Linear Strain Tensors on Hyperbolic Surfaces and Asymptotic Theories for Thin Shells
aa r X i v : . [ m a t h - ph ] D ec Linear Strain Tensors on Hyperbolic Surfaces andAsymptotic Theories for Thin Shells
Peng-Fei YAOKey Laboratory of Systems and ControlInstitute of Systems Science, Academy of Mathematics and Systems ScienceChinese Academy of Sciences, Beijing 100190, P. R. ChinaSchool of Mathematical SciencesUniversity of Chinese Academy of Sciences, Beijing 100049, Chinae-mail: [email protected]
Abstract
We perform a detailed analysis of the solvability of linear strain equationson hyperbolic surfaces under a technical assumption (noncharacteristic). For regularenough hyperbolic surfaces, it is proved that smooth infinitesimal isometries are densein the W , infinitesimal isometries and that smooth enough infinitesimal isometriescan be matched with higher order infinitesimal isometries. Then those results areapplied to elasticity of thin shells for the Γ-limits. The recovery sequences (Γ-limsup inequlity) are obtained for dimensionally-reduced shell theories, when the elasticenergy density scales like h β , β ∈ (2 , , that is, intermediate regime between purebending ( β = 2) and the von-Karman regime ( β = 4), where h is thickness of a shell. Keywords hyperbolic surface, shell, nonlinear elasticity, Riemannian geometry
Mathematics Subject Classifications (2010)
Let M ⊂ IR be a surface with a normal ~n and let the middle surface of a shell be anopen set Ω ⊂ M. Let T k ( M ) denote all the k -order tensor fields on M for an integer k ≥ . Let T ( M ) be all the 2-order symmetrical tensor fields on M. For y ∈ H (Ω , IR ) , wedecompose it into y = W + w~n, where w = h y, ~n i and W ∈ T (Ω) . For U ∈ T (Ω) given, This work is supported by the National Science Foundation of China, grants no. 61473126 and no.61573342, and Key Research Program of Frontier Sciences, CAS, no. QYZDJ-SSW-SYS011. y ∈ W , (Ω , IR ) of the middle surface Ω takes theform sym DW + w Π = U for x ∈ Ω , (1.1)where D is the connection of the induced metric in M, DW = DW + D T W, andΠ is the second fundamental form of M. Equation (1.1) plays a fundamental role in thetheory of thin shells, see [5, 6, 7, 8] . When U = 0 , a solution y to (1.1) is referredto as an infinitesimal isometry. Under a technical assumption (noncharacteristic) onhyperbolic surfaces, we establish existence, uniqueness, and regularity of solutions for(1.1). For regular enough hyperbolic surface that satisfies a noncharacteristic assumption,it is proved that smooth infinitesimal isometries are dense in the W , (Ω , IR ) infinitesimalisometries (Theorem 1.2) and that smooth enough infinitesimal isometries can be matchedwith higher order infinitesimal isometries (Theorem 1.3). This matching property is animportant tool in obtaining recovery sequences (Γ-lim sup inequlity) for dimensionally-reduced shell theories in elasticity, when the elastic energy density scales like h β , β ∈ (2 , , that is, intermediate regime between pure bending ( β = 2) and the von-Karman regime( β = 4). Such results have been obtained for elliptic surfaces [8] and developable surfaces[5]. A survey on this topic is presented in [6]. Here we shall establish the similar resultsfor hyperbolic surfaces in Theorems 1.6-1.7.We state our main results for the hyperbolic surfaces as follows.A region Ω ⊂ M is said to be hyperbolic if its Gaussian curvature κ is strictly negative.We assume throughout this paper that κ ( x ) < x ∈ Ω . We introduce the notion of a noncharacteristic region below, subject to the second funda-mental form Π of the surface M. Definition 1.1
A region Ω ⊂ M is said to be noncharacteristic if Ω = { α ( t, s ) | ( t, s ) ∈ (0 , a ) × (0 , b ) } , where α : [0 , a ] × [0 , b ] → M is an imbedding map which is a family of regular curves withtwo parameters t, s such that Π( α t ( t, s ) , α t ( t, s )) = 0 , for all ( t, s ) ∈ [0 , a ] × [0 , b ] , Π( α s (0 , s ) , α s (0 , s )) = 0 , Π( α s ( a, s ) , α s ( a, s )) = 0 , for all s ∈ [0 , b ] , Π( α t (0 , s ) , α s (0 , s )) = Π( α t ( a, s ) , α s ( a, s )) = 0 , for all s ∈ [0 , b ] . h : IR → IR,M = { ( x, h ( x )) | x = ( x , x ) ∈ IR } . Under the coordinate system ψ ( p ) = x for p = ( x, h ( x )) ∈ M,∂x = (1 , , h x ( x )) , ∂x = (0 , , h x ( x )) , ~n = 1 p |∇ h | ( −∇ h, , Π = − p |∇ h | ∇ h, κ = h x x h x x − h x x (1 + |∇ h | ) . (i) Let h ( x ) = h ( x )+ h ( x ) where h i : IR → IR are C functions with h ′′ ( x ) h ′′ ( x ) < . Let σ i ∈ IR for 1 ≤ i ≤ σ < σ and σ < σ . ThenΩ = { ( x, h ( x )) | σ < x < σ , σ < x < σ } is noncharacteristic.(ii) Let h ( x ) = x − x x . Then κ ( p ) < p = ( x, h ( x )) , x ∈ IR , | x | > . For ε > σ < σ givenΩ = { ( x, h ( x )) | ε < x < ε , σ x < x < σ x } is noncharacteristic. However, there exists a region on M for the h, that is not a nonchar-acteristic. For example, a regionΩ = { ( x, h ( x )) | a < | x | < b } is not noncharacteristic, where 0 < a < b and | x | = x + x , since its boundaries | x | = a and | x | = b are not noncharacteristic curves.Moreover, if Ω is given by a single principal coordinate, i.e.,Ω = { α ( t, s ) | ( t, s ) ∈ ( a, b ) × ( c, d ) } , where ∇ ∂t ~n = λ ∂t, ∇ ∂s ~n = λ ∂s, λ > , λ < , and ~n is the normal of M, then Ω is a noncharacteristic region clearly. Since a principalcoordinate exists locally [14], a noncharacteristic region also exists locally.The notion of the noncharacteristic region is a technical assumption and a differentregion is given in [13]. In general, for U ∈ T (Ω) given, there are many solutions to(1.1). The aim of this assumption is to help us choose a regular solution for each U. In3act, equation (1.1) can be translated into a scalar second order partial differential equa-tion (see Theorem 2.1 later), which is elliptic for the elliptic surface [8], parabolic for thedevelopable surface with no flat part [5], and hyperbolic for the hyperbolic surface, respec-tively. Here we assume Ω to be a noncharacteristic region in order to set up appropriateboundary conditions such that the scalar equation is to be well-posedness (see Theorem4.1 and (4.1)-(4.5)). The main observation is that if the values of a solution v to the hyper-bolic equation (2.25) and its derivatives along a noncharacteristic curve are preset, thenthe solution v is uniquely determined in some neighborhood of this curve. We first solve(2.25) locally and then paste up the local solutions to yield a global one (see Lemma 4.4),where the noncharacteristic assumption is such that this produce is successful. We believethe corresponding results hold true for a more general region but some more complicateddiscusses may be involved.We say that a noncharacteristic region Ω ⊂ M is of class C m, for some integer m ≥ M is of class C m, and all the curves α (0 , · ) , α ( a, · ) , and α ( · , s ) for each s ∈ [0 , b ] are of class C m, . The points α (0 , , α ( a, , α (0 , b ) , and α ( a, b ) are angularpoints of Ω even if Ω is smooth. Theorem 1.1
Let Ω be a noncharacterisic region of class C , . For U ∈ C , (Ω , T ) , there exists a solution y = W + w~n ∈ C , (Ω , IR ) to equation (1 . satisfying the bounds k W k C , (Ω ,T ) + k w k C , (Ω) ≤ C k U k C , (Ω ,T ) . (1.2) If, in addition, Ω ∈ C m +2 , , U ∈ C m +1 , (Ω , T ) for some m ≥ , then k W k C m +1 , (Ω ,T ) + k w k C m, (Ω) ≤ C k U k C m +1 , (Ω ,T ) . (1.3) Remark 1.1
For the solvability of (1 . , the noncharacterisic assumption of Ω canbe relaxed. Let Ω be not a noncharacterisic region but there be a noncharacterisic one ˆΩ such that Ω ⊂ ˆΩ . Then Theorem . still holds true. In fact, we can extend U such that ˆ U ∈ C m +1 , ( ˆΩ , T ) with the estimate k ˆ U k C m +1 , (ˆΩ ,T ) ≤ C k U k C m +1 , (Ω ,T ) . Then we solve (1 . on ˆΩ to obtain a solution y for which (1 . still holds. For y ∈ W , (Ω , IR ) , we denote the left hand side of equation (1.1) by sym ∇ y. Let V (Ω , IR ) = { V ∈ W , (Ω , IR ) | sym ∇ V = 0 } . Theorem 1.2
Let Ω be a noncharacteristic region of class C m +2 , for some inte-ger m ≥ . Then, for every V ∈ V (Ω , IR ) there exists a sequence { V k } ⊂ V (Ω , IR ) ∩ C m, (Ω , IR ) such that lim k →∞ k V − V k k W , (Ω ,IR ) = 0 . (1.4)4 one parameter family { u ε } ε> ⊂ C , (Ω , IR ) is said to be a (generalized) m thorder infinitesimal isometry if the change of metric induced by u ε is of order ε m +1 , that k∇ T u ε ∇ u ε − g k L ∞ (Ω ,T ) = O ( ε m +1 ) as ε → , where g is the induced metric of M from IR , see [5]. A given m th order infinitesimalisometry can be modified by higher order corrections to yield an infinitesimal isometryof order m > m, a property to which we refer to by matching property of infinitesimalisometries , [5, 8]. This property plays an important role in the construction of a recoversequence in the Γ-limit for thin shells. Theorem 1.3
Let Ω be a noncharacteristic region of class C m +1 , . Given V ∈V (Ω , IR ) ∩ C m − , (Ω , IR ) , there exists a family { w ε } ε> ⊂ C , (Ω , IR ) , equiboundedin C , (Ω , IR ) , such that for all small ε > the family: u ε = id + εV + ε w ε is a m th order infinitesimal isometry of class C , . Let A : Ω → IR × be a matrix field. We define A ∈ T (Ω) by A ( α, β ) = h A ( x ) α, β i for α, β ∈ T x Ω , x ∈ Ω . For V ∈ V (Ω , IR ) given, there exists a unique A ∈ W , (Ω , T ) such that ∇ α V = A ( x ) α for α ∈ T x M, A ( x ) = − A T ( x ) , x ∈ Ω . (1.5)The finite strain space is the following closed subspace of L (Ω , T ) B (Ω , T ) = { lim h → sym ∇ w h | w h ∈ W , (Ω , R ) } where limits are taken in L (Ω , T ) , see [3, 7, 9]. B (Ω , T ) and V (Ω , IR ) are twobasic spaces for the Γ-limit functional. A region Ω ⊂ M is said to be approximately robust if ( A ) tan ∈ B (Ω , T ) for V ∈ V (Ω , IR ) , where ( A ) tan ( α, β ) = h A α, β i for α, β ∈ T x Ω , x ∈ Ω . If Ω is approximately robust, then the Γ-limit functional can be simplified to the bendingenergy. An approximately robust surface exhibits a better capacity to resist stretching sothat the limit functional consists only of a bending term, see [7].
Theorem 1.4
Let Ω ⊂ M be a noncharacteristic region of class C , . Then Ω isapproximately robust. pplication to elasticity of thin shells Let ~n be the normal field of surface M. Consider a family { Ω h } h> of thin shells of thickness h around Ω , Ω h = { x + t~n ( x ) | x ∈ Ω , | t | < h/ } , < h < h , where h is small enough so that the projection map π : Ω h → Ω , π ( x + t~n ) = x is welldefined. For a W , deformation u h : Ω h → IR , we assume that its elastic energy (scaledper unit thickness) is given by the nonlinear functional: E h ( u h ) = 1 h Z Ω h W ( ∇ u h ) dz. The stored-energy density function W : IR × IR → IR is C in an open neighborhoodof SO(3), and it is assumed to satisfy the conditions of normalization, frame indifferenceand quadratic growth: For all F ∈ IR × IR , R ∈ SO (3) ,W ( R ) = 0 , W ( RF ) = W ( F ) , W ( F ) ≥ C dist ( F, SO (3)) , with a uniform constant C > . The potential W induces the quadratic forms ([1]) Q ( F ) = D W ( Id )( F, F ) , Q ( x, F tan ) = min { Q ( ˆ F ) | ˆ F = F tan } . We shall consider a sequence e h > < lim h → e h /h β < ∞ for some 2 < β ≤ . (1.6)Let β m = 2 + 2 /m. Recall the following result.
Theorem 1.5 [ ] Let Ω be a surface embedded in IR , which is compact, connected,oriented, of class C , , and whose boundary ∂ Ω is the union of finitely many Lipschitzcurves. Let u h ∈ W , (Ω h , IR ) be a sequence of deformations whose scaled energies E h ( u h ) /e h are uniformly bounded. Then there exist a sequence Q h ∈ SO (3) and c h ∈ IR such that for the normalized rescaled deformations y h ( z ) = Q h u h ( x + hh t~n ( x )) − c h , z = x + t~n ( x ) ∈ Ω h , the following holds. ( i ) y h converge to π in W , (Ω h , IR ) . ( ii ) The scaled average displacements V h ( x ) = hh √ e h Z h / − h / [ y h ( x + t~n ) − x ] dt onverge to some V ∈ V (Ω , IR ) . ( iii ) lim inf h → E h ( u h ) /e h ≥ I ( V ) , where I ( V ) = 124 Z Ω Q (cid:16) x, ( ∇ ( A~n ) − A ∇ ~n ) tan (cid:17) dg, (1.7) where A is given in (1 . . The above result proves the lower bound for the Γ-convergence. We now state theupper bound in the Γ-convergence result for a smooth noncharacteristic region.Since Ω is approximately robust (Theorem 1.4), Theorem 1.6 below follows from [7,Theorem 2.3] immediately.
