.16) g l δ = n (cid:88) j =1 (2 + κ ( q l )( q l − p j ) · n l ) ¯ R δ ( q l , p j ) g ( q l ) A j . (7.17)Now we start our second numerical example, where non-homogeneous Dirichletboundary condition is imposed. Still ,we let the manifold and the boundary to be thesame hemisphere as the first example, and the sample points are randomly given byMatlab. we choose δ = ( n ) as well, where n denotes the number of interior samplepoints.In this example, we consider the function u ( x, y, z ) = x . By calculation, f ( x, y, z ) = ∆ M u = 2 . x + 1 . y ) x (1 + 8 x + 0 . y ) . (7.18)Now u is no longer zero along the boundary circle. Still, we record the l error ofinterior and boundary as previous. Applying the same implementation on the system(7.15) , we record the following results on the error: nterior points boundary points δ e rate e b rate512 64 0.250 0.0409 N/A 0.0538 N/A1250 100 0.200 0.0299 1.4039 0.0250 3.43452592 144 0.125 0.0188 2.5450 0.0107 4.65464802 196 0.143 0.0132 2.2941 0.0089 1.19498192 256 0.125 0.0080 3.7502 0.0055 3.604413122 324 0.111 0.0066 1.6333 0.0039 2.918720000 400 0.100 0.0054 1.9046 0.0036 0.759729282 484 0.909 0.0043 2.3899 0.0027 3.1084 Fig. 7.4 . Diagram: Convergence of PIM: Non-Hom case
Fig. 7.5 . Error: x -axis: log δ ; blue line: log e ; red line: y = 2 x − . . ig. 7.6 . Error: x -axis: log δ ; blue line: log e b ; red line: y = 1 . x − . . The above numerical simulation indicates that the discrete solution generated byPIM converges to the exact solution in a rate of O ( δ ) in the discrete l norm, which is O ( h ) where h represents the average distance between each adjancent sample points.One advantage of PIM is that only local mesh is required so that we do not need aglobal mesh as the surface finite element method. Moreover, PIM can be efficientlyapplied when the explicit formulation of the manifold is not known but only a set ofsample points, which is often occurred in data mining and machine learning models.Nevertheless, the quadrature rule we used in the point integral method is of lowaccuracy. If we have more information, such as the local mesh or local hypersurface,we could use high order quadrature rule to improve the accuracy of the point integralmethod.
8. Conclusion.
In this work, we have constructed a class of integral models thatapproximate the Poisson equation on a bounded two dimensional manifold embeddedin R d with Dirichlet boundary. Our calculation indicates that the convergence rateis O ( δ ) in H norm. To the author’s best knowledge, all previous studies haveprovided at most linear convergence rate in the Dirichlet case. Having a Dirichlet-type constraint with second order convergence to the local limit in two dimensionalmanifold would be mathematically interesting and of important practical interests. n fact, our model can be generalize into three dimensional case with more complicatecalculation.Similar to the integral approximation of Poisson models, the integral approxi-mation of some other types of PDEs are also of great interest. In our subsequentialpaper, we will introduce how to approximate the elliptic equation with discontinuouscoefficients in high dimensional manifolds. Our future plan is to extend our resultsinto a two dimensional polygonal domain where singularity appears near each vertex.The integral approximation for Stokes equation with Dirichlet boundary will also beanalyzed.
9. Appendix.9.1. Proof of Lemma 3.3.
This lemma is a direct corollary of the followingtwo equalities1. ∆ M u ( y ) = n i n j ( ∇ i ∇ j u ( y )) + τ i τ j ( ∇ i ∇ j u ( y )) , ∀ y ∈ ∂ M .2. τ i τ j ( ∇ i ∇ j u ( y )) = κ ( y )( n b ( y ) · n ( y )) ∂u∂ n ( y ) , ∀ y ∈ ∂ M .