Theorem 1.6
Let Ω ⊂ M be a noncharacteristic region of class C , . Assume that (1 . holds for β = 4 . Then for every V ∈ V (Ω , IR ) there exists a sequence of deformations { u h } ⊂ W , (Ω , IR ) such that ( i ) and ( ii ) of Theorem . hold. Moreover, lim h → e h E h ( u h ) = I ( V ) , (1.8) where I ( V ) is given in (1 . . Theorem 1.7
Let assumption (1 . hold with < β < . Let Ω ⊂ M be a noncharac-teristic region of class C m +1 , , where m ≥ is given such that e h = o ( h β m ) . Then the results in Theorem . hold. The rest of the paper is organized as follows.Section 2 reduces the linear strain equations (1.1) into one scalar second order equation(2.25) (Theorem 2.1).Sections 3 makes preparations to solve problem (2.25). The main observation is thatunder an asymptotic coordinate system, this equation locally takes a normal form (Propo-sition 4.1). Thus we studies solvability regions for the normal equation, in where existence,uniqueness and estimates for solutions are presented.Section 4 is devoted to solvability of the scalar equation (2.25). Using solvability of anormal equation in Section 3, we first solve the scalar equation (2.25) locally and then paththe local solutions together (Theorems 4.1-4.4), where the noncharacteristic assumptionof the region Ω is needed to guarantee this process to be successful.Section 5 returns to the main theorems in Section 1, and provides proofs for them,using the main results in Section 4. 7
Linear Strain Equations
We reformulate some expressions from [4, Section 9.2] to reduce (1.1) to a coordinatefree, scalar equation which can be solved by selecting special charts.Let k ≥ T ∈ T k ( M ) be a k th order tensor field and let X ∈ T ( M )be a vector field. We define a k − X T ( X , · · · , X k − ) = T ( X, X , · · · , X k − ) for X , · · · , X k − ∈ T ( M ) , which is called an inner product of T with X. For any T ∈ T ( M ) and α ∈ T x M, tr g i α DT is a linear functional on T x M, where tr g i α DT is the trace of the 2-order tensor fieldi α DT in the induced metric g. Thus there is a vector, denoted by Λ( T ) , such that h Λ( T ) , α i = tr g i α DT for α ∈ T x M, x ∈ M. (2.1)Clearly, the above formula defines a vector field Λ( T ) ∈ T ( M ) . We also need another linear operator Q as follows. Let M be oriented and E be thevolume element of M with the positive orientation. Let x ∈ M be given and let e , e bean orthonormal basis of T x M with the positive orientation, that is, E ( e , e ) = 1 at x. We define Q : T x M → T x M by Qα = h α, e i e − h α, e i e for all α ∈ T x M. (2.2) Q is well defined in the following sense: Let ˆ e , ˆ e be a different orthonormal basis of T x M with the positive orientation, E (ˆ e , ˆ e ) = 1 . Let ˆ e i = X j =1 α ij e j for i = 1 , . Then 1 = E (ˆ e , ˆ e ) = α α − α α . Using the above formula, a simple computation yields h α, ˆ e i ˆ e − h α, ˆ e i ˆ e = h α, e i e − h α, e i e . Clearly, Q : T x M → T x M is an isometry and Q T = − Q, Q = − Id . emark 2.1 Q : T x M → T x M is the rotation by π/ along the clockwise direction. The operator, defined above, defines an operator, still denoted by Q : T ( M ) → T ( M ) , by( QX )( x ) = QX ( x ) , x ∈ M, X ∈ T ( M ) . For each k ≥ , the operator Q further induces an operator, denoted by Q ∗ : T k ( M ) → T k ( M ) by( Q ∗ T )( X , · · · , X k ) = T ( QX , · · · , QX k ) , X , · · · , X k ∈ T ( M ) , T ∈ T k ( M ) . (2.3)Notice that orientability of M is necessary to operators Q or Q ∗ . Let x ∈ Ω be given and let y ∈ W , (Ω , IR ) . Set p ( y )( x ) = 12 [ ∇ y ( e , e ) − ∇ y ( e , e )] for x ∈ Ω , (2.4)where ∇ y ( α, β ) = h∇ β y, α i for α, β ∈ T x M, ∇ is the differential in the Euclidean space IR , and e , e is an orthonormal basis of T x M with the positive orientation. It is easy tocheck that the value of the right hand side of (2.4) is independent of choice of a positivelyorientated orthonormal basis. Thus p : W , (Ω , IR ) → L (Ω)is a linear operator.For U ∈ T ( M ) given, consider problemsym ∇ y ( α, β ) = U ( α, β ) for α, β ∈ T x M, x ∈ M, (2.5)where y ∈ W , (Ω , IR ) . Let x ∈ Ω be given. To simplify computation we use many times the following specialframe field: Let E , E be a positively orientated frame field normal at x with followingproperties h E i , E j i = δ ij in some neighbourhood of x,D E i E j = 0 , ∇ E i ~n = λ i E i at x for 1 ≤ i, j ≤ , (2.6)where ∇ is the connection of the Euclidean space IR , D is the connection of M in theinduced metric, ~n is the normal field of M, and λ λ = κ is the Gaussian curvature. Itfollows from (2.6) thatΠ( E i , E j ) = λ i δ ij , ∇ E i E j = − λ i δ ij ~n at x for 1 ≤ i, j ≤ , (2.7)where Π( α, β ) = h∇ α ~n, β i is the second fundamental form of M. We need to deal with therelation between the connections ∇ and D, carefully.9et y ∈ W , (Ω , IR ) be a solution to problem (2.5). Then (2.5) reads ∇ y ( E , E ) = U ( E , E ) , ∇ y ( E , E ) + ∇ y ( E , E ) = 2 U ( E , E ) , ∇ y ( E , E ) = U ( E , E ) , in some neighbourhood of x. (2.8)Let v = p ( y )and define u = ∇ y ( ~n, E ) E + ∇ y ( ~n, E ) E . (2.9)We can check easily that u is a globally defined vector field on Ω . Moreover, v satisfies v + U ( E , E ) = ∇ y ( E , E ) , v − U ( E , E ) = −∇ y ( E , E ) (2.10)in some neighbourhood of x. Therefore, { v, u } determines ∇ α y for α ∈ T x M, that is, ( ∇ E y = U ( E E ) E + [ v + U ( E , E )] E + h u, E i ~n, ∇ E y = [ − v + U ( E , E )] E + U ( E , E ) E + h u, E i ~n. (2.11)The relation (2.11) can be rewritten as in a form of coordinate free ∇ α y = i α U − vQα + h u, α i ~n for α ∈ T x M, x ∈ Ω . The function v and the vector field u are the new dependent variables and we proceed tofind the differential equations they satisfy.Differentiating the first equation in (2.10) with respect to E and using the relations(2.6) and (2.7), we have E ( v ) + DU ( E , E , E ) = ∇ y ( E , E , E ) + ∇ y ( ∇ E E , E )= E [ ∇ y ( E , E )] − λ ∇ y ( ~n, E )= DU ( E , E , E ) − λ h u, E i at x, (2.12)where the following formula has been used ∇ y ( E , E , E ) = ∇ y ( E , E , E ) at x. Similarly, we obtain E ( v ) − DU ( E , E , E ) = − DU ( E , E , E ) + λ h u, E i at x. (2.13)10ombining (2.12), (2.13) and (2.6) yields Dv = [ DU ( E , E , E ) − DU ( E , E , E )] E + λ h u, E i E +[ DU ( E , E , E ) − DU ( E , E , E )] E − λ h u, E i E = Q { [ DU ( E , E , E ) + DU ( E , E , E )] E − [ DU ( E , E , E ) + DU ( E , E , E )] E +[ DU ( E , E , E ) + DU ( E , E , E )] E − [ DU ( E , E , E ) + DU ( E , E , E )] E } + ∇ ~nQu = Q [Λ( U ) − D ( tr g U )] + ∇ ~nQu for x ∈ Ω , (2.14)where the operator Q : T x M → T x M is defined in (2.2), Λ( U ) ∈ X (Ω) is given in (2.1),and ∇ ~n : T x M → T x M is the shape operator, defined by ∇ ~nα = ∇ α ~n for α ∈ T x M, x ∈ M. Now we proceed to derive the differential equations for which the function v satisfies.Since κ = Π( E , E )Π( E , E ) − Π ( E , E ) in a neighbourhood of x, from (2.6) and (2.7) we compute Dκ = [ D Π( E , E , E ) λ + λ D Π( E , E , E )] E +[ D Π( E , E , E ) λ + λ D Π( E , E , E )] E at x. (2.15)Using (2.14), (2.6) and (2.7), we have D ( ∇ ~nQu )( E , E ) = E h∇ ~nQu, E i = E h u, Q T ∇ E ~n i = Du ( Q T ∇ E ~n, E ) + h u, D E ( Q T ∇ E ~n ) i = λ Du ( E , E ) + D Π( E , E , E ) h u, E i − D Π( E , E , E ) h u, E i at x, (2.16)where the symmetry of D Π is used. A similar computation yields D ( ∇ ~nQu )( E , E ) = − λ Du ( E , E )+ D Π( E , E , E ) h u, E i − D Π( E , E , E ) h u, E i at x. (2.17)Multiplying (2.16) by λ and (2.17) by λ , respectively, summing them, and using (2.15),we obtain h D ( ∇ ~nQu ) , Q ∗ Π i = κ [ Du ( E , E ) − Du ( E , E )] + h Qu, Dκ i . (2.18)Note that the function Du ( E , E ) − Du ( E , E ) is globally defined on Ω which is inde-pendent of choice of a positively orientated orthonormal basis when the vector field u isgiven. From (2.14) and (2.18), we obtain h D v, Q ∗ Π i = h D { Q [Λ( U ) − D ( tr g U )] } , Q ∗ Π i + κ [ Du ( E , E ) − Du ( E , E )] + h Qu, Dκ i . y to satisfy (2.11)exists when the function v and the vector field u are given to satisfy (2.14). We define B : T x M → T x M for x ∈ Ω by Bα = i α U − vQα + h u, α i ~n for α ∈ T x M. (2.19)It is easy to check that there is a y : Ω → IR such that ∇ α y = Bα for α ∈ T x M, x ∈ Ωif and only if the operator B satisfies ∇ X ( BY ) = ∇ Y ( BX ) + B [ X, Y ] for
X, Y ∈ X (Ω) . (2.20)Using (2.6), (2.7), (2.13), and (2.19), we have ∇ E ( BE ) = [ DU ( E , E , E ) − E ( v ) + λ h u, E i ] E + DU ( E , E , E ) E +[ Du ( E , E ) − λ U ( E , E ) + vλ ] ~n = DU ( E , E , E ) E + DU ( E , E , E ) E +[ Du ( E , E ) − λ U ( E , E ) + vλ ] ~n at x. (2.21)Similarly, we obtain ∇ E ( BE ) = DU ( E , E , E ) E + DU ( E , E , E ) E +[ Du ( E , E ) − λ U ( E , E ) − vλ ] ~n at x. (2.22)It follows from (2.21) and (2.22) that the relation (2.20) holds if and only if Du ( E , E ) − Du ( E , E ) + tr g U ( Q ∇ ~n · , · ) + v tr g Π = 0 for x ∈ Ω . Moreover, we assume that κ ( x ) = 0 for all x ∈ Ω . (2.23)From (2.14), we obtain u = Q ( ∇ ~n ) − Q [Λ( U ) − D ( tr g U )] − Q ( ∇ ~n ) − Dv for x ∈ Ω . (2.24)The above derivation yields the following. Theorem 2.1 ([ ]) Suppose that (2 . holds. Let v be a solution to problem h D v, Q ∗ Π i = P ( U ) − vκ tr g Π + X ( v ) for x ∈ Ω , (2.25)12 here P ( U ) = h D { Q [Λ( U ) − D ( tr g U )] } , Q ∗ Π i − h Q [Λ( U ) − D ( tr g U )] , ( ∇ ~n ) − Dκ i− κ tr g U ( Q ∇ ~n · , · ) , (2.26) X = ( ∇ ~n ) − Dκ. (2.27)
Let u be given by (2 . . Then there is a y to satisfy (2 . such that (2 . holds. Moreover, |∇ y | ( x ) = | U | ( x ) + 2 v ( x ) + | u ( x ) | for x ∈ Ω . If, in addition, y = W + w~n, w = h y, ~n i , then u = Dw − i W Π ,Dw = i W Π − Q ( ∇ ~n ) − Dv + Q ( ∇ ~n ) − Q [Λ( U ) − D ( tr g U )] . Remark 2.2
A solution y, modulo a constant vector, in Theorem . is unique whena solution v to (2 . is given. Remark 2.3 If Ω is elliptic and Π > , then ˆ g = Π is another metric on Ω . From [ ] we have h D v, Q ∗ Π i = κ ∆ ˆ g v + 12 κ Π( QDκ, QDv ) for x ∈ Ω , where ∆ ˆ g is the Laplacion of the metric ˆ g. Thus, in this case equation (2 . becomes ∆ g v = 1 κ P ( U ) − v tr g Π + 12 κ X ( v ) for x ∈ Ω . Under an asymptotic coordinate system, equation (2.25) on a hyperbolic surface takesthe form of a normal equation in IR locally, such as in (3.1) below. Thus the localsolvability of equation (2.25) transfers to that of equation (3.1) in the Euclidean space IR . We study the solvability of the normal equation (3.1) in the space IR in this section.We consider the following normal equation w x x ( x ) = η ( f, w ) for x = ( x , x ) ∈ IR (3.1)where η ( f, w ) = f + f ( x ) w ( x ) + X ( w ) ,f is a function, and X = ( X , X ) is a vector field on IR . Regions E ( γ ) , R ( z, a, b ) , P i ( β ) , Ξ i ( β, γ ) and Φ( β, γ, ˆ β ) In appropriate asymptotic co-ordinate systems, the problem (2.25) can transfer to solvability of (3.1) on E ( γ ) , R ( z, a, b ) ,P i ( β ) , Ξ i ( β, γ ) or Φ( β, γ, ˆ β ) with appropriate boundary data. We now introduce those re-gions to establish the corresponding solvability.13 .1 Regions E ( γ ) and R ( z, a, b ) Let k ≥ f and X be of class C k − , , where C − , = L ∞ . A curve γ ( t ) = ( γ ( t ) , γ ( t )) : [ a, b ] → IR is said to be noncharacteristic if γ ′ ( t ) γ ′ ( t ) = 0 for t ∈ [ a, b ] . We define a linear operator F : IR → IR by F x = ( x , − x ) for x = ( x , x ) ∈ IR . (3.2)Let γ ( t ) = ( γ ( t ) , γ ( t )) : [0 , t ] → IR be a noncharacteristic curve with γ ′ (0) γ ′ (0) < . We assume that γ ′ ( t ) > , γ ′ ( t ) < t ∈ [0 , t ] . (3.3)Otherwise, we consider the curve z ( t ) = γ ( − t + t ). Set E ( γ ) = { ( x , x ) ∈ IR | γ ◦ γ − ( x ) < x < γ ( t ) , γ ( t ) < x < γ (0) } . (3.4)Consider the boundary data w ◦ γ ( t ) = q ( t ) , h∇ w, F ˙ γ i ◦ γ ( t ) = q ( t ) for t ∈ (0 , t ) . (3.5)Next, we consider a rectangle. For z = ( z , z ) ∈ IR , a > , and b > R ( z, a, b ) = ( z , z + a ) × ( z , z + b ) . (3.6)Consider the boundary data w ( x , z ) = p ( x ) , w ( z , x ) = p ( x ) (3.7)for x ∈ [ z , z + a ] and x ∈ [ z , z + b ] , respectively.Let f be a function with its domain E. For simplicity, we denote k f k C k, = k f k C k, ( E ) , k f k W k, = k f k W k, ( E ) , and so on. Proposition 3.1
Let q be of C k, and q , f be of C k − , , respectively. Then problem (3 . admits a unique solution w ∈ C k, ( E ( γ )) with the data (3 . . Moreover, there is a
C > , independent of solutions w, such that k w k C k, ≤ C ( k q k C k, + k q k C k − , + k f k C k − , ) . (3.8) Proposition 3.2
Let p and p be of class C k, with p ( z ) = p ( z ) . Let f be of class C k − , . Then there is a unique solution w ∈ C k, ( R ( z, a, b )) to (3 . with the data (3 . satisfying k w k C k, ≤ C ( k p k C k, + k p k C k, + k f k C k − , ) . Lemma 3.1
Let
T > be given. There is a ε T > such that if | γ (0) | ≤ T and max { γ ( t ) − γ (0) , γ (0) − γ ( t ) } < ε T , Theorem . holds true. Proof.
The proof is broken into several steps as follows.
Step 1.
Let k = 0 and let w ∈ C , ( E ( γ )) be a solution to (3.1) with the data (3.5).It follows from (3.5) that w x ◦ γ ( t ) = 1 | γ ′ ( t ) | [ γ ′ ( t ) q ′ ( t ) + γ ′ ( t ) q ( t )] , w x ◦ γ ( t ) = 1 | γ ′ ( t ) | [ γ ′ ( t ) q ′ ( t ) − γ ′ ( t ) q ( t )] . Let x = ( x , x ) ∈ E ( γ ) be given. We integrate (3.1) with respect to the first variable ζ over ( γ ◦ γ − ( ζ ) , x ) for ζ ∈ ( γ ◦ γ − ( x ) , x ) to have w x ( x , ζ ) = w x ◦ γ ( γ − ( ζ )) + Z x γ ◦ γ − ( ζ ) η ( f, w )( ζ , ζ ) dζ . (3.9)Then integrating the above identity over ( γ ◦ γ − ( x ) , x ) with respect to the secondvariable ζ yields w ( x , x ) = B ( q , q ) + Z E ( x ) η ( f, w ) dζ, where B ( q , q ) = q ◦ γ − ( x ) + Z γ − ( x ) γ − ( x ) γ ′ ( t ) | γ ′ ( t ) | [ γ ′ ( t ) q ′ ( t ) − γ ′ ( t ) q ( t )] dt, (3.10) E ( x ) = { ( ζ , ζ ) | γ ◦ γ ( ζ ) < ζ < x , γ ◦ γ − ( x ) < ζ < x } . (3.11) Step 2.
We define an operator I : C , ( E ( γ )) → C , ( E ( γ )) by I ( w ) = B ( q , q ) + Z E ( x ) η ( f, w ) dζ for w ∈ C , ( E ( γ )) . (3.12)It is easy to check that w ∈ C , ( E ( γ )) solves (3.1) with the data (3.5) if and only if I ( w ) = w. Next, we show that there is a 0 < ε T ≤ | γ (0) | ≤ T and 0 < max { γ ( t ) − γ (0) , γ (0) − γ ( t ) } < ε T , the map I : C , ( E ( γ )) → C , ( E ( γ )) is con-tractible. Thus the existence and uniqueness of solutions in the case k = 0 follows fromBanach’s fixed point theorem.A simple computation shows that for w ∈ C , ( E ( γ ))[ I ( w )] x = 1 | γ ′ ( t ) | [ γ ′ ( t ) q ′ ( t ) + γ ′ ( t ) q ( t )] (cid:12)(cid:12)(cid:12) t = γ − ( x ) + Z x γ ◦ γ − ( x ) η ( f, w )( x , ζ ) dζ , [ I ( w )] x = 1 | γ ′ ( t ) | [ γ ′ ( t ) q ′ ( t ) − γ ′ ( t ) q ( t )] (cid:12)(cid:12)(cid:12) t = γ − ( x ) + Z x γ ◦ γ − ( x ) η ( f, w )( ζ , x ) dζ . w , w ∈ C , ( E ( γ )) , k I ( w ) − I ( w ) k C , ( E ( γ )) ≤ C T max { λ, λ }k w − w k C , ( E ( γ )) , where λ = max { γ ( t ) − γ (0) , γ ( t ) − γ (0) } , C T = k f k L ∞ ( | x |≤ T ) + k X k L ∞ ( | x |≤ T ) . Thus, the map I : C , ( E ( γ )) → C , ( E ( γ )) is contractible if λ > Step 3.
Consider the case k = 1 . Let q ∈ C , [0 , t ] , q ∈ C , [0 , t ] , and f ∈ C , ( E ( γ )) be given. By Step 2, there is a ε T > | γ (0) | ≤ T and 0 < λ < ε T , problem (3 .
1) has a unique solution w ∈ C , ( E ( γ )) with the data (3.5). A formalcomputation shows that u = w x solves problem u x x = η ( ˆ f , u ) for x ∈ E ( γ ) , (3.13)with the data u ◦ γ ( t ) = ˆ q ( t ) , h∇ u, F ˙ γ i ◦ γ ( t ) = ˆ q ( t ) for t ∈ (0 , t ) , (3.14)where ˆ f = f x + f x w + ∇ ∂x X ( w ) , ˆ q ( t ) = 1 | γ ′ ( t ) | [ γ ′ ( t ) q ′ ( t ) + γ ′ ( t ) q ( t )] , ˆ q ( t ) = γ ′ ( t ) γ ′ ( t ) ˆ q ′ ( t ) − | γ ′ ( t ) | γ ′ ( t ) η ( f, w ) ◦ γ ( t ) . We apply Step 2 to problem (3.13) and (3.14) to obtain u = w x ∈ C , ( E ( γ )) when0 < λ < ε T . A similar argument yields w x ∈ C , ( E ( γ )) . Thus w ∈ C , ( E ( γ )) . By repeating the above procedure, the existence and uniqueness of the solutions in thecases k ≥ Step 4.
Let map I : C k, ( E ( γ )) → C k, ( E ( γ )) be defined in Step 2 and let w ∈ C k, ( E ( γ )) be the solution to problem (3 .
1) with the data (3.5). Then k w k C k, = k I ( w ) k C k, ≤ k I (0) k C k, + k I ( w ) − I (0) k C k, ≤ C ( k q k C k, + k q k C k − , + k f k C k − , ) + C T max { λ, λ }k w k C k, . Thus, the estimate (3.8) follows if λ > ✷ By a similar argument as for Lemma 3.1, we have the following lemmas.
Lemma 3.2
Let
T > . There is ε T > such that if | z | ≤ T and < max { a, b } < ε T , then Proposition . holds. roof of Proposition 3.1. We shall show that the assumptions | γ (0) | ≤ T andmax { γ ( t ) − γ (0) , γ ( t ) − γ (0) } < ε T in Lemma 3.1 are unnecessary. Let T > E ( γ ) ⊂ { x ∈ IR | | x | ≤ T } . Let ε T > γ into m parts with the points τ = 0 , τ < τ < · · · < τ m = t such that | γ ( τ i +1 ) − γ ( τ i ) | = ε T , ≤ i ≤ m − , | γ ( t ) − γ ( τ m − ) | ≤ ε T . For simplicity, we assume that m = 3 . The other cases can be treated by a similar argu-ment.In the case of m = 3 , we have E ( γ ) = ( ∪ i =0 E i ) ∪ ( ∪ i =1 R i ) (3.15)where E i = { x ∈ E ( γ ) | γ ( τ i ) ≤ x ≤ γ ( τ i +1 ) , γ ( τ i +1 ) ≤ x ≤ γ ( τ i ) } i = 0 , , ,R = [ γ ( τ ) , γ ( τ )] × [ γ ( τ ) , γ (0)] , R = [ γ ( τ ) , γ ( t )] × [ γ ( τ ) , γ ( τ )] ,R = [ γ ( τ ) , γ ( t )] × [ γ ( τ ) , γ (0)] . From Lemma 3.1, problem (3.1) admits a unique solution w i ∈ C k, ( E i ) for each i = 0 , , and 2 , respectively, with the corresponding data and the corresponding estimates. Wedefine w ∈ C k, ( ∪ i =0 E i ) by w ( x ) = w i ( x ) for x ∈ E i for i = 0 , , . We extend the domain of w from ∪ i =0 E i to E ( γ ) by the following way. By Lemma 3.2,we define w ∈ C k, ( R i ) to be the solution u i ∈ C k, ( R i ) to problem (3.1) with the data u i ( γ ( τ i ) , x ) = w i − ( γ ( τ i ) , x ) for x ∈ [ γ ( τ i ) , γ ( τ i − )] ,u i ( x , γ ( τ i )) = w i ( x , γ ( τ i )) for x ∈ [ γ ( τ i ) , γ ( τ i +1 )] , for i = 1 , and 2 , respectively. Then we extend w on C k, ( R ) to be the solution u of(3.1) with the data u ( γ ( τ ) , x ) = u ( γ ( τ ) , x ) for x ∈ [ γ ( τ ) , γ (0)] ,u ( x , γ ( τ )) = u ( x , γ ( τ )) for x ∈ [ γ ( τ ) , γ ( t )] . To complete the proof, we have to show that w is a C k, solution on all the connectionsegments between any two subregions above. Consider the subregion˜ E = E ∪ E ∪ R . | γ ( τ ) − γ (0) | ≤ ε T , Lemma 3.1 insures that problem (3.1) admits a unique solution˜ w ∈ C k, ( ˜ E ) with the corresponding data. Then the uniqueness implies that w ( x ) = ˜ w ( x )for x ∈ ˜ E. In particular, w is C k, on the segments { ( γ ( τ ) , x ) | x ∈ [ γ ( τ ) , γ (0)] } and { ( x , γ ( τ )) | x ∈ [ γ ( τ ) , γ ( τ )] } , respectively. By a similar argument, we show that w is also C k, on all the other segments.The estimates in (3.8) follow from the ones in Lemmas 3.1 and 3.2. ✷ Proof of Proposition 3.2.
We divided R ( z, a, b ) into a sum of small rectangles andapply Lemma 3.2 to paste the solutions together. ✷ To have density results in Theorem 1.2, we also need estimates of some (boundary)traces of the solutions. For σ ∈ (0 , t ) , let β σ ( t ) = γ ( σ ) − t F ˙ γ ( σ ) for t ∈ (0 , t σ ) , where t σ > β σ ( t σ ) ∈ ∂E ( γ ) . Proposition 3.3
Let f and X be of class C , . Let q be of class W , and q , f beof class W , , respectively. Then problem (3 . admits a unique solution w ∈ W , withthe data (3 . . Moreover, there is a
C > , independent of solutions w, such that k w k , + k w x ◦ β σ k , ≤ C ( k q k , + k q k , + k f k W , ) , (3.16) where W i, = W i, ( E ( γ )) for ≤ i ≤ . Proof
A similar argument as for Theorem 3.1 shows that a unique solution w ∈ W , ( E ( γ )) with the data (3.5) exists, and the estimate k w k , ≤ C ( k q k , + k q k , + k f k W , ) (3.17)holds.Let β σ ( t ) = ( β σ ( t ) , β σ ( t )) . Using equation (3.1), we have w x x ◦ β σ ( t ) = w x x ◦ γ ◦ γ − ◦ β σ ( t ) + Z β σ ( t ) γ ◦ γ − ◦ β σ ( t ) [ η ( f, w )] x ( ζ , β σ ( t )) dζ , which yields | w x x ◦ β σ ( t ) | ≤ | w x x ◦ γ ◦ γ − ◦ β σ ( t ) | +2[ γ ( t ) − γ (0)] C Z β σ ( t ) γ ◦ γ − ◦ β σ ( t ) ( | f | + |∇ f | + | w | + |∇ w | + |∇ w | )( ζ , β σ ( t )) dζ . Integrating the above inequality over (0 , t σ ) with respect to t, we obtain k w x x ◦ β σ k L ≤ C ( k f k , + k q k , + k q k , + k w k , ) .
18 similar computation shows that k∇ w x ◦ β σ k L , k∇ w ◦ β σ k L , and k w ◦ β k L can bebounded also by the right hand side of the above inequality. Thus estimate (3.16) followsfrom (3.17). ✷ LetΓ( γ, w ) = X j =0 k∇ j w ◦ γ k L (0 ,t ) + Z t [ | w x x ◦ γ ( t ) | t + | w x x ◦ γ ( t ) | ( t − t )] dt. (3.18) Proposition 3.4
Let f and X be of class C , . Then there are < c < c such thatfor all solutions w ∈ W , to problem (3 . c Γ( γ, w ) ≤ k f k , + k w k , ≤ c [ k f k , + Γ( γ, w )] , (3.19) k w ( · , γ (0)) k , ( γ (0) ,γ ( t )) + Z γ ( t ) γ (0) | w x x ( x , γ (0)) | ( x − γ (0)) dx ≤ c [ k f k , +Γ( γ, w )] , k w ( γ ( t ) , · ) k , ( γ ( t ) ,γ (0)) + Z γ (0) γ ( t ) | w x x ( γ ( t ) , x ) | ( x − γ ( t )) dx ≤ c [ k f k , +Γ( γ, w )] , where W i, = W i, ( E ( γ )) for ≤ i ≤ . Proposition 3.5
Let f and X be of class C , . Then there is
C > such that forall solutions w ∈ W , ( R ( z, a, b )) to problem (3 . k w k , ≤ C ( k f k , + k p k , ( z ,z + a ) + k p k , ( z ,z + b ) ) . (3.20)The proofs of the above two propositions will complete from Lemmas 3.3 and 3.4 belowby an argument as for Proposition 3.1. We omit the details. Lemma 3.3
Let
T > be given. There is ε T > such that if | γ (0) | ≤ T and max { γ ( t ) − γ (0) , γ (0) − γ ( t ) } < ε T , Proposition . holds. Proof Step 1
Using (3.1) we have w x x ( x ) = w x x ◦ γ ◦ γ − ( x ) + Z x γ ◦ γ − ( x ) [ η ( f, w )] x ( x , ζ ) dζ , which yields | w x x ( x ) | ≤ | w x x ◦ γ ◦ γ − ( x ) | +2[ x − γ ◦ γ − ( x )] Z γ (0) γ ◦ γ − ( x ) | [ η ( f, w )] x ( x , ζ ) | dζ , and | w x x ◦ γ ◦ γ − ( x ) | ≤ | w x x ( x ) | +2[ x − γ ◦ γ − ( x )] Z γ (0) γ ◦ γ − ( x ) | [ η ( f, w )] x ( x , ζ ) | dζ , x over ( γ ◦ γ − ( x ) , γ (0)) and then with respect to x over ( γ (0) , γ ( t )) respectively, we obtain k w x x k L ≤ σ Z t | w x x ◦ γ ( t ) | tdt + ε T C T ( k f k , + k w k , )and σ Z t | w x x ◦ γ ( t ) | tdt ≤ k w x x k L + ε T C T ( k f k , + k w k , ) , where σ = inf t ∈ (0 ,t ) [ γ (0) − γ ( t )] γ ′ ( t ) /t, σ = sup t ∈ (0 ,t ) [ γ (0) − γ ( t )] γ ′ ( t ) /t,C T = sup | x |≤ T (1 + f + |∇ f | + | X | + |∇ X | ) . By similar arguments, we establish the following k w x x k L ≤ σ Z t | w x x ◦ γ ( t ) | ( t − t ) dt + ε T C T ( k f k , + k w k , ) ,σ Z t | w x x ◦ γ ( t ) | ( t − t ) dt ≤ k w x x k L + ε T C T ( k f k , + k w k , ) , where σ = inf t ∈ (0 ,t ) [ γ ( t ) − γ ( t )][ − γ ′ ( t )] / ( t − t ) , σ = sup t ∈ (0 ,t ) [ γ ( t ) − γ ( t )][ − γ ′ ( t )] / ( t − t ) . Step 2
As in Step 1, we have k w x k L ≤ σ Z t | w x ◦ γ ( t ) | tdt + ε T C T ( k f k L + k w k , ) ≤ σ t k w x ◦ γ k L (0 ,t ) + ε T C T ( k f k L + k w k , ) ,σ Z t | w x ◦ γ ( t ) | tdt ≤ k w x k L + ε T C T ( k f k L + k w k , ) . (3.21)In addition, since w x ◦ γ ◦ γ − ( x ) = w x ( x ) − Z x γ ◦ γ − ( x ) w x x ( ζ , x ) dζ for x ∈ ( γ ( t ) , γ (0)) , it follows that σ Z t | w x ◦ γ ( t ) | ( t − t ) dt ≤ k w x k L + ε T k w x x k L . (3.22)Combing (3.21) and (3.22), we havemin { σ , σ }k w x ◦ γ k L (0 ,t ) ≤ t ( σ Z t / | w x ◦ γ ( t ) | ( t − t ) dt + σ Z t t / | w x ◦ γ ( t ) | tdt ) ≤ t [4 + ε T ( C T + 1)]( k f k L + k w k , ) .
20y a similar computation, we obtain k w x k L ≤ σ t k w x ◦ γ k L (0 ,t ) + C T ε T ( k f k L + k w k , ) , min { σ , σ }k w x ◦ γ k L (0 ,t ) ≤ t [4 + ε T ( C T + 1)]( k f k L + k w k , ) , k w k L ≤ σ t k w ◦ γ k L (0 ,a ) + ε T k w x k L , min { σ , σ }k w ◦ γ k L (0 ,t ) ≤ t [4 + ε T ( C T + 1)] k w k , . Step 3
From Steps 1 and 2, we obtain[1 − (4 C T + 1) ε T ] k w k , ≤ σ (1 + t ) + σ ]Γ( γ, w ) + (4 C T + 1) ε T k f k W , , when λ is small, andmin { σ , σ } Γ( γ, w ) ≤ { C T ε T + 3 t [4 + ( C T + 1) ε T ] } ( k f k , + k w k , ) , respectively. Thus (3.19) follows. Step 4
We have w x x ( x , γ (0)) = w x x ◦ γ ◦ γ − ( x ) + Z γ (0) γ ◦ γ − ( x ) [ η ( f, w )] x ( x , ζ ) dζ , which gives, by (3.19), Z γ ( t ) γ (0) | w x x ( x , γ (0)) | ( x − γ (0)) dx ≤ Z t | w x x ◦ γ ( t ) | [ γ ( t ) − γ (0)] dt + C ( k f k , + k w k , ) ≤ C [ k f k , + Γ( γ, w )] . A similar argument completes the proof of the third inequality in Proposition 3.4. ✷ A similar argument yields the following.
Lemma 3.4
Let
T > be given. There is ε T > such that if | z | ≤ T and < max { a, b } < ε T , then Proposition . holds. P i ( β ) Let β = ( β , β ) : [0 , t ] → IR be a noncharcteristic curve with β ′ (0) β ′ (0) > . Weassume β ′ i ( t ) > t ∈ [0 , t ] , i = 1 , . (3.23)Otherwise, we consider the curve z ( t ) = β ( − t + t ) . Set P ( β ) = { ( x , x ) | β ◦ β − ( x ) < x < β ( t ) , β (0) < x < β ( t ) } , (3.24)21nd consider the boundary data w x ◦ β ( t ) = p ( t ) , t ∈ (0 , t ); w ( x , β (0)) = p ( x ) , x ∈ ( β (0) , β ( t )) . (3.25)Set P ( β ) = { ( x , x ) | β (0) < x < β ◦ β − ( x ) , β (0) < x < β ( t ) } , (3.26)and consider the boundary data w x ◦ β ( t ) = p ( t ) , t ∈ (0 , t ); w ( β (0) , x ) = p ( x ) , x ∈ ( β (0) , β ( t )) . (3.27)By similar arguments for the region E ( γ ) , we establish Propositions 3.6-3.8 below. Thedetails are omitted. Proposition 3.6
Let the curve β be of class C k − , . Let p ( or p ) be of class C k, and let p, f be of class C k − , . Then problem (3 . admits a unique solution w ∈ C k, ( P ( β )) ( or C k, ( P ( β )) with the data (3 . ( or (3.27)) to satisfy k w k C k, ≤ C ( k p k C k − , + k p k C k, + k f k C k − , )( or ( k p k C k − , + k p k C k, + k f k C k − , )) . Proposition 3.7
Let the curve β be of class C . Let f and X be of class C , . Let p ( or p ) be of class W , and let p, f be of class W , . Then problem (3 . admitsa unique solution w ∈ W , ( P ( β )) ( or W , ( P ( β )) with the data (3 .
25) ( or (3 . tosatisfy k w k W , ≤ C ( k p k W , + k p k W , + k f k W , )( or ( k p k W , + k p k W , + k f k W , )) . LetΓ( P i , w ) = Z t | p ′ ( t ) | ( t − t ) dt + k p i k , + Z β i ( t ) β i (0) | p ′′ i ( x i ) | ( x i − z i )] dx i , i = 1 , . Proposition 3.8
Let the curve β be of class C . Let f and X be of class C , . Thenthere are < c < c such that for all solutions w ∈ W , ( P i ( β )) to problem (3 . withthe corresponding boundary data satisfy c Γ( P i , w ) ≤ k w k , + k f k , ≤ c [Γ( P i , w ) + k f k , ] ,c k w | x i = β i ( t ) k , ≤ Z t | p ′ ( t ) | dt + Z β i ( t ) β i (0) | p ′′ i ( x ) | ( x i − β i (0))] dx i + k p i k , + k f k , ≤ c ( k w | x i = β i ( t ) k , + Z β i ( t ) β i (0) | p ′′ i ( x i ) | ( x i − β i (0))] dx i + k p i k , + k f k , ) , (3.28) for i = 1 , and , respectively. Remark 3.1 (3 . implies that p ∈ W , if and only if w | x i = β i ( t ) ∈ W , . However,the case of p / ∈ W , may happen. .3 Regions Ξ i ( β, γ ) Let γ : [0 , t ] → IR and β : [0 , t ] → IR be two noncharacterstic curves with γ (0) = β (0) such that γ ( t ) ≤ β ( t ) , γ ′ ( t ) > , γ ′ ( t ) < , β ′ ( t ) > , β ′ ( t ) > ( β, γ ) = P ( β ) ∪ R ( z, a, b ) ∪ E ( γ ) , (3.29)where P ( β ) , R ( z, a, b ) , and E ( γ ) are given in (3.24), (3.6), and (3.4), respectively, with z = ( β ( t ) , γ (0)) , a = γ ( t ) − β ( t ) , and b = β ( t ) − γ ( t ) . Consider the boundarydata w x ◦ β ( t ) = p ( t ) for t ∈ [0 , t ] , (3.30) w ◦ γ ( t ) = q ( t ) , h∇ w, F ˙ γ i ◦ γ ( t ) = q ( t ) for t ∈ (0 , t ) , (3.31)where F is given by (3.2).Let γ : [0 , t ] → IR and β : [0 , t ] → IR be two noncharacterstic curves with γ ( t ) = β (0) such that γ (0) ≥ β ( t ) , γ ′ ( t ) > , γ ′ ( t ) < , β ′ ( t ) > , β ′ ( t ) > ( β, γ ) = E ( γ ) ∪ R ( z, a, b ) ∪ P ( β ) , (3.32)where E ( γ ) , R ( z, a, b ) , and P ( β ) are given in (3.4), (3.6), and (3.26), respectively, with z = ( γ ( t ) , γ ( t )) , a = β ( t ) − γ ( t ) , and b = γ (0) − β ( t ) . Consider the data w x ◦ β ( t ) = p ( t ) for t ∈ [0 , t ] , (3.33) w ◦ γ ( t ) = q ( t ) , h∇ w, F ˙ γ i ◦ γ ( t ) = q ( t ) for t ∈ (0 , t ) , (3.34)where F is given by (3.2).We consider solvability of (3.1) on Ξ ( β, γ ) . To have a C k, solution on Ξ ( β, γ ) , weneed some kind of compatibility conditions at the point γ (0) = β (0) . From Proposition3.1, problem (3.1) admits a unique solution u ∈ C k, ( E ( γ )) with the data (3.31). FromProposition 3.6, there is a unique solution v ∈ C k, ( P ( β )) to problem (3.1) with the data v x ◦ β ( t ) = p ( t ) , t ∈ (0 , t ) , v ( x , β (0)) = u ( x , β (0)) , x ∈ [ β (0) , β ( t )] . (3.35)In terms of the uniqueness, if problem (3.1) has a unique solution w ∈ C k, (Ξ ( β, γ )) withthe data (3.30) and (3.31) together, then w ( x ) = ( v ( x ) for x ∈ P ( β ) ,u ( x ) for x ∈ E ( γ ) . (3.36)23onversely, if we define w by the formula (3.36), then whether it is a C k, solution to (3.1)on Ξ ( β, γ ) depends on the C k, regularity of w at the point β (0) . Thus, compatibilityconditions are something which can guarantee that w is C k, at γ (0) = β (0) , that is ∇ j u ◦ γ (0) = ∇ j v ◦ β (0) for 0 ≤ j ≤ k. (3.37)The solution u with the data (3.31) yields ∇ u ◦ γ ( t ) = 1 | γ ′ ( t ) | ( γ ′ ( t ) q ′ ( t ) + γ ′ ( t ) q ( t ) , γ ′ ( t ) q ′ ( t ) − γ ′ ( t ) q ( t )) (3.38)for t ∈ [0 , t ] . Using (3.1) and (3.38), we have u x x ◦ γ ( t ) = f ◦ γ ( t ) + 1 | γ ′ ( t ) | [ γ ′ ( t ) X ◦ γ ( t ) − γ ′ ( t ) X ◦ γ ( t )] q ( t )+ f ◦ γ ( t ) q ( t ) + 1 | γ ′ ( t ) | [ γ ′ ( t ) X ◦ γ ( t ) + γ ′ ( t ) X ◦ γ ( t )] q ′ ( t ) (3.39)for t ∈ (0 , t ) . Next, differentiating the second component in (3.38) with respect to variable t and using (3.39), we obtain u x x ◦ γ ( t ) = − γ ′ γ ′ f ◦ γ ( t ) − γ ′ γ ′ f ◦ γq − [ 2 h γ ′′ , γ ′ i| γ ′ | + γ ′ | γ ′ | γ ′ ( γ ′ X ◦ γ + γ ′ X ◦ γ ) − γ ′′ | γ ′ | γ ′ ] q ′ + 1 | γ ′ | q ′′ +[ 2 h γ ′′ , γ ′ i γ ′ | γ ′ | γ ′ − γ ′ | γ ′ | γ ′ ( γ ′ X ◦ γ − γ ′ X ◦ γ ) − γ ′′ | γ ′ | γ ′ ] q − γ ′ | γ ′ | γ ′ q ′ . (3.40)By repeating the above procedure, we have shown that, for 1 ≤ j ≤ k − , there are j order tensor fields A αβ ( t ) , A α ( t ) , and A α ( t ) such that ∇ j u x ◦ γ ( t ) = X α + β ≤ j − ∂ αx ∂ βx f ◦ γ ( t ) A αβ ( t ) + X α ≤ j q ( α )1 ( t ) A α ( t )+ X α ≤ j +1 q ( j )0 ( t ) A α ( t ) for t ∈ [0 , t ] . (3.41)Let v ∈ C k, ( P ( β )) be the solution to (3.1) with the data (3.35). Then p ′ ( t ) = h∇ v x ( β ( t )) , ˙ β ( t ) i , p ′′ ( t ) = h∇ v x ( β ( t )) , ˙ β ( t ) ⊗ ˙ β ( t ) i + h∇ v x ( β ( t )) , ¨ β ( t ) i for t ∈ [0 , t ] . Some computations show that p ( l ) ( t ) = h∇ l v x ( β ( t )) , ˙ β ( t ) ⊗ · · · ⊗ ˙ β ( t ) i + X j + ··· + j i = l, ≤ i ≤ l − a j ··· j i h∇ i v x ( β ( t )) , β ( j ) ( t ) ⊗ · · · ⊗ β ( j i ) ( t ) i (3.42)for t ∈ [0 , t ] , and 1 ≤ l ≤ k, where a j ··· j i are positive integers. Then assumption (3.37) isstated as the following. 24 efinition 3.1 Let the curves β and γ be of class C k, . Let q be of class C k, and p, q , f of class C k − , , respectively. It is said that the k th order compatibility conditionshold at γ (0) = β (0) if | γ ′ (0) | p (0) = γ ′ (0) q ′ (0) − γ ′ (0) q (0) and p ( l ) (0) = h∇ l u x ◦ γ (0) , ˙ β (0) ⊗ · · · ⊗ ˙ β (0) i + X j + ··· + j i = l, ≤ i ≤ l − a j ··· j i h∇ i u x ◦ γ (0) , β ( j ) (0) ⊗ · · · ⊗ β ( j i ) (0) i (3.43) for ≤ l ≤ k − , where ∇ i u x ◦ γ (0) and a j ··· j i are given in (3 . and (3 . , respectively. Proposition 3.9
Let the curves β and γ be of class C k, . Let q be of class C k, and p, q , f of class C k − , , respectively. If k ≥ , we assume that the k th order com-patibility conditions hold at γ (0) = β (0) . Then problem (3 . admits a unique solution w ∈ C k, (Ξ ( β, γ )) with the data (3 . and (3 . . Moreover, the following estimateshold k w k C k, ≤ C ( k p k C k − , + k q k C k, + k q k C k − , + k f k C k − , ) . Proof
The uniqueness and the estimate follows from Propositions 3.1, 3.2, and 3.6. Itis remaining to show the existence. Let u and v be given in (3.36) with the correspondingboundary date. Let h be the solution to (3 .
1) on R ( z, a, b ) with the data h ( x , γ (0)) = u ( x , γ (0)) for x ∈ [ β ( t ) , γ ( t )] ,h ( β ( t ) , x ) = v ( β ( t ) , x ) for x ∈ [ γ (0) , β ( t )] , where R ( z, a, b ) is given in (3.29). We now define w ( x ) = v for x ∈ P ( β ) ,u for x ∈ E ( γ ) ,h for x ∈ R ( z, a, b ) . Then w is a solution to (3.1) with the data (3.30) and (3.31). Next we shall show w ∈ C k, (Ξ ( β, γ )) . We proceed by induction in k ≥ . The definition of w guarantees w ∈ C , (Ξ ( β, γ )) . Let w ∈ C k, (Ξ ( β, γ )) . Next we show that the k + 1th order compatibility conditionsimply w ∈ C k +1 , (Ξ ( β, γ )) . For this purpose it is enough to show that w is C k +1 on the segments ϑ = { ( x , β (0)) , ( x , β ( t )) | x ∈ [ β (0) , γ ( t )] , x ∈ [ γ (0) , β ( t )] } . By the induction assumptions, we have ∂ ix ∂ jx v ( x , β (0)) = ∂ ix ∂ jx u ( x , β (0)) for β (0) ≤ x ≤ β ( t ) , (3.44)25or 0 ≤ i + j ≤ k. Next we show that (3.44) are true with i + j = k + 1 . Since v ( x , β (0)) = u ( x , β (0)) for all x ∈ [ β (0) , β ( t )] , it follows that ∂ k +1 x v ( x , β (0)) = ∂ k +1 x u ( x , β (0)) for all x ∈ [ β (0) , β ( t )] . Let i + j = k + 1 with j ≥ . If i ≥ , then j = k + 1 − i ≤ k and, by the inductionassumptions, ∂ jx v ( x , β (0)) = ∂ jx u ( x , β (0)) for all x ∈ [ β (0) , β ( t )] , which yield ∂ ix ∂ jx v ( x , β (0)) = ∂ ix ∂ jx u ( x , β (0)) for all x ∈ [ β (0) , β ( t )] . (3.45)Next we check the case of i = 0 and j = k + 1 . Using (3.1), we have (cid:16) ∂ k +1 x v ( x , β (0)) (cid:17) x = ∂ kx ( v x x )( x , β (0)) = ∂ kx [ f + f v + X v x + X v x ]( x , β (0))= X ( x , β (0)) ∂ k +1 x v ( x , β (0)) + ∂ kx [ f + f v + X v x ]( x , β (0))+[ k X i =1 C ik ∂ ix X ∂ k − i +1 x v ]( x , β (0)) . (3.46)Let ρ ( x ) = ∂ kx [ f + f v + X v x ]( x , β (0)) + [ k X i =1 C ik ∂ ix X ∂ k − i +1 x v ]( x , β (0))for x ∈ [ β (0) , β ( t )] . It follows from (3.46) that τ ( x ) = ∂ k +1 x v ( x , β (0)) is the solutionto problem ( τ ′ ( x ) = X ( x , β (0)) τ ( x ) + ρ ( x ) for x ∈ [ β (0) , β ( t )] ,τ ( β (0)) = ∂ k +1 x v ( β (0)) . (3.47)Moreover, the induction assumptions, w ∈ C k, (Ξ ( β, γ ))) , yield h∇ i v x ( z ) , β ( j ) (0) ⊗ · · · β ( j i ) (0) i = h∇ i u x ( z ) , β ( j ) (0) ⊗ · · · β ( j i ) (0) i for j + · · · + j i = l, ≤ i ≤ l − , and 1 ≤ l ≤ k. Then the k + 1th order compatibilityconditions imply h∇ k v x ◦ β (0) , ˙ β (0) ⊗ · · · ⊗ ˙ β (0) i = h∇ k u x ◦ γ (0) , ˙ β (0) ⊗ · · · ⊗ ˙ β (0) i . Using (3.45) and β ′ (0) > , we obtain ∂ k +1 x u ◦ γ (0) = ∂ k +1 x v ◦ β (0) . (3.48)26n addition, it follows from the induction assumptions and (3.45) that ρ ( x ) = ∂ kx [ f + f u + X u x ]( x , β (0)) + [ k X i =1 C ik ∂ ix X ∂ k − i +1 x u ]( x , β (0)) , for x ∈ [ β (0) , β ( t )] . By a similar computation as in (3.46), ∂ k +1 x u ( x , β (0)) is also asolution to problem (3.47) with the same initial date (3.48). The uniqueness of solutionsof problem (3.47) yields ∂ k +1 x v ( x , β (0)) = ∂ k +1 x u ( x , β (0)) , x ∈ [ β (0) , β ( t )] . Thus w is C k +1 on the segment { ( x , β (0)) | x ∈ [ β (0) , β ( t )] } . A similar argument shows that w is C k +1 on the rest of ϑ. The induction is complete. ✷ By similar arguments, we have Propositions 3.10-3.12 below. The details are omitted.
Proposition 3.10
Let the curves β and γ be of class C . Let f and X be of class C , . Let q be of class W , and p, q , f of class W , , respectively, such that the th order compatibility conditions hold true at γ (0) . Then problem (3 . admits a uniquesolution w ∈ W , (Ξ ( β, γ )) with the data (3 . and (3 . . Moreover, the followingestimates hold k w k W , ≤ C ( k p k W , + k q k W , + k q k W , + k f k W , ) . Let Γ i ( β, w ) = Z t | [ w x i ◦ β ( s )] ′ | ( t − s ) ds for s ∈ (0 , t ) , i = 1 , . (3.49) Proposition 3.11
Let the curves β and γ be of class C . Let f and X be of class C , . Then there are < c < c such that for all solutions w ∈ W , (Ξ ( β, γ )) to problem (3 . c [Γ( γ, w ) + Γ ( β, w )] ≤ k w k , + k f k , ≤ c [Γ( γ, w ) + Γ ( β, w ) + k f k , ] , where Γ( γ, w ) is given in (3 . . Proposition 3.12
The corresponding results as in Propositions . , . , and . hold where Ξ ( β, γ ) and Γ ( β, w ) are replaced with Ξ ( β, γ ) and Γ ( β, w ) , respectively. .4 Region Φ( β, γ, ˆ β ) Let β : [0 , t ] → IR , γ : [0 , t ] → IR , and ˆ β : [0 , t ] → IR be noncharacteristiccurves with β (0) = γ (0) and γ ( t ) = ˆ β (0) such that γ ( t ) ≥ β ( t ) , γ ′ ( t ) > , γ ′ ( t ) < , β ′ ( t ) > , β ′ ( t ) > ,γ ( t ) ≥ ˆ β ( t ) , ˆ β ′ ( t ) > , ˆ β ′ ( t ) > . We define Φ( β, γ, ˆ β ) = Ξ ( β, γ ) ∪ R ( z, a, b ) ∪ P ( ˆ β ) , where Ξ ( β, γ ) , R ( z, a, b ) , P ( ˆ β ) are given in (3.29), (3.6), and (3.26), respectively, with z = ( γ ( t ) , ˆ β ( t )) , a = ˆ β ( t ) − γ ( t ) , and b = β ( t ) − ˆ β ( t ) . Consider the boundarydata w x ◦ β ( t ) = p ( t ) , t ∈ [0 , t ] , w x ◦ ˆ β ( t ) = p ( t ) , t ∈ (0 , t ) , (3.50) w ◦ γ ( t ) = q ( t ) , h∇ w, F ˙ γ i ◦ γ ( t ) = q ( t ) for t ∈ (0 , t ) . (3.51)By similar arguments as for Ξ ( β, γ ) , we have Propositions 3.13-3.15 below. The detailsare omitted. Proposition 3.13
Let the curves β, γ, and ˆ β be of class C k, . Let q be of class C k, , and p , p , q , f of class C k − , such that the k th order compatibility conditionshold true at γ (0) and γ ( t ) , respectively. Then problem (3 . admits a unique solution w ∈ C k, (Φ( β, γ, ˆ β )) with the data (3 . and (3 . . Moreover, the following estimateshold k w k k, ≤ C ( k p k k − , + k p k k − , + k q k k, + k q k k − , + k f k k − , ) . Proposition 3.14
Let the curves β, γ, and ˆ β be of class C . Let f and X be of class C , . Let q be of class W , , and p , p , q , f of class W , , such that the th ordercompatibility conditions hold true at γ (0) and γ ( t ) , respectively. Then problem (3 . admits a unique solution w ∈ W , (Φ( β, γ, ˆ β )) with the data (3 . and (3 . . Moreover,the following estimates hold k w k , ≤ C ( k p k , + k p k , + k q k , + k q k , + k f k , ) . Proposition 3.15
Let the curves β, γ, and ˆ β be of class C . Let f and X be ofclass C , . Then there are < c < c such that for all solutions w ∈ W , (Φ( β, γ, ˆ β )) toproblem (3 . c [Γ( γ, w )+Γ ( ˆ β, w )+Γ ( β, w )] ≤ k w k , + k f k , ≤ c [Γ( γ, w )+Γ ( ˆ β, w )+Γ ( β, w )+ k f k , ] , where Γ( γ, w ) , Γ ( ˆ β, w ) , and Γ ( β, w ) are given in (3 . and (3 . , respectively. Solvability for Hyperbolic Surfaces
Let M ⊂ IR be a hyperbolic surface with the normal field ~n and let Ω ⊂ M be anoncharacteristic region, whereΩ = { α ( t, s ) | ( t, s ) ∈ (0 , a ) × (0 , b ) } . We consider solvability of problem under appropriate part boundary data h D w, Q ∗ Π i = f + f w + X ( w ) for x ∈ Ω , (4.1)where f is a function on M and X ∈ T ( M ) is a vector field on M. Clearly, equation(2.25) takes the form of (4.1).To set up boundary data, we consider some boundary operators. Let x ∈ ∂ Ω be given. µ ∈ T x M with | µ | = 1 is said to be the noncharacteristic normal outside Ω if there is acurve ζ : (0 , ε ) → Ω such that ζ (0) = x, ζ ′ (0) = − µ, Π( µ, X ) = 0 for X ∈ T x ( ∂ Ω) . Let µ be the the noncharacteristic normal field along ∂ Ω . Let the linear operator Q : T x M → T x M be given in (2.2) for x ∈ M. Recall that the shape operator ∇ ~n : T x M → T x M is defined by ∇ ~nX = ∇ X ~n ( x ) for X ∈ T x M. We define boundary operators T i : T x M → T x M by T i X = 12 h X + ( − i χ ( µ, X ) ρ ( X ) Q ∇ ~nX for X ∈ T x M, i = 1 , , (4.2)where χ ( µ, X ) = sign det (cid:16) µ, X, ~n (cid:17) , ̺ ( X ) = 1 √− κ sign Π( X, X ) , (4.3)and sign is the sign function.We shall consider the part boundary data h Dw, T α s i ◦ α (0 , s ) = p ( s ) , h Dw, T α s i ◦ α ( a, s ) = p ( s ) for s ∈ (0 , b ) , (4.4) w ◦ α ( t,
0) = q ( t ) , √ h Dw, ( T − T ) α t i ◦ α ( t,
0) = q ( t ) for t ∈ (0 , a ) . (4.5)To have a smooth solution, we need some kind of compatibility conditions as follows.Let A and B be k th order and m th order tensor fields on M, respectively, with k ≥ m. We define A ( i − ) B to be a ( k − m )th order tensor field by A ( i − ) B ( X , · · · , X k − m ) = h i X k − m · · · i X A, B i ( x ) for x ∈ M, (4.6)where X , · · · , X k − m are vector fields on M. | α t ( t, | = 1 for t ∈ [0 , a ] . Then Qα t , α t forms an orthonormbal basis of T α ( t, M with the positive orientation forall t ∈ [0 , a ] and Q ∇ ~nα t = Π( α t , α t ) Qα t − Π( α t , Qα t ) α t for t ∈ [0 , a ] . Let k ≥ w is a C k, solution to (4.1) in aneighborhood of the curve α ( t,
0) with the data (4.5). Then Dw ( α ( t, B ( t ) q ′ ( t ) + C ( t ) q ( t ) for t ∈ [0 , a ] , where B ( t ) = [ α t + Π( α t , Qα t )Π( α t , α t ) Qα t ] , C ( t ) = √ ̺ ( α t )Π( α t , α t ) Qα t , are vector fields along the curve α ( t, , from which we obtain D α t Dw ( α ( t, D α t B q ′ ( t ) + B ( t ) q ′′ ( t ) + D α t C q ( t ) + C ( t ) q ′ ( t ) . Using (4.1) and the above formula, we compute along the curve α ( t,
0) to have D w ( Qα t , Qα t )Π( α t , α t ) = f + f w + h Dw, X i − h D α t Dw, α t i Π( Qα t , Qα t )+2 h D α t Dw, Qα t i Π( Qα t , α t )= f + f q ( t ) + [ h X, B ( t ) i + h D α t B , Z ( t ) i ] q ′ ( t ) + h B ( t ) , Z ( t ) i q ′′ ( t )+[ h X, C ( t ) i + h D α t C , Z ( t ) i ] q ( t ) + h C ( t ) , Z ( t ) q ′ ( t ) for t ∈ [0 , a ] , (4.7)where Z ( t ) = 2Π( Qα t , α t ) Qα t − Π( Qα t , Qα t ) α t . Since Π( α t , α t ) = 0 for all t ∈ [0 , a ] , we have obtained two order tensor fields, A ( t ) , B i ( t ) , and C i ( t ) , that are given by f , X, Π , Qα t , α t , and their differentials, such that D w ( α ( t, A ( t ) f + X i =0 B i ( t ) q ( i )0 ( t ) + X i =0 C i ( t ) q ( i )1 ( t ) for t ∈ [0 , a ] . By repeating the above procedure, we obtain ( k + i )th order tensors fields A k + ii ( t ) , and k th order tensor fields B ki ( t ) , C ki ( t ) , such that D k w ( α ( t, Q k ( q , q , f )( t ) for t ∈ [0 , a ] , where Q k ( q , q , f )( t ) = k − X i =0 A k + ii ( t )( i − ) D i f ( α ( t, k X i =0 B ki ( t ) q ( i )0 ( t ) + k − X i =0 C ki ( t ) q ( i )1 ( t ) (4.8)for t ∈ [0 , a ] and k ≥ , where “( i − )” is defined in (4.6).30 efinition 4.1 Let q be of class C k, , and p , p , q , f of class C k − , to be said tosatisfy the k th order compatibility conditions at α (0 , and α ( a, if p j ( t j ) = h B ( t j ) , T α s i q ′ ( t j ) + h C ( t j ) , T α s i q ( t j ) , (4.9) p ( l ) j (0) = hQ l ( q , q , f )( t j ) , ˙ γ j (0) ⊗ · · · ⊗ ˙ γ j (0) i + X j + ··· + j i = l, ≤ i ≤ l − a j ··· j i hQ i ( q , q , f )( t j ) , γ ( j ) j (0) ⊗ · · · ⊗ γ ( j i ) j (0) i (4.10) for ≤ l ≤ k − , where a j ··· j i are positive integers given in (3 . , j = 1 , , γ ( s ) = α (0 , s ) ,γ ( s ) = α ( a, s ) , t = 0 , and t = a. Our main task in this section is to establish the following.
Theorem 4.1
Let Ω be a noncharacteristic region of class C m +2 , and let f and X be of class C m − , . Let q be of class C m, , and p , p , q , f be of C m − , , respectively.If m ≥ , we assume that the m th compatibility conditions holds. Then there is a uniquesolution w ∈ C m, (Ω) to problem (4 . with the data (4 . and (4 . satisfying k w k C m, (Ω) ≤ C ( k q k C m − , [0 ,a ] + k q k C m, [0 ,a ] + k p k C m − , [0 ,b ] + k p k C m − , [0 ,b ]) + k f k C m − , (Ω) ) . (4.11) Remark 4.1 If p , p ∈ C m − , (0 , b ) , q ∈ C m, (0 , a ) , q ∈ C m − , (0 , a ) , and f ∈ C m − , (Ω) for an integer m ≥ , then the m th order compatibility conditions are clearlytrue. Theorem 4.2
Let Ω be a noncharacteristic region of class C , and let f and X beof class C , . Let q be of class W , , and p , p , q , f of class W , to satisfy the thorder compatibility conditions. Then there is a unique solution w ∈ W , (Ω) to problem (4 . with the data (4 . and (4 . . Moreover, there is
C > , in dependent of solution w, such that k w k , (Ω) ≤ C ( k q k , (0 ,a ) + k q k , (0 ,a ) + k p k , (0 ,b ) + k p k , (0 ,b ) + k f k W , (Ω) ) . (4.12)We define Γ(Ω , w ) = Z b ( | p ′ ( s ) | + | p ′ ( s ) | )( b − s ) ds + Γ( α ( · , , w ) , (4.13)where p , p are given in (4.4), andΓ( α ( · , , w ) = X j =0 k∇ j w ◦ α ( · , k L (0 ,a ) + Z a [ | D w ( T α t , T α t ) | t + | D w ( T α t , T α t ) | ( a − t )] dt. (4.14)31 heorem 4.3 Let Ω be a noncharacteristic region of class C , and let f and X beof class C , . Then there are < c < c such that for all solutions w ∈ W , (Ω) toproblem (4 . c Γ(Ω , w ) ≤ k w k , (Ω) + k f k , (Ω) ≤ c ( k f k , (Ω) + Γ(Ω , w )) . (4.15)Next, we assume that f = 0 to consider problem h D w, Q ∗ Π i = f w + X ( w ) for x ∈ Ω . (4.16)Denote by Υ(Ω) all the solutions w ∈ W , (Ω) to problem (4.16). For w ∈ Υ(Ω) , we letΓ( w ) = Z b ( | p ′ ( s ) | + | p ′ ( s ) | ) ds + k q k , (0 ,a ) + k q k , (0 ,a ) , where p , p , q , and q are given in (4.4) and (4.5), respectively. We define H (Ω) = { w ∈ Υ(Ω) with the 1th order compatibility conditions | Γ( w ) < ∞ } . Theorem 4.4
Let Ω be a noncharacteristic region of class C , and X of class C , . For each w ∈ Υ(Ω) , there exists a sequence w n ∈ H (Ω) such that lim n →∞ k w n − w k W , (Ω) = 0 . The remains of this section is devoted to the proofs of Theorems 4.1-4.4. The proofsof Theorems 4.1-4.2 and 4.3-4.4 are given after Lemma 4.5 and Lemma 4.7, respectively.We shall solve (4.1) locally in asymptotic coordinate systems and then paste the localsolutions together. A chart ψ ( p ) = ( x , x ) on M is said to be an asymptotic coordinatesystem if Π( ∂x , ∂x ) = Π( ∂x , ∂x ) = 0 . (4.17)Let p ∈ M. Then κ ( p ) < p ([10]). In this system κ ( q ) = − Π ( ∂x , ∂x )det G , det G = | ∂x | | ∂x | − h ∂x , ∂x i . In an asymptotic coordinate system, equation (4.1) takes a normal form. We have thefollowing.
Proposition 4.1
Let M be a hyperbolic orientated surface and let ψ ( p ) = ( x , x ) : U ( ⊂ M ) → IR be an asymptotic coordinate system on M with the positive orientation.Then h D w, Q ∗ Π i = ± r − κ det G w x x ( x ) + the first order terms , (4.18) where w ( x ) = w ◦ ψ − ( x ) and the sign takes − if Π( ∂x , ∂x ) > and + if Π( ∂x , ∂x ) < , respectively. roof Let p ∈ U be fixed. Let α i = ( α i , α i ) T ∈ R be such that( α , α ) ∈ SO (2) , det( α , α ) = 1 , G ( p ) α i = η i α i for i = 1 , , where η i > G ( p ) . Set E i = α i ∂x + α i ∂x for i = 1 , . (4.19)Since h E i , E j i = α Ti G ( p ) α j = η j δ ij for 1 ≤ i, j ≤ ,E √ η , E √ η forms an orthonormal basis of M p . Moreover, E √ η , E √ η is of the positive orien-tation due todet (cid:16) E √ η , E √ η , ~n (cid:17) = det[ (cid:16) ∂x , ∂x , ~n (cid:17) α √ η α √ η α √ η α √ η
00 0 1 ] = det (cid:16) ∂x , ∂x , ~n (cid:17) √ η η = 1 . It follows from (2.2) that
Q E √ η = − E √ η , Q E √ η = E √ η . Using the above relations and the formulas (4.17), we have at pη η h D w, Q ∗ Π i = D w ( E , E )Π( E , E ) − D w ( E , E )Π( E , E )+ D w ( E , E )Π( E , E )= 2[ α α D w ( E , E ) − ( α α + α α ) D w ( E , E )+ α α D w ( E , E )]Π( ∂x , ∂x )= 2 D w ( α E − α E , α E − α E )Π( ∂x , ∂x )= − α α − α α ) D w ( ∂x , ∂x )Π( ∂x , ∂x )= − w x x − D ∂x ∂x ( w )]Π( ∂x , ∂x ) , (4.20)where the formula α α − α α = det( α , α ) = 1 , has been used.(4.18) follows from (4.20) since κ = − Π ( ∂x , ∂x ) η η . ✷ Lemma 4.1
There is a σ > such that, for all p ∈ Ω , there exist asymptotic coordi-nate systems ψ : B ( p, σ ) → IR with ψ ( p ) = (0 , , where B ( p, σ ) is the geodesic plate in M centered at p with radius σ . roof. For p ∈ Ω , let σ ( p ) denote the least upper bound of the radii σ for whichan asymptotic systems ψ = x : B ( p, σ ) → IR with ψ ( p ) = (0 ,
0) exists. From theexistence of local asymptotic coordinate systems, σ ( p ) > p ∈ Ω . Let p, q ∈ Ω , and q ∈ B ( p, σ ( p )) . Let σ ( q ) = inf z ∈ M, d ( z,p )= σ ( p ) d ( q, z ) , where d ( · , · ) is the distance function on M × M in the induced metric. Then σ ( q ) > B ( q, σ ( q )) ⊂ B ( p, σ ( p )) , since q ∈ B ( p, σ ( p )) . For any 0 < ˆ σ < σ ( q ) , B ( q, ˆ σ ) ⊂ B ( p, σ ( p )) . Thus, there is a 0 < σ < σ ( p ) such that B ( q, ˆ σ ) ⊂ B ( p, σ ) . Let ψ = x : B ( p, σ ) → IR be an asymptotic system with ψ ( p ) = (0 , . Set ˆ ψ ( z ) = ψ ( z ) − ψ ( q ) for z ∈ B ( q, ˆ σ ) . Then ˆ ψ : B ( q, ˆ σ ) → IR is an asymptotic coordinatesystem with ˆ ψ ( q ) = (0 , , that is, σ ( q ) ≥ σ ( q ) for q ∈ B ( p, σ ( p )) . Thus, σ ( p ) is lower semi-continuous in Ω and min p ∈ Ω σ ( p ) > ✷ Lemma 4.2
Let γ : [0 , a ] → M be a regular curve without self intersection points.Then there is a σ > such that, for all p ∈ { γ ( t ) | t ∈ (0 , a ) } , S ( p, σ ) has at most twointersection points with { γ ( t ) | t ∈ [0 , a ] } , where S ( p, σ ) is the geodesic circle centered at p with radius σ . If p = γ (0) , or γ ( a ) , then S ( p, σ ) has at most one intersection pointwith { γ ( t ) | t ∈ [0 , a ] } . Proof.
By contradiction. Let the claim in the lemma be not true. For each integer k ≥ , there exists t k < t k < t k (or t k > t k > t k ) in [0 , a ] such that d ( γ ( t k ) , γ ( t k )) = d ( γ ( t k ) , γ ( t k )) = 1 k for k ≥ . (4.21)We may assume that t k → t , t k → t , t k → t as k → ∞ , for certain points t , t , t ∈ [0 , a ] . Then 0 ≤ t ≤ t ≤ t ≤ a and γ ( t ) = γ ( t ) = γ ( t ) . The assumption that the curve γ has no self intersection point implies that t = t = t . For k ≥ , let f k ( t ) = 12 ρ k ( γ ( t )) , for t ∈ [0 , a ] , ρ k ( p ) = d ( γ ( t k ) , p ) for p ∈ M. It follows from (4.21) that there is a ζ k with t k <ζ k < t k such that f ′ k ( ζ k ) = 0 . On the other hand, the formula f ′ k ( t ) = ρ k ( γ ( t )) h Dρ k ( γ ( t )) , ˙ γ ( t ) i implies that f ′ k ( t k ) = 0 . Thus, we obtain η k ∈ ( t k , ζ k ) such that f ′′ k ( η k ) = 0 for k ≥ . Since f ′′ k ( t ) = D ( ρ k Dρ k )( ˙ γ ( t ) , ˙ γ ( t )) + ρ k ( γ ( t )) h Dρ k ( γ ( t )) , D ˙ γ ( t ) ˙ γ i , we have | ˙ γ ( t ) | = f ′′ ( t ) = lim k →∞ f ′′ k ( η k ) = 0 , which contradicts the regularity of the curve γ, where f ( t ) = 12 d ( γ ( t ) , γ ( t )) for t ∈ [0 , a ] . ✷ We need the following.
Proposition 4.2 ( i ) det (cid:16) Q ∇ ~nX, X, ~n ( x ) (cid:17) = Π( X, X )( x ) for X ∈ T x M, x ∈ M. ( ii ) Π( Q ∇ ~nX, Q ∇ ~nX ) = κ Π( X, X ) for X ∈ T x M, x ∈ M. Proof
Let x ∈ M be given. Let e , e be an orthonormal basis of T x M with thepositive orientation such thatΠ( e i , e j )( x ) = λ i δ ij for 1 ≤ i, j ≤ . (4.22)Then det (cid:16) Q ∇ ~nX, X, ~n (cid:17) = det (cid:16) e , e , ~n (cid:17) λ h X, e i h X, e i − λ h X, e i h X, e i
00 0 1 = Π(
X, X ) . (4.23)In addition, using (4.22), we haveΠ( Q ∇ ~nX, Q ∇ ~nX ) = Π (cid:16) − λ h X, e i e + λ h X, e i e , − λ h X, e i e + λ h X, e i e (cid:17) = λ λ h X, e i + λ λ h X, e i = κ Π( X, X ) . ✷ emma 4.3 Let p ∈ M and let B ( p , σ ) be the geodesic ball centered at p with radius σ > . Let γ : [ − a, a ] → B ( p , σ ) and β : [ − b, b ] → B ( p , σ ) be two noncharacteristic curvesof class C , respectively, with γ (0) = β (0) = p , Π( ˙ γ (0) , ˙ β (0)) = 0 . Let ˆ ψ : B ( p , σ ) → IR be an asymptotic coordinate system. Then there exists an asymp-totic coordinate system ψ : B ( p , σ ) → IR with ψ ( p ) = (0 , such that ψ ( γ ( t )) = ( t, − t ) for t ∈ [ − a, a ] , (4.24) β ′ ( s ) > , β ′ ( s ) > for s ∈ [ − b, b ] , (4.25) where ψ ( β ( s )) = ( β ( s ) , β ( s )) . Moreover, for X = X ∂x + X ∂x with Π( X, X ) = 0 , wehave ̺ ( X ) Q ∇ ~nX = χ (cid:16) γ ′ (0) , β ′ (0) (cid:17) ( X ∂x − X ∂x , X X > , − X ∂x + X ∂x , X X < , (4.26) where ̺ ( X ) is given in (4 . and χ (cid:16) γ ′ (0) , β ′ (0) (cid:17) = sign det (cid:16) γ ′ (0) , β ′ (0) , ~n ( p ) (cid:17) . Proof.
Let ˆ ψ ( p ) = (0 ,
0) andˆ ψ ( γ ( t )) = ( γ ( t ) , γ ( t )) for t ∈ [ − a, a ] . Since γ is noncharacteristic,Π( ˙ γ ( t ) , ˙ γ ( t )) = 2 γ ′ ( t ) γ ′ ( t )Π( ∂x , ∂x ) = 0 for t ∈ [ − a, a ] . Without loss of generality, we assume that γ ′ ( t ) > , γ ′ ( t ) < t ∈ [ − a, a ] . (4.27)We extend the domain [ − a, a ] of γ ( t ) to IR such thatlim t →±∞ γ ( t ) = ±∞ , lim t →±∞ γ ( t ) = ∓∞ , and the relations (4.27) hold for all t ∈ IR.
Consider a diffeomorphism ϕ ( x ) = y : R → IR given by ϕ ( x ) = ( γ − ( x ) , − γ − ( x )) for x = ( x , x ) ∈ IR . (4.28)Then ϕ ◦ ˆ ψ : B ( p , σ ) → IR is an asymptotic coordinate system such that ϕ ◦ ˆ ψ ( γ ( t )) = ( t, − t ) for t ∈ [ − a, a ] . (4.29)36et ϕ ◦ ˆ ψ ( β ( s )) = ( β ( s ) , β ( s )) . Since β is noncharacteristic, β ′ ( s ) β ′ ( s ) = 0 for s ∈ [ − b, b ] . In addition, the assumption Π( ˙ γ (0) , ˙ β (0)) = 0 and the relation (4.29) imply that0 = Π( ˙ γ (0) , ˙ β (0)) = Π( ∂x − ∂x , β ′ (0) ∂x + β ′ (0) ∂x ) = [ β ′ (0) − β ′ (0)]Π( ∂x , ∂x ) , that is, β ′ (0) = β ′ (0) . If β ′ (0) > , we let ψ ( p ) = ϕ ◦ ˆ ψ ( p ) to have (4.25). If β ′ (0) < , we define instead of (4.28) ϕ ( x ) = ( γ − ( x ) , − γ − ( x )) for x = ( x , x ) ∈ IR . Thus (4.25) follows again.Next, we prove (4.26). Let Q ∇ ~nX = Y ∂x + Y ∂x . Since ( Y X + Y X )Π( ∂x , ∂x ) =Π (cid:16) Q ∇ ~nX, X (cid:17) = h Q ∇ ~nX, ∇ ~nX i = 0 , we have Q ∇ ~nX = σ ( X ∂x − X ∂x ) , where σ is a function. Using Proposition 4.2 (ii), we obtain σ = − κ. Next, from (4.24) and (4.25), we have (cid:16) γ ′ (0) , β ′ (0) , ~n (cid:17) = (cid:16) ∂x , ∂x , ~n (cid:17) β ′ (0) 0 − β ′ (0) 00 0 1 , which yields sign det (cid:16) γ ′ (0) , β ′ (0) , ~n (cid:17) = sign det (cid:16) ∂x , ∂x , ~n (cid:17) . Thus (4.26) follows from Proposition 4.2 (i). ✷ Denote Ω(0 , s ) = { α ( t, s ) | t ∈ (0 , a ) , s ∈ (0 , s ) } for s ∈ [0 , b ] . (4.30)Then Ω = Ω(0 , b ) . Lemma 4.4
Let the assumptions in Theorem . hold. Then there is a < ω ≤ b such that problem (4 . admits a unique solution w ∈ C m, (Ω(0 , ω )) with the data (4 . where s ∈ [0 , ω ] , and (4 . to satisfy k w k C m, (Ω(0 ,ω )) ≤ C ( k p k C m − , [0 ,b ] + k p k C m − , [0 ,b ] + k q k C m, [0 ,a ] + k q k C m − , [0 ,a ] + k f k C m − , (Ω) ) . (4.31)37 roof. Let σ > γ ( t ) = α ( t,
0) in Lemma 4.2. We divide the curve α ( t,
0) into m parts with thepoints λ i = α ( t i ,
0) such that λ = α (0 , , λ m = α ( a, , d ( λ i , λ i +1 ) = σ , ≤ i ≤ m − , d ( λ m − , λ m ) ≤ σ , where t = 0 , t > , t > t , · · · , and t m = a > t m − . For simplicity, we assume that m = 3 . The other cases can be treated by a similar argument.We shall construct a local solution in a neighborhood of α ( t,
0) by the following steps.
Step 1.
Let ˆ s > α (0 , s ) ∈ B ( λ , σ ) for s ∈ [0 , s ] . From Lemma 4.3, there is asymptotic coordinate system ψ ( p ) = x : B ( λ , σ ) → IR with ψ ( λ ) = (0 ,
0) such that ψ ( α ( t, t, − t ) for t ∈ [0 , t ] , (4.32) β ′ ( s ) > , β ′ ( s ) > s ∈ [0 , s ] , where β ( s ) = ψ ( α (0 , s )) = ( β ( s ) , β ( s )) . Let γ ( t ) = ( t, − t ) . We may assume that s > β ( s ) ≤ t since β (0) = 0 . SetΞ ( β , γ ) = P ( β ) ∪ R (( β ( s ) , , c , d ) ∪ E ( γ ) , (4.33)as in (3.29) with c = t − β ( s ) and d = β ( s ) . Then we letΩ = Ω ∩ ψ − [Ξ ( β , γ )] . Noting that for the region Ω χ ( µ ( α ( t, , α t ( t, χ ( − α s (0 , , α t (0 , t ∈ (0 , t ) ,χ ( µ ( α (0 , s )) , α s (0 , s )) = χ ( − α t (0 , , α s (0 , s ∈ (0 , s ) , from (4.26), we obtain T α s (0 , s ) = β ′ ( s ) ∂x , T α s (0 , s ) = β ′ ( s ) ∂x for s ∈ (0 , s ) , (4.34) T α t ( t,
0) = ∂x , T α t ( t,
0) = − ∂x for t ∈ (0 , t ) . (4.35)From Proposition 4.1, solvability of problem (4.1) on Ω ∩ ψ − (Ξ ( β , γ )) is equivalentto that of problem (3.1) over the region Ξ ( β , γ ) . Next, we consider the transfer of theboundary data under the chart ψ . The corresponding part data are w x ◦ β ( s ) = h Dw, T α s i ◦ α (0 , s ) /β ′ ( s ) = p ( s ) /β ′ ( s ) for s ∈ [0 , s ] , ( t, − t ) = w ◦ ψ − ( t, − t ) = w ( α ( t, q ( t ) for t ∈ [0 , t ] ,∂∂ν w ( t, − t ) = 1 √ h Dw, ( T − T ) α t i ◦ α ( t,
0) = q ( t ) for t ∈ [0 , a ] , where w ( x ) = w ◦ ψ − ( x ) . It is easy to check that p /β ′ , q , q , and f are m th order compatible at α (0 ,
0) in the senseof Definition 4.1 is equivalent to that p /β ′ , q , q , and ˆ f do in the sense of Definition3.1, where ˆ f = f ◦ ψ − ( x )2 s det G ( x ) − k ◦ ψ − ( x ) for x ∈ ψ ( B ( λ , σ )) ⊂ IR , where det G ( x ) = det( h ∂x i , ∂x j i ) . From Proposition 3.9, problem (3.1) admits a unique solution w ∈ C m, (Ξ ( β , γ ))with the corresponding boundary data. Thus, we have obtained a solution, denoted by w ∈ C m, (Ω ) , to problem (4.1) with the data h Dw , T α s i ◦ α (0 , s ) = p ( s ) for s ∈ [0 , s ] ,w ◦ α ( t,
0) = q ( t ) , √ h Dw , ( T − T ) α t i ◦ α ( t,
0) = q ( t ) for t ∈ [0 , t ] , where Ω = Ω ∩ ψ − [Ξ ( β , γ )] . It follows from the estimate in Proposition 3.9 that k w k C m, (Ω ) ≤ C Γ m C ( p , p , q , q , f ) , (4.36)where Γ m C ( p , p , q , q , f ) = k p k C m − , (0 ,b ) + k p k C m − , (0 ,b ) + k q k C m, (0 ,a ) + k q k C m − , (0 ,a ) + k f k C m − , (Ω) . We define a curve on Ω by ζ ( s ) = ψ − ◦ γ t ( s ) for s ∈ [0 , s t ] , (4.37)where γ t ( s ) = ( s + t , s − t ) , s t = t if t ∈ (0 , t ,t − t if t ∈ ( t , t ) . Then ζ ( s ) is noncharacteristic andΠ( ˙ ζ (0) , α t ( t , ∂x + ∂x , ∂x − ∂x ) = 0 . (4.38)39 tep 2. Let the curve ζ be given in (4.37). Let s > ζ ( s ) ∈ B ( λ , σ ) for s ∈ [0 , s ] . From the noncharacteristicness of ζ ( s ) and the relation (4.38) and Lemma 4.3 again, thereexists an asymptotic coordinate system ψ ( p ) = x : B ( λ , σ ) → IR with ψ ( λ ) = (0 , ψ ( α ( t + t , t, − t ) for t ∈ [0 , t − t ] ,β ′ ( s ) > , β ′ ( s ) > s ∈ [0 , s ] , where β ( s ) = ψ ( ζ ( s )) = ( β ( s ) , β ( s )) . We also assume that s > β ( s ) ≤ t . This time, we setΞ ( β , γ ) = P ( β ) ∪ R (( β ( s ) , , c , d ) ∪ E ( γ ) , where c = t − t − β ( s ), d = β ( s ) , and γ ( t ) = ψ ( α ( t + t , . Next, letΩ = Ω ∩ ψ − [Ξ ( β , γ )] . Since for the region Ω χ ( µ ( ζ ( s )) , ζ ′ ( s )) = χ ( − α t (0 , t ) , ζ ′ (0)) for s ∈ (0 , s ) ,χ ( µ ( α ( t + t, , α t ( t + t, χ ( − ζ ′ (0) , α t ( t , t ∈ (0 , t − t ) , it follows from (4.26) that T ζ ′ ( s ) = β ′ ( s ) ∂x , T ζ ′ ( s ) = β ′ ( s ) ∂x for s ∈ (0 , s ) , T α t ( t + t,
0) = ∂x , T α t ( t + t,
0) = − ∂x for t ∈ (0 , t − t ) . By some similar arguments in Step 1, we obtain a unique solution w ∈ C m, (Ω ) toproblem (4.1) with the data h Dw , T ˙ ζ i ◦ β ( s ) = h Dw , T ˙ ζ ( s ) i ◦ β ( s ) for s ∈ [0 , s ] ,w ( α ( t, q ( t ) , √ h Dw , ( T − T ) α t i ◦ α ( t,
0) = q ( t ) for t ∈ [ t , t ] , where w is the solution of (4.1) on Ω , given in Step 1. The following estimate also holds k w k C m, (Ω ) ≤ C ( kh Dw , T ˙ ζ i ◦ ζ k C m − , [0 ,s ] + k q k C m, [0 ,a ] + k q k C m − , [0 ,a ] + k f k C m − , (Ω) ) ≤ C Γ m C ( p, q , q , h ) . (4.39)As in Step 1, we define a curve on Ω by ζ ( s ) = ψ − ( s + t − t , s + t − t ) for s ∈ [0 , s t ] , (4.40)40here s t = t − t if t − t ≤ t − t s t = t − t if t − t > t − t . Then ζ ( s ) is noncharacteristic andΠ( ˙ ζ (0) , α t ( t , ∂x + ∂x , ∂x − ∂x ) = 0 . (4.41) Step 3.
Let the curve ζ be given in (4.40). Let s > ζ ( s ) , α ( a, s ) ∈ B ( λ , σ ) for s ∈ [0 , s ] . Let ψ ( p ) = x : B ( λ , σ ) → IR be an asymptotic coordinate system with ψ ( λ ) = (0 , ,ψ ( α ( t + t , t, − t ) for t ∈ [0 , a − t ] , (4.42)and β ′ ( s ) > , β ′ ( s ) > s ∈ [0 , s ] , where β ( s ) = ψ ( ζ ( s )) = ( β ( s ) , β ( s )) . Let β ( s ) = ψ ( α ( a, s )) = ( β ( s ) , β ( s )) . Next, we prove that β ′ ( s ) > , β ′ ( s ) > s ∈ [0 , s ] , (4.43)by contradiction. Since α ( a, s ) is noncharacteristic, using (4.42) and the assumptionΠ( α t ( a, , α s ( a, , we have β ′ (0) = β ′ (0); thus β ′ ( s ) β ′ ( s ) > s ∈ [0 , s ] . Let p ( t, s ) = α ( t, s ) + α ( t, s ) , ψ ( α ( t + t , s )) = ( α ( t, s ) , α ( t, s )) . Let (4.43) be not true, that is, β ′ ( s ) < , β ′ ( s ) < s ∈ [0 , s ] . Thus p (0 , s ) = β ( s ) + β ( s ) > β (0) + β (0) = 0 for s ∈ (0 , s ] ,p ( a − t , s ) = β ( s ) + β ( s ) < β (0) + β (0) = 0 for s ∈ (0 , s ] . Let t ( s ) ∈ (0 , a − t ) be such that α ( t ( s ) , s ) + α ( t ( s ) , s ) = 0 for s ∈ (0 , s ) . (4.44)Since α t (0 ,
0) = 1 and α ( t + t , s ) are noncharacteristic for all s ∈ [0 , s ] , we have α t ( t, s ) > < α (0 , s ) < α ( t ( s ) , s ) < α ( a − t , s ) = β ( s ) < β (0) = a − t . α ( α ( t ( s ) , s )+ t ,
0) = α ( t ( s ) , s ) , which is a contradictionsince α : [0 , a ] × [ a, b ] → M is an imbedding map.We also assume that s has been taken so small such that β ( s ) < a − t , β ( s ) < , since β (0) = 0 and β (0) = − ( a − t ) < . Let γ ( t ) = ψ ( α ( t + t , t, − t ) . Wenow set Φ( β , γ , β ) = Ξ ( β , γ ) ∪ R (( a − t , β ( s )) , c , d ) ∪ P ( β ) , where Ξ ( β , γ ) , R (( a − t , β ( s )) , c , d ) , and P ( β ) are given in (3.29), (3.6), and(3.26), respectively, with c = β ( s ) − a + t and d = β ( s ) − β ( s ) . LetΩ = Ω ∩ ψ − [Φ( β , γ , β )] . This time we use (4.26) to obtain, for the region Ω , T ζ ′ ( s ) = β ′ ( s ) ∂x , T ζ ′ ( s ) = β ′ ( s ) ∂x for s ∈ (0 , s ) , T α t ( t + t,
0) = ∂x , T α t ( t + t,
0) = − ∂x for t ∈ (0 , a − t ) , T α s ( a, s ) = β ′ ( s ) ∂x , T α s ( a, s ) = β ′ ( s ) ∂x for s ∈ (0 , s ) . Applying Proposition 3.13, problem (4.1) admits a unique solution w ∈ C m, (Ω )with the data h Dw , T ˙ ζ i ◦ β ( s ) = h Dw , T ˙ ζ i ◦ β ( s ) , h Dw , T α s i ◦ α ( a, s ) = p ( s ) for s ∈ [0 , s ] ,w ( α ( t, q ( t ) , √ h Dw , ( T − T ) α t i ◦ α ( t,
0) = q ( t ) for t ∈ [ t , a ] . Using the estimates in Proposition 3.13 and (4.39), we obtain k w k C m, (ˆΩ ) ≤ C ( kh Dw , T ˙ ζ i ◦ ζ k C m − , [0 ,s ] + k p k C m − , [0 ,b ] + k q k C m, [0 ,a ] + k q k C m − , [0 ,a ] + k f k C m − , (Ω) ) ≤ C Γ m C ( p , p , q , q , f ) . (4.45) Step 4.
We define w = w i for p ∈ Ω i for i = 0 , , . Let ω > α ( t, s ) ∈ Ω ∪ Ω ∪ Ω for ( t, s ) ∈ (0 , a ) × (0 , ω ) . Then w ∈ C m, (Ω(0 , ω )) will be a solution to (4.1) with the corresponding data if weshow that w ( p ) = w ( p ) for p ∈ Ω ∩ Ω ; w ( p ) = w ( p ) for p ∈ Ω ∩ Ω . (4.46)42ince w x ◦ β ( s ) = w x ◦ β ( s ) for s ∈ [0 , s ] ,w ( t, − t ) = w ( t, − t ) , ∂w ∂ν ( t, − t ) = ∂w ∂ν ( t, − t ) for t ∈ [ t , t ] , from the uniqueness in Proposition 3.9, we have w ( x ) = w ( x ) for x ∈ Ξ ( β , γ ) ∩ Ξ ( β , γ ) , which yields the first identity in (4.46). A similar argument shows that the second identityin (4.46) is true.Finally, the estimate (4.31) follows from (4.36), (4.39), and (4.45). ✷ From a similar argument as for the proof of Lemma 4.4, we obtain the following.
Lemma 4.5
Let the assumptions in Theorem . hold. Then there is a < ω ≤ b such that problem (4 . admits a unique solution w ∈ W , (Ω(0 , ω )) with the data (4 . where s ∈ (0 , ω ) , and (4 . to satisfy k w k , (Ω(0 ,ω )) ≤ C ( k q k , (0 ,a ) + k q k , (0 ,a ) + k p k , (0 ,b ) + k p k , (0 ,b ) + k f k W , (Ω) ) . (4.47)We are now ready to prove Theorems 4.1 and 4.2. Proof of Theorem 4.1
Let ℵ be the set of all 0 < ω ≤ b such that the claims inLemma 4.4 hold. We shall prove b ∈ ℵ . Let ω = sup ω ∈ℵ ω. Then 0 < ω ≤ b. Thus there is a unique solution w ∈ C m, (Ω(0 , ω ))to (4.1) with the data (4 . , where s ∈ [0 , ω ) , and (4 . . Next we show that ω = b by contradiction. Let 0 < ω < b. By an argumentas for Lemma 4.4, the solution w ∈ C m, (Ω(0 , ω )) can be extended such that w ∈ C m, (Ω(0 , ω )) . Then by Lemma 4.4 again, w can be extend outside C m, (Ω(0 , ω )) , which contradicts with the definition of ω . Let λ = α (0 , ω ) , σ , t = 0 , t , t , and t = a be given as in the proof of Lemma 4.4.Let ψ ( p ) = x : B ( λ , σ ) → IR be an asymptotic coordinate system with ψ ( λ ) =(0 ,
0) such that ψ ( α ( t, ω )) = ( t, − t ) for t ∈ [0 , t ] ,ζ ′ ( s ) > , ζ ′ ( s ) > s ∈ [ ω − ε , ω ] , where ψ ( α (0 , s )) = ( ζ ( s ) , ζ ( s )) . For ε > , let β ( s ) = ψ ( α (0 , s + ω − ε )) = ( β ( s ) , β ( s )) , γ ( t ) = ψ ( α ( t, ω − ε )) = ( γ ( t ) , γ ( t )) , s ∈ [0 , ε ] , where β i ( s ) = ζ i ( s + ω − ε ) for i = 1 , . We fixed ε > ω + ε ≤ b, β ( ε ) ≤ γ ( t ) , γ ′ ( t ) > , γ ′ ( t ) < . Let Ξ ( β , γ ) be given as in (4.33). Clearly, { α ( t, ω ) | t ∈ [0 , t ] } ⊂ ψ − (Ξ ( β , γ )) . From Proposition 3.9, we can extend a solution w such that w is C k, on the segment { α ( t, ω + ε ) | t ∈ [0 , t ] } . Repeating Steps 2-4 in the proof of Lemma 4.4, the solution w can be extended such that w is C k, on the segment { α ( t, ω + ε ) | t ∈ [0 , a ] } , whichcontradicts the definition of ω . The proof is complete. ✷ Proof of Theorem 4.2
A similar argument as in the proof of Theorem 4.1 completesthe proof. ✷ To prove Theorems 4.3 and 4.4, we need the following lemmas.
Lemma 4.6
Let the assumptions in Theorem . hold. Then there are < ω ≤ b and C > such that for all solutions w ∈ W , (Ω) to problem (4 . k w k , (Ω(0 ,ω )) ≤ C [ k f k , (Ω) + Γ(Ω , w )] , (4.48) where Ω(0 , ω ) and Γ(Ω , w ) is given in (4 . and (4 . , respectively. Proof
We keep all the notion in the proof of Lemma 4.4. Let ω > w ( x ) = w ◦ ψ − ( x )is a solution to problem (3.1) on the region Ξ ( β , γ ) , where Ξ ( β , γ ) is given in (4.33)and β ( s ) = ψ ( α (0 , s )) = ( β ( s ) , β ( s )) for s ∈ (0 , s ) , γ ( t ) = ψ ( α ( t, t, − t )for t ∈ [0 , t ] . It follows from (4.34) and (4.35) that (cid:12)(cid:12)(cid:12) | D w ( T α t ( t, , T α t ( t, | − | w x x ◦ γ ( t ) | (cid:12)(cid:12)(cid:12) ≤ C |∇ w ◦ γ ( t ) | , Similarly, we have (cid:12)(cid:12)(cid:12) | D w ( T α t ( t, , T α t ( t, | − | w x x ◦ γ ( t ) | (cid:12)(cid:12)(cid:12) ≤ C |∇ w ◦ γ ( t ) | , (cid:12)(cid:12)(cid:12) | D w ( T α s (0 , s ) , T α s (0 , s )) | − | w x x ◦ β ( s ) | β ′ ( s ) (cid:12)(cid:12)(cid:12) ≤ C |∇ w ◦ β ( s ) | , (cid:12)(cid:12)(cid:12) | D w ( T α s (0 , s ) , T α s (0 , s )) | − | w x x ◦ β ( s ) | β ′ ( s ) (cid:12)(cid:12)(cid:12) ≤ C |∇ w ◦ β ( s ) | . γ , w ) + Γ ( β , w ) ≤ C Γ(Ω , w ) , (4.49)where Γ( γ , w ) and Γ ( β , w ) are given in (3.18) and (3.49), respectively.Applying Proposition 3.11 to Ξ ( β , γ ) and using (4.49), we have k w k , (Ω ) ≤ C k w k , (Ξ ( β ,γ )) ≤ C ( k f k , + Γ(Ω , w )) . Using (3.16) by a similar argument as for the above estimates, we obtain k w k , (Ω i ) ≤ C ( k f k , + Γ(Ω , w )) , for i = 1 , . Thus the estimate (4.48) follows. ✷ Lemma 4.7
Let the assumptions in Theorem . hold. Then there is C > such thatfor all solutions w ∈ W , (Ω) to problem (4 . , w ) ≤ C ( k w k , (Ω) + k f k , (Ω) ) . (4.50) Proof Step 1
We claim that for each ε > C ε > X j =0 Z a − εε | D j w ◦ α ( t, | dt ≤ C ε ( k w k , (Ω) + k f k , (Ω) ) . (4.51)Let t ∈ (0 , a ) be fixed and let p = α ( t , . Let ζ : (0 , ǫ ) → Ω be such that ζ (0) = p , ζ ′ (0) = − µ ( p ) , where µ ( p ) is the noncharacteristic normal at the boundary point p outside Ω . FromLemma 4.3, there are 0 < σ < min { t , a − t } and an an asymptotic coordinate system ψ : B ( p , σ ) → IR with ψ ( p ) = (0 ,
0) such that ψ ( α ( t + t , t, − t ) for t ∈ ( − σ , σ ) , ζ ′ ( s ) > , ζ ′ ( s ) > s ∈ (0 , ε ) , where ψ ( ζ ( s )) = ( ζ ( s ) , ζ ( s )) . SetΩ p = Ω ∩ ψ − [ E ( γ )] , where γ ( t ) = ( t, − t ) , E ( γ ) = { x | − x < x < σ , − σ < x < σ } . Using (4.26) for the region Ω p , we obtain T α t ( t + t , ∂x , T α t ( t + t ,
0) = − ∂x for t ∈ ( − σ , σ ) , T i are given in (4 . . Observe that w ( x ) = w ◦ ψ − ( x ) is a solution to problem (3.1) on the region E ( γ ) . Applying Proposition 3.4, we have X j =0 Z σ / − σ / | D j w ◦ α ( t + t , s ) | dt ≤ C X j =0 Z σ / − σ / |∇ j w ( t, − t ) | dt ≤ C ( k w k , ( E ( γ )) + k f ◦ ψ − k , ( E ( γ )) ) ≤ C ( k w k , (Ω) + k f k , (Ω) ) . Thus the estimates (4.51) follows from the finitely covering theorem. By a similarargument, we have X j =0 Z b − εε | D j w ◦ α ( t k , s ) | ds ≤ C ε ( k w k , (Ω) + k f k , (Ω) ) , k = 1 , , where t = 0 and t = a, which particularly imply that Z b − εε | p ′ k ( s ) | ( b − s ) ds ≤ C ε ( k w k , (Ω) + k f k , (Ω) ) , k = 1 , , . (4.52) Step 2
We treat the estimates at the angular points α (0 , , α (0 , b ) , α ( a, , and α ( a, b ) , respectively.Consider the angular α (0 , b ) first. Let ε > ψ : B ( α (0 , b ) , σ ) → IR with ψ ( α (0 , b )) = (0 ,
0) suchthat γ ( t ) = ψ ( α ( t, b )) = ( t, − t ) for t ∈ [0 , ε ] ,β ( s ) = ψ ( α (0 , b − s )) = ( β ( s ) , β ( s )) , β ′ ( s ) > , β ′ ( s ) > s ∈ [0 , ε ] . Consider the region Ω α (0 ,b ) = Ω ∩ ψ − [Ξ ( β, γ )] . From (4.26), we have T α s (0 , b − s ) = − β ′ ( s ) ∂x for s ∈ (0 , ε ) . It follows from Proposition 3.11 that Z bb − ε | p ′ ( s ) | ( b − s ) ds ≤ C ( k f ◦ ψ − k , (Ξ ( β,γ )) + k w ◦ ψ − k , (Ξ ( β,γ )) ) ≤ C ( k f k , + k w k , ) . Similarly, we can treat the estimates at the other angular points. Thus estimate (4.50)follows by combing the above estimates with those in Step 1. ✷ Proof of Theorem 4.3
Let R be the set of all 0 < ω ≤ b such that estimate (4 . ω = sup ω ∈R ω. By Lemmas 4.6 and 4.7, it is sufficient to prove ω ∈ R .
46y following the proof of Theorem 4.1, we obtain a ε > k w k , (Ω( ω − ε,ω )) ≤ C [ Z ω ω − ε ( | p ′ ( s ) | + | p ′ ( s ) | )( ω − s ) ds + Γ( α ( · , ω − ε ) , w ) + k f k , ] , where Γ( α ( · , ω − ε ) , w ) is given in (4.14). On the other hand, we fix 0 < ε < ε and applyLemma 4.7 to the region Ω( ω − ε, ω − ε ) to obtainΓ( α ( · , ω − ε ) , w ) ≤ C [ k w k , (Ω( ω − ε,ω − ε )) + k f k , (Ω( ω − ε,ω − ε )) ] ≤ C [ k w k , (Ω(0 ,ω − ε )) + k f k , (Ω(0 ,ω − ε )) ] ( by (4.48)) ≤ C [ k f k , (Ω) + Γ(Ω , w )] . By Lemma 4.6, we have ω ∈ R . By Lemma 4.7, we obtain ω = b. ✷ Proof of Theorem 4.4
Let w ∈ Υ(Ω) and ε > w ∈ H (Ω)such that k w − ˆ w k , (Ω) < ε. Let p , p , and q , q be given in (4.4) and (4.5), respectively. Towards approximating w by H (Ω) functions, we first approximate its traces q , q , p , and p . From Theorem 4.3,those traces are regular except for the angular points α (0 , , α (0 , b ) , α ( a, , and α ( a, b ) . Next, we change their values near those angular points to make them regular and to letthe 1th order compatibility conditions hold at α (0 ,
0) and α ( a, . Step 1
Consider the point α (0 , . Let σ > ψ : B ( α (0 , , σ ) → IR with ψ ( α (0 , , ψ ( α ( t, t, − t ) for t ∈ [0 , t ) ,β ( s ) = ψ ( α (0 , s )) = ( β ( s ) , β ( s )) , β ′ (0) = β ′ (0) , β ′ ( s ) > , β ′ ( s ) > s ∈