1. The equation (30) of [32] gives us∆ M u ( y ) = ∇ j ( ∇ j ( u ( y )) = ∆ M u ( y ) = ( ∂ k (cid:48) φ j ) g k (cid:48) l (cid:48) ∂ l (cid:48) (cid:0) ( ∂ m (cid:48) φ j ) g m (cid:48) n (cid:48) ( ∂ n (cid:48) u ) (cid:1) . (9.1)Since ∂ k (cid:48) φ lies on the tangent plane of M at y , we have ∂ k (cid:48) φ j = ( ∂ k (cid:48) φ · n ) n j + ( ∂ k (cid:48) φ · τ ) τ j = ∂ k (cid:48) φ l n l n j + ∂ k (cid:48) φ l τ l τ j , (9.2)hence we can calculate( ∂ k (cid:48) φ j ) g k (cid:48) l (cid:48) ∂ l (cid:48) (cid:0) ( ∂ m (cid:48) φ j ) g m (cid:48) n (cid:48) ( ∂ n (cid:48) u ) (cid:1) =( ∂ k (cid:48) φ l n l n j ) g k (cid:48) l (cid:48) ∂ l (cid:48) (cid:0) ( ∂ m (cid:48) φ j ) g m (cid:48) n (cid:48) ( ∂ n (cid:48) u ) (cid:1) + ( ∂ k (cid:48) φ l τ l τ j ) g k (cid:48) l (cid:48) ∂ l (cid:48) (cid:0) ( ∂ m (cid:48) φ j ) g m (cid:48) n (cid:48) ( ∂ n (cid:48) u ) (cid:1) =( ∂ k (cid:48) φ l ) g k (cid:48) l (cid:48) ∂ l (cid:48) (cid:0) ( ∂ m (cid:48) φ l ) g m (cid:48) n (cid:48) ( ∂ n (cid:48) u ) (cid:1) n l n j + ( ∂ k (cid:48) φ l ) g k (cid:48) l (cid:48) ∂ l (cid:48) (cid:0) ( ∂ m (cid:48) φ l ) g m (cid:48) n (cid:48) ( ∂ n (cid:48) u ) (cid:1) τ l τ j = n i n j ( ∇ i ∇ j u ( y )) + τ i τ j ( ∇ i ∇ j u ( y )) , (9.3)and this give rise to∆ M u ( y ) = n i n j ( ∇ i ∇ j u ( y )) + τ i τ j ( ∇ i ∇ j u ( y )) . (9.4)2. We let γ : ( a, b ) → ∂ M be the equation of the curve in a neighborhood of y .Without loss of generality, we let γ ( t ) = y and (cid:12)(cid:12) d γ dt (cid:12)(cid:12) ≡ t ∈ ( a, b ). ince u = u ( γ ( t )) ≡ ∂ M , we have the following twoequalities0 = ddt u ( φ ( γ ( t ))) = ∇ u ( y ) · ddt φ ( γ ( t )) = ( ∇ j u ( y )) τ j ( y )=( ∂ m (cid:48) φ j ) g m (cid:48) n (cid:48) ( ∂ n (cid:48) u ( y )) τ j ( y ); (9.5)and0 = d dt u ( φ ( γ ( t ))) = ∇ u ( y ) · d dt φ ( γ ( t )) + ddt ( ∇ u ( y )) · ddt φ ( γ ( t ))=( ∇ j u ( y )) κ ( y ) n jb ( y ) + τ i τ j ( ∇ i ∇ j u ( y )) . (9.6)Still, as ∂ m (cid:48) φ lies on the tangent plane of M at y , We discover( ∇ j u ( y )) n jb ( y ) = ( ∂ m (cid:48) φ j ) g m (cid:48) n (cid:48) ( ∂ n (cid:48) u ( y )) n jb ( y ) = ( n b · ∂ m (cid:48) φ ) g m (cid:48) n (cid:48) ( ∂ n (cid:48) u ( y ))= ( n b · ( ∂ m (cid:48) φ · n ) n + n b · ( ∂ m (cid:48) φ · τ ) τ ) g m (cid:48) n (cid:48) ( ∂ n (cid:48) u ( y ))= ( n b · n )( ∂ m (cid:48) φ j ) g m (cid:48) n (cid:48) ( ∂ n (cid:48) u ( y )) n j ( y ) + ( n b · τ )( ∂ m (cid:48) φ j ) g m (cid:48) n (cid:48) ( ∂ n (cid:48) u ( y )) τ j ( y )= ( n b · n )( ∂ m (cid:48) φ j ) g m (cid:48) n (cid:48) ( ∂ n (cid:48) u ( y )) n j ( y ) = ( n b · n ) ∂u∂ n ( y ) , (9.7)where the second last equality comes from (9.5). Back to (9.6), this implies τ i τ j ( ∇ i ∇ j u ( y )) = − κ ( y )( ∇ j u ( y )) n jb ( y ) = − κ ( y )( n b ( y ) · n ( y )) ∂u∂ n ( y ) . (9.8)Hence we have completed our proof. By the definition of η , we can do the followingexpansion: (cid:90) ∂ M (cid:90) M f ( x ) g ( y ) · ( x − y − η ( x , y )) R δ ( x , y ) dµ x dτ y = (cid:90) ∂ M (cid:90) M f ( x ) g j ( y ) ξ i (cid:48) ξ j (cid:48) (cid:90) (cid:90) s∂ j (cid:48) ∂ i (cid:48) φ j ( α + τ s ξ ) dτ ds R δ ( x , y ) dµ x dτ y = (cid:90) ∂ M (cid:90) M f ( x ) g j ( y ) ξ i (cid:48) ξ j (cid:48) ξ k (cid:48) ( (cid:90) (cid:90) (cid:90) s τ ∂ j (cid:48) ∂ i (cid:48) ∂ k (cid:48) φ j ( α + τ st ξ ) dtdτ ds ) R δ ( x , y ) dµ x dτ y + (cid:90) ∂ M (cid:90) M f ( x ) g j ( y ) 12 ξ i (cid:48) ξ j (cid:48) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) R δ ( x , y ) dµ x dτ y = (cid:90) ∂ M g j ( y ) (cid:90) M f ( x ) 12 ξ i (cid:48) ξ j (cid:48) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) R δ ( x , y ) dµ x dτ y + r , (9.9) here r is a lower order term and will be mentioned later. Next, we will use thefact that2 δ ∂ x l ¯ R δ ( x , y ) =2 δ ∂ l φ ( β ) · ∇ x M ¯ R δ ( x , y ) = ∂ l φ ( β ) · η ( y , x ) R δ ( y , x ) + r = ∂ l φ k (cid:48) ( β ) ξ i (cid:48) ( y , x ) ∂ i (cid:48) φ k (cid:48) ( β ) R δ ( x , y ) + r = − ∂ l φ k (cid:48) ( β ) ∂ i (cid:48) φ k (cid:48) ( β ) ξ i (cid:48) ( x , y ) R δ ( x , y ) + r = − g i (cid:48) l ( x ) ξ i (cid:48) R δ ( x , y ) + r , (9.10)where r l ( x , y ) = ∂ l φ ( β ) · (2 δ ∇ x M ¯ R δ ( x , y ) − η ( y , x ) R δ ( y , x ))= ∂ l φ ( β )( ∂ m (cid:48) φ l ( α ) g m (cid:48) n (cid:48) ∂ n (cid:48) φ i ( α )( x i − y i ) ¯ R δ ( x , y ) − ∂ m (cid:48) φ l ( α ) g m (cid:48) n (cid:48) ∂ n (cid:48) φ i ( α ) ξ i (cid:48) ∂ i (cid:48) φ i ¯ R δ ( x , y ))= ∂ l φ ( β ) ∂ m (cid:48) φ l ( α ) g m (cid:48) n (cid:48) ∂ n (cid:48) φ i ( α )( x i − y i − ξ i (cid:48) ∂ i (cid:48) φ i ) ¯ R δ ( x , y )= ∂ l φ ( β ) ∂ m (cid:48) φ l ( α ) g m (cid:48) n (cid:48) ∂ n (cid:48) φ i ( α ) ¯ R δ ( x , y ) ξ i (cid:48) ξ j (cid:48) (cid:90) (cid:90) s∂ j (cid:48) ∂ i (cid:48) φ j ( α + τ s ξ ) dτ ds, (9.11)is obviously an O ( δ ) term, hence (cid:90) M f ( x ) ξ i (cid:48) ξ j (cid:48) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) R δ ( x , y ) dµ x = − δ (cid:90) Ω J ( y ) f ( x ) ξ j (cid:48) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) ∂ x l ¯ R δ ( x , y ) (cid:112) det G ( β ) d β + r = − δ (cid:90) ∂ Ω J ( y ) f ( x ) ξ j (cid:48) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) n l M ( β ) ¯ R δ ( x , y ) (cid:112) det G ( β ) dL x + 2 δ (cid:90) Ω J ( y ) ∂ l (cid:0) f ( x ) ξ j (cid:48) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) (cid:112) det G ( β ) (cid:1) ¯ R δ ( x , y ) d β + r = − δ (cid:90) ∂ Ω J ( y ) f ( x ) ξ j (cid:48) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) n l M ( β ) ¯ R δ ( x , y ) (cid:112) det G ( β ) dL x + 2 δ (cid:90) Ω J ( y ) ∂ l (cid:0) f ( x ) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) (cid:112) det G ( β ) (cid:1) ξ j (cid:48) ¯ R δ ( x , y ) d β − δ (cid:90) Ω J ( y ) f ( x ) ∂ l ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) (cid:112) det G ( β ) ¯ R δ ( x , y ) d β + r = − δ (cid:90) ∂ Ω J ( y ) f ( x ) ξ j (cid:48) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) n l M ( β ) ¯ R δ ( x , y ) (cid:112) det G ( β ) dL x − δ (cid:90) M f ( x ) ∂ l ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) ¯ R δ ( x , y ) dµ x + r + r . (9.12)Here n Ω ( β ) is the unit outward normal vector of Ω J ( y ) at β ∈ ∂ Ω J ( y ) , (cid:112) det G ( β )is the Jacobian matrix defined in (1.3) such that dµ x = (cid:112) det G ( β ) d β . To simplifythe notations, we always use Ω to denote Ω J ( y ) in the following content of this lemma. or the boundary term of (9.12), we aim to present that it is of lower order by asymmetry property. First, we decompose it into (cid:90) ∂ Ω f ( x ) ξ j (cid:48) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) n l M ( β ) ¯ R δ ( x , y ) (cid:112) det G ( β ) dL x = (cid:90) ∂ Ω f ( x )( ξ · n M ( β )) n j (cid:48) M ( x ) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) n l M ( β ) ¯ R δ ( x , y ) (cid:112) det G ( β ) dL x + (cid:90) ∂ Ω f ( x )( ξ · τ M ( x )) τ j (cid:48) M ( x ) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) n l M ( β ) ¯ R δ ( x , y ) (cid:112) det G ( β ) dL x = r + r (9.13)the term ξ · n M ( β ) is O ( δ ) since x , y both lie on ∂ M and | x − y | ≤ δ . Thisobservation implies that r is of lower order. For the term r , we have r = (cid:90) ∂ Ω f ( x )( ξ · τ M ( x )) τ j (cid:48) M ( x ) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) n l M ( β ) ¯ R δ ( x , y ) (cid:112) det G ( β ) dL x = (cid:90) ∂ Ω f ( x ) τ j (cid:48) M ( x ) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) n l M ( β ) (cid:112) det G ( β ) ξ ¯ R δ ( x , y ) · d γ =2 δ (cid:90) ∂ Ω f ( x ) τ j (cid:48) M ( x ) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) n l M ( β ) (cid:112) det G ( β ) g il (cid:48) ( x ) ∂ x l (cid:48) = R δ ( x , y ) · d γ i + r =2 δ (cid:90) ∂ Ω ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) ∂ x l (cid:48) ( f ( x ) τ j (cid:48) M ( x ) g i (cid:48) l ( x ) n l M ( β ) (cid:112) det G ( β ) g il (cid:48) ( x )) = R δ ( x , y ) · d γ i + r =2 δ (cid:90) ∂ Ω ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) ∂ x l (cid:48) ( f ( x ) τ j (cid:48) M ( x ) g i (cid:48) l ( x ) n l M ( β ) (cid:112) det G ( β ) g il (cid:48) ( x )) τ M ( x ) = R δ ( x , y ) dL x + r , (9.14)where γ represents the equation of the curve ∂ Ω with the parameter | γ (cid:48) | ≡ r is also a lower order term. For theinterior term of (9.12), we multiply by g and integrate over ∂ M to discover (cid:90) ∂ M g j ( y ) (cid:90) M f ( x ) ∂ l ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) ¯ R δ ( x , y ) dµ x dτ y = (cid:90) ∂ M g j ( y ) (cid:90) M f ( x ) ∂ l ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( y ) ¯ R δ ( x , y ) dµ x dτ y + r = (cid:90) ∂ M ∂ l ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( y ) g j ( y ) (cid:0) (cid:90) M f ( x ) ¯ R δ ( x , y ) dµ x (cid:1) dτ y + r = (cid:90) ∂ M ∂ l ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( y ) g j ( y ) ¯ f ( y ) dτ y + r ≤ C (cid:107) g (cid:107) L ( ∂ M ) (cid:13)(cid:13) ¯ f (cid:13)(cid:13) L ( ∂ M ) + r ≤ C (cid:107) g (cid:107) H ( M ) (cid:13)(cid:13) ¯ f (cid:13)(cid:13) H ( M ) + r (9.15) ere the lower order terms r = (cid:90) ∂ M (cid:90) M f ( x ) g j ( y ) ξ i (cid:48) ξ j (cid:48) ξ k (cid:48) ( (cid:90) (cid:90) (cid:90) s τ ∂ j (cid:48) ∂ i (cid:48) ∂ k (cid:48) φ j ( α + τ st ξ ) dtdτ ds ) R δ ( x , y ) dµ x dτ y , (9.16) r = 2 δ (cid:90) Ω ∂ l (cid:0) f ( x ) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) (cid:112) det G ( β ) (cid:1) ξ j (cid:48) ¯ R δ ( x , y ) d β , (9.17)is obviously O ( δ ), for the other error terms, recall the expression r = − δ (cid:90) Ω ( f ( x ) ξ j (cid:48) ∂ j (cid:48) ∂ i (cid:48) φ j ( α )) r l (cid:112) det G ( β ) d β , (9.18) r = 2 δ (cid:90) ∂ Ω f ( x ) τ j (cid:48) M ( x ) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) n l M ( β ) (cid:112) det G ( β ) r d γ i , (9.19)which indicates that they are O ( δ ) terms due to the estimate (9.11). For the term r , we have r = (cid:90) ∂ M g j ( y ) (cid:90) M f ( x ) ∂ l ∂ i (cid:48) φ j ( α )( g i (cid:48) l ( x ) − g i (cid:48) l ( y )) ¯ R δ ( x , y ) dµ x dτ y (9.20)which is of lower order by the continuity of the matrix g i (cid:48) l . Now we combine all theabove estimates to conclude (cid:90) ∂ M (cid:90) M f ( x ) g j ( y ) 12 ξ i (cid:48) ξ j (cid:48) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) R δ ( x , y ) dµ x dτ y = 12 (cid:90) ∂ M g j ( y ) (cid:90) M f ( x ) ξ i (cid:48) ξ j (cid:48) ∂ j (cid:48) ∂ i (cid:48) φ j ( α ) R δ ( x , y ) dµ x dτ y = − δ (cid:90) ∂ M g j ( y )( (cid:90) M f ( x ) ∂ l ∂ i (cid:48) φ j ( α ) g i (cid:48) l ( x ) ¯ R δ ( x , y ) dµ x + r + r + r + r ) dτ y ≤ Cδ ( (cid:107) g (cid:107) H ( M ) ( (cid:13)(cid:13) ¯ f (cid:13)(cid:13) H ( M ) + (cid:107) f (cid:107) H ( M ) ) . (9.21) r . This subsection is about the estimate on r in theproof of theorem 3.1. We denote d ( x , y ) = u ( x ) − u ( y ) − ( x − y ) · ∇ u ( y ) − η i η j ( ∇ i ∇ j u ( y )) , (9.22)then r ( x ) = 1 δ (cid:90) M d ( x , y ) R δ ( x , y ) dy. (9.23) sing Newton-Leibniz formula, we can discover d ( x , y ) = u ( x ) − u ( y ) − ( x − y ) · ∇ u ( y ) − η i η j ( ∇ i ∇ j u ( y ))= ξ i ξ i (cid:48) (cid:90) (cid:90) (cid:90) s dds (cid:16) ∂ i φ j ( α + s s ξ ) ∂ i (cid:48) φ j (cid:48) ( α + s s s ξ ) ∇ j (cid:48) ∇ j u ( φ ( α + s s s ξ )) (cid:17) ds ds ds = ξ i ξ i (cid:48) ξ i (cid:48)(cid:48) (cid:90) (cid:90) (cid:90) s s ∂ i φ j ( α + s s ξ ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α + s s s ξ ) ∇ j (cid:48) ∇ j u ( φ ( α + s s s ξ )) ds ds ds + ξ i ξ i (cid:48) ξ i (cid:48)(cid:48) (cid:90) (cid:90) (cid:90) s ∂ i (cid:48)(cid:48) ∂ i φ j ( α + s s ξ ) ∂ i (cid:48) φ j (cid:48) ( α + s s s ξ ) ∇ j (cid:48) ∇ j u ( φ ( α + s s s ξ )) ds ds ds + ξ i ξ i (cid:48) ξ i (cid:48)(cid:48) (cid:90) (cid:90) (cid:90) s s ∂ i φ j ( α + s s ξ ) ∂ i (cid:48) φ j (cid:48) ( α + s s s ξ ) ∂ i (cid:48)(cid:48) φ j (cid:48)(cid:48) ( α + s s s ξ ) ∇ j (cid:48)(cid:48) ∇ j (cid:48) ∇ j u ( φ ( α + s s s ξ )) ds ds ds = d ( x , y ) + d ( x , y ) + d ( x , y ) + d + d + d , (9.24)where φ = φ J ( x ) for simplicity as we mentioned, and d ( x , y ) = 16 ξ i ξ i (cid:48) ξ i (cid:48)(cid:48) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) , (9.25) d ( x , y ) = 13 ξ i ξ i (cid:48) ξ i (cid:48)(cid:48) ∂ i (cid:48)(cid:48) ∂ i φ j ( α ) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) , (9.26) d ( x , y ) = 16 η i η i (cid:48) η i (cid:48)(cid:48) ( ∇ i ∇ i (cid:48) ∇ i (cid:48)(cid:48) u ( y )) , (9.27)with the lower order terms d = ξ i ξ i (cid:48) ξ i (cid:48)(cid:48) (cid:90) (cid:90) (cid:90) (cid:90) s s dds ( ∂ i φ j ( α + s s s ξ ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α + s s s s ξ ) ∇ j (cid:48) ∇ j u ( φ ( α + s s s s ξ )) ds ds ds ds , (9.28) d = ξ i ξ i (cid:48) ξ i (cid:48)(cid:48) (cid:90) (cid:90) (cid:90) (cid:90) s dds ∂ i (cid:48)(cid:48) ∂ i φ j ( α + s s s ξ ) ∂ i (cid:48) φ j (cid:48) ( α + s s s s ξ ) ∇ j (cid:48) ∇ j u ( φ ( α + s s s s ξ )) ds ds ds ds , (9.29) d = ξ i ξ i (cid:48) ξ i (cid:48)(cid:48) (cid:90) (cid:90) (cid:90) (cid:90) s s dds ∂ i φ j ( α + s s s ξ ) ∂ i (cid:48) φ j (cid:48) ( α + s s s s ξ ) ∂ i (cid:48)(cid:48) φ j (cid:48)(cid:48) ( α + s s s s ξ ) ∇ j (cid:48)(cid:48) ∇ j (cid:48) ∇ j u ( φ ( α + s s s s ξ )) ds ds ds ds , (9.30) ame as the control of d ( x , y ) in the page 11-14 of [32], we can calculate (cid:90) M ( (cid:90) M d i ( x , y ) R δ ( x , y ) dµ y ) dµ x ≤ C (cid:90) M ( (cid:90) M d i ( x , y ) R δ ( x , y ) dµ y )( (cid:90) M R δ ( x , y ) dµ y ) dµ x ≤ C (cid:90) M (cid:90) M d i ( x , y ) R δ ( x , y ) dµ y dµ x = C N (cid:88) j =1 (cid:90) O j (cid:90) M d i ( x , y ) R δ ( x , y ) dµ y dµ x ≤ C N (cid:88) j =1 (cid:90) O j (cid:90) B δ q j d i ( x , y ) R δ ( x , y ) dµ y dµ x ≤ Cδ N (cid:88) j =1 (cid:90) B δ q j | D , u ( y ) | dµ y ≤ Cδ (cid:107) u (cid:107) H ( M ) , (9.31)where i = 1 , , D , u ( y ) = d (cid:88) j,j (cid:48) ,j (cid:48)(cid:48) ,j (cid:48)(cid:48)(cid:48) =1 |∇ j (cid:48)(cid:48)(cid:48) ∇ j (cid:48)(cid:48) ∇ j (cid:48) ∇ j u ( x ) | + d (cid:88) j,j (cid:48) ,j (cid:48)(cid:48) =1 |∇ j (cid:48)(cid:48) ∇ j (cid:48) ∇ j u ( x ) | . Also (cid:90) M ( ∇ x M (cid:90) M d i ( x , y ) R δ ( x , y ) dµ y ) dµ x ≤ Cδ (cid:107) u (cid:107) H ( M ) , (9.32)where i = 1 , , . Hence we have completed the control of d i , i = 1 , ,
3. The termsremaining is d , d , d . By a simple observation we see d = 2 d . We now start toanalyze d . From (9.23), the term we need to control is (cid:90) M d ( x , y ) R δ ( x , y ) dµ y = (cid:90) M ξ i ξ i (cid:48) ξ i (cid:48)(cid:48) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) R δ ( x , y ) dµ y . (9.33)We aim to control it by integration by parts. The calculation of the following term isneeded: ∂ y l ¯ R δ ( x , y ) = ∂ l φ ( α ) · ∇ y ¯ R δ ( x , y ) = 12 δ ∂ l φ ( α ) · η ( x , y ) R δ ( x , y ) + d l = 12 δ ∂ l φ k (cid:48) ( α ) ξ i (cid:48) ( x , y ) ∂ i (cid:48) φ k (cid:48) ( α ) R δ ( x , y ) + d l = − δ ∂ l φ k (cid:48) ( α ) ∂ i (cid:48) φ k (cid:48) ( α ) ξ i (cid:48) R δ ( x , y ) + d l = − δ g i (cid:48) l ( α ) ξ i (cid:48) R δ ( x , y ) + d l , (9.34)where d l is a lower order term with the form d l = ∂ l φ ( α ) · ( 12 δ η ( x , y ) R δ ( x , y ) − ∇ y ¯ R δ ( x , y ))= − δ ∂ l φ i ( α ) ∂ m (cid:48) φ i g m (cid:48) n (cid:48) ∂ n (cid:48) φ j ( x j − y j − ξ i (cid:48) ∂ i (cid:48) φ j ) R δ ( x , y )= − δ ξ i (cid:48) ξ j (cid:48) ∂ l φ i ( α ) ∂ m (cid:48) φ i g m (cid:48) n (cid:48) ∂ n (cid:48) φ j ( (cid:90) (cid:90) s∂ j (cid:48) ∂ i (cid:48) φ j ( α + τ s ξ ) dτ ds ) R δ ( x , y ) . (9.35) y a similar argument, we have ∂ y l = R δ ( x , y ) = − δ g i (cid:48) l ( α ) ξ i (cid:48) ¯ R δ ( x , y ) + d l , (9.36)where d l = − δ ξ i (cid:48) ξ j (cid:48) ∂ l φ i ( α ) ∂ m (cid:48) φ i g m (cid:48) n (cid:48) ∂ n (cid:48) φ j ( (cid:90) (cid:90) s∂ j (cid:48) ∂ i (cid:48) φ j ( α + τ s ξ ) dτ ds ) ¯ R δ ( x , y ) . (9.37)Back to (9.33), we apply integration by parts on the parametric plane Ω J ( x ) of M : (cid:90) M ξ i ξ i (cid:48) ξ i (cid:48)(cid:48) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) R δ ( x , y ) dµ y = − δ (cid:90) M ξ i ξ i (cid:48) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α )( ∂ y l ¯ R δ ( x , y ) − r ) dµ y = − δ (cid:90) Ω J ( x ) ξ i ξ i (cid:48) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) ∂ y l ¯ R δ ( x , y ) (cid:112) Det G ( α ) d α + 2 δ (cid:90) M ξ i ξ i (cid:48) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) d l dµ y = − δ (cid:90) Ω J ( x ) ξ i ξ i (cid:48) ∂ l (cid:0) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) (cid:1) ¯ R δ ( x , y ) d α + 2 δ (cid:90) Ω J ( x ) ξ i (cid:48) ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) ¯ R δ ( x , y ) (cid:112) det G ( α ) d α + 2 δ (cid:90) Ω J ( x ) ξ i ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ l φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) ¯ R δ ( x , y ) (cid:112) det G ( α ) d α − δ (cid:90) ∂ Ω J ( x ) ξ i ξ i (cid:48) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) n l Ω ( α ) ¯ R δ ( x , y ) d α + 2 δ (cid:90) M ξ i ξ i (cid:48) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) d l dµ y = d + d + d + d + d = d + 2 d + d + d , (9.38)Still, n Ω ( α ) is the unit outward normal vector of Ω J ( x ) at α ∈ ∂ Ω J ( y ) , (cid:112) det G ( α )is the Jacobian matrix defined in (1.3) such that dµ y = (cid:112) det G ( α ) d α . We are thenready to control the error terms d i , i = 1 , , , ,
5. For d , we have d (cid:107) L ( M ) =4 δ (cid:90) M (cid:16) (cid:90) Ω J ( x ) ξ i ξ i (cid:48) ∂ l (cid:0) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) (cid:1) ¯ R δ ( x , y ) d α (cid:17) dµ x =4 δ N (cid:88) p =1 (cid:90) O p (cid:16) (cid:90) Ω p ξ i ξ i (cid:48) ∂ l (cid:0) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) (cid:1) ¯ R δ ( x , y ) d α (cid:17) dµ x ≤ δ N (cid:88) p =1 (cid:90) O p (cid:16) (cid:90) Ω p max i,i (cid:48) (cid:12)(cid:12) ∂ l (cid:0) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) (cid:1)(cid:12)(cid:12) ¯ R δ ( x , y ) d α (cid:17) dµ x ≤ δ N (cid:88) p =1 (cid:90) O p (cid:16) (cid:90) Ω p max i,i (cid:48) (cid:12)(cid:12) ∂ l (cid:0) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) (cid:1)(cid:12)(cid:12) ¯ R δ ( x , y ) d α (cid:17)(cid:16) (cid:90) Ω J ( x ) ¯ R δ ( x , y ) d α (cid:17) dµ x ≤ δ N (cid:88) p =1 (cid:90) O p (cid:16) (cid:90) Ω p max i,i (cid:48) (cid:12)(cid:12) ∂ l (cid:0) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) (cid:1)(cid:12)(cid:12) ¯ R δ ( x , y ) d α (cid:17) dµ x ≤ Cδ N (cid:88) p =1 (cid:90) Ω p (cid:0) (cid:90) O p ¯ R δ ( x , y ) dµ x (cid:1) max i,i (cid:48) (cid:12)(cid:12) ∂ l (cid:0) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) (cid:1)(cid:12)(cid:12) d α ≤ Cδ N (cid:88) p =1 (cid:90) Ω p max i,i (cid:48) (cid:12)(cid:12) ∂ l (cid:0) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) (cid:1)(cid:12)(cid:12) d α ≤ Cδ N (cid:88) p =1 (cid:107) u (cid:107) H (Ω p ) ≤ Cδ (cid:107) u (cid:107) H ( M ) , (9.39)by the similar argument, we discover (cid:107)∇ d (cid:107) L ( M ) ≤ Cδ (cid:90) M (cid:16) (cid:90) Ω J ( x ) ( ∇ x M ξ i ξ i (cid:48) ) ∂ l (cid:0) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) (cid:1) ¯ R δ ( x , y ) d α (cid:17) dµ x + (cid:90) M (cid:90) Ω J ( x ) (cid:16) ξ i ξ i (cid:48) ∂ l (cid:0) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) (cid:1) ( ∇ x M ¯ R δ ( x , y )) d α (cid:17) dµ x ≤ Cδ (cid:90) M (cid:16) (cid:90) Ω J ( x ) δ max i,i (cid:48) (cid:12)(cid:12) ∂ l (cid:0) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) (cid:1)(cid:12)(cid:12) ¯ R δ ( x , y ) d α (cid:17) dµ x + Cδ (cid:90) M (cid:16) (cid:90) Ω J ( x ) δ max i,i (cid:48) (cid:12)(cid:12) ∂ l (cid:0) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) (cid:1)(cid:12)(cid:12) δ R δ ( x , y ) d α (cid:17) dµ x ≤ Cδ N (cid:88) p =1 (cid:90) O p (cid:16) (cid:90) Ω p max i,i (cid:48) (cid:12)(cid:12) ∂ l (cid:0) ∂ i φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) (cid:112) det G ( α ) (cid:1)(cid:12)(cid:12) ( ¯ R δ ( x , y ) + R δ ( x , y )) d α (cid:17) dµ x ≤ Cδ (cid:107) u (cid:107) H ( M ) . (9.40) or d we apply integration by parts again: d ( x ) =2 δ (cid:90) Ω J ( x ) ξ i (cid:48) ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) ¯ R δ ( x , y ) (cid:112) det G ( α ) d α = − δ (cid:90) ∂ Ω J ( x ) ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) n l (cid:48) Ω ( α ) = R δ ( x , y ) (cid:112) det G ( α ) d α + 4 δ (cid:90) Ω J ( x ) ∂ l (cid:48) (cid:16) ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) (cid:112) det G ( α ) (cid:17) = R δ ( x , y ) d α + 4 δ (cid:90) Ω J ( x ) ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) d l (cid:48) ( α ) (cid:112) det G ( α ) d α = − δ (cid:90) ∂ Ω J ( x ) ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) n l (cid:48) Ω ( α ) = R δ ( x , y ) (cid:112) det G ( α ) d α + 4 δ (cid:90) M (cid:112) det G ( α ) ∂ l (cid:48) (cid:16) ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) (cid:112) det G ( α ) (cid:17) = R δ ( x , y ) dµ y + 4 δ (cid:90) M ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) d l (cid:48) ( α ) dµ y = d + d + d . (9.41)Here d is a term supported in the layer of ∂ M with width 2 δ due to the supportof ¯ R , and (cid:107) d (cid:107) L ( M ) =16 δ (cid:90) M (cid:16) (cid:90) ∂ Ω J ( x ) ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) n l (cid:48) Ω ( α ) = R δ ( x , y ) (cid:112) det G ( α ) d α (cid:17) dµ x ≤ Cδ (cid:90) M (cid:90) ∂ Ω J ( x ) ( ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) n l (cid:48) Ω ( α )) R δ ( x , y ) det G ( α ) d α ( (cid:90) ∂ Ω J ( x ) = R δ ( x , y ) dα ) dµ x ≤ Cδ N (cid:88) p =1 (cid:90) O p (cid:90) ∂ Ω p ( ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) n l (cid:48) Ω ( α )) R δ ( x , y ) det G ( α ) d α dµ x , ≤ Cδ N (cid:88) p =1 (cid:90) ∂ Ω p ( (cid:90) O p = R δ ( x , y ) dµ x )( ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) n l (cid:48) Ω ( α )) det G ( α ) d α ≤ Cδ N (cid:88) p =1 (cid:90) ∂ Ω p ( ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) n l (cid:48) Ω ( α )) det G ( α ) d α ≤ Cδ N (cid:88) p =1 (cid:107) u (cid:107) H (Ω p ) ≤ Cδ (cid:107) u (cid:107) H ( M ) , (9.42) imilarly, we have (cid:107)∇ d (cid:107) L ( M ) =16 δ N (cid:88) p =1 (cid:90) O p (cid:90) ∂ Ω p ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) n l (cid:48) Ω ( α )( ∇ x M = R δ ( x , y )) (cid:112) det G ( α ) d α (cid:17) dµ x ≤ Cδ (cid:107) u (cid:107) H ( M ) . (9.43)In addition, let us verify the property (3.6) of d . In fact, for any f ∈ H ( M ),we have (cid:90) M d ( x ) f ( x ) dµ x = − δ (cid:90) M f ( x ) (cid:90) ∂ Ω J ( x ) ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) n l (cid:48) Ω ( α ) = R δ ( x , y ) (cid:112) det G ( α ) d α dµ x = − δ N (cid:88) p =1 (cid:90) O p (cid:90) ∂ Ω p ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) n l (cid:48) Ω ( α ) (cid:112) det G ( α )( f ( x ) = R δ ( x , y )) d α dµ x = − δ N (cid:88) p =1 (cid:90) ∂ Ω p ( (cid:90) O p f ( x ) = R δ ( x , y ) dµ x ) ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) n l (cid:48) Ω ( α ) (cid:112) det G ( α ) d α = − δ N (cid:88) p =1 (cid:90) ∂ Ω p = f ( y ) ∂ l φ j ( α ) ∂ i (cid:48)(cid:48) ∂ i (cid:48) φ j (cid:48) ( α ) ∇ j (cid:48) ∇ j u ( α ) g i (cid:48)(cid:48) l ( α ) g i (cid:48) l (cid:48) ( α ) n l (cid:48) Ω ( α ) (cid:112) det G ( α ) d α ≤ Cδ (cid:90) ∂ M = f ( y ) |∇ u ( y ) | dµ y ≤ Cδ (cid:13)(cid:13)(cid:13)(cid:13) = f (cid:13)(cid:13)(cid:13)(cid:13) L ( ∂ M ) (cid:107) u (cid:107) H ( ∂ M ) ≤ Cδ (cid:13)(cid:13)(cid:13)(cid:13) = f (cid:13)(cid:13)(cid:13)(cid:13) H ( M ) (cid:107) u (cid:107) H ( M ) . (9.44)For the next terms, d and d are two lower order terms defined on the interior,where the control is by the similar argument of d .The control for d is similar as d , where two more integration by parts than d are needed, thus heavy calculations are need in the control of d . However, asa consequence, d satisfy the same property of d . d is obviously a lower order term due to the calculation of d in (9.35).Hence we have finished the control on the term d and d . For the term d , wedenote r = 1 δ (cid:90) M d ( x , y ) R δ ( x , y ) d y = 16 δ (cid:90) M η i η j η k ( ∇ i ∇ j ∇ k u ( y )) R δ ( x , y ) dµ y (9.45)Similar as r , we can decompose r into r = (cid:80) i =2 r i , where = 16 δ ( (cid:90) M η i η j η k ( ∇ i ∇ j ∇ k u ( y )) R δ ( x , y ) dµ y − (cid:90) M η i η j ( ∇ i ∇ j ∇ k u ( y )) ∇ k ¯ R δ ( x , y ) dµ y ) , (9.46) r ( x ) = 12 δ ( (cid:90) M η i η j ( ∇ i ∇ j ∇ k u ( y )) ∇ k ¯ R δ ( x , y ) dµ y + (cid:90) M div ( η i η j ( ∇ i ∇ j ∇ u ( y ))) ¯ R δ ( x , y ) dµ y ) , (9.47) r ( x ) = 12 δ ( − (cid:90) M div ( η i η j ( ∇ i ∇ j ∇ u ( y ))) ¯ R δ ( x , y ) dµ y +2 (cid:90) M η i ( ∇ i ∇ j ∇ j u ( y )) ¯ R δ ( x , y ) dµ y ) , (9.48) r ( x ) = − δ ( (cid:90) M η i ( ∇ i ∇ j ∇ j u ( y )) ¯ R δ ( x , y ) dµ y − (cid:90) M ( ∇ i ∇ j ∇ j u ( y )) ∇ i = R δ ( x , y ) dµ y ) , (9.49) r ( x ) = 1 δ ( (cid:90) M ( ∇ i ∇ j ∇ j u ( y )) ∇ i = R δ ( x , y ) dµ y + (cid:90) M ( ∇ i ∇ i ∇ j ∇ j u ( y )) = R δ ( x , y ) dµ y ) , (9.50) r ( x ) = − δ (cid:90) M ( ∇ i ∇ i ∇ j ∇ j u ( y )) = R δ ( x , y ) dµ y . (9.51)The control for r and r is the same as r in [32]. For r , we can calculate r = (cid:90) ∂ M ( η i η j n k ( ∇ i ∇ j ∇ k u ( y ))) ¯ R δ ( x , y ) dτ y (9.52)Since η i = ξ i (cid:48) ∂ i (cid:48) φ i , we can refer to the control of the term (9.9) in the proof of lemma3.4 to deduce (cid:90) M r ( x ) f ( x ) dµ x ≤ Cδ ( (cid:107) f (cid:107) H ( M ) + (cid:13)(cid:13) ¯ f (cid:13)(cid:13) H ( M ) ) (cid:107) u (cid:107) H ( M ) , (9.53) nd the same argument hold for r . For r , since ∂ i (cid:48) ξ l = − δ li (cid:48) , we have div ( η i η j ( ∇ i ∇ j ∇ u ( y ))= 1 √ det G ∂ i (cid:48) ( √ det Gg i (cid:48) j (cid:48) ( ∂ j (cid:48) φ k ) η i η j ∇ i ∇ j ∇ k u ( y ))= ξ l √ det G ∂ i (cid:48) ( √ det Gg i (cid:48) j (cid:48) ( ∂ j (cid:48) φ k ) η i ( ∂ l φ j ) ∇ i ∇ j ∇ k u ( y )) − g i (cid:48) j (cid:48) ( ∂ j (cid:48) φ k ) η i ( ∂ i (cid:48) φ j )( ∇ i ∇ j ∇ k u ( y ))= ξ l ξ k (cid:48) √ det G ∂ i (cid:48) ( √ det Gg i (cid:48) j (cid:48) ( ∂ j (cid:48) φ k )( ∂ k (cid:48) φ i )( ∂ l φ j ) ∇ i ∇ j ∇ k u ( y )) − g i (cid:48) j (cid:48) ( ∂ j (cid:48) φ k ) η i ( ∂ i (cid:48) φ j )( ∇ i ∇ j ∇ k u ( y ))= ξ l ξ k (cid:48) √ det G ∂ i (cid:48) ( √ det Gg i (cid:48) j (cid:48) ( ∂ j (cid:48) φ k )( ∂ k (cid:48) φ i )( ∂ l φ j ) ∇ i ∇ j ∇ k u ( y )) − η i ( ∇ i ∇ j ∇ j u ( y )) , (9.54)where the last inequality results from the equation (31) of [32]. This implies r ( x ) = (cid:90) M div ( η i η j ( ∇ i ∇ j ∇ u ( y )) ¯ R δ ( x , y ) dµ y + 2 (cid:90) M η i ( ∇ i ∇ j ∇ j u ( y )) ¯ R δ ( x , y ) dµ y = (cid:90) M ξ l ξ k (cid:48) √ det G ∂ i (cid:48) ( √ det Gg i (cid:48) j (cid:48) ( ∂ j (cid:48) φ k )( ∂ k (cid:48) φ i )( ∂ l φ j ) ∇ i ∇ j ∇ k u ( y )) ¯ R δ ( x , y ) dµ y . (9.55)By the same calculation as the term d we can conclude (cid:107) r (cid:107) L ( M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) , (9.56) (cid:107)∇ r (cid:107) L ( M ) ≤ Cδ (cid:107) u (cid:107) H ( M ) . (9.57)And similarly, we have the same bound for r as r .Thereafter, we have completed the control on all the terms that composes r . REFERENCES[1] B. Alali and M. Gunzburger. Peridynamics and material interfaces.
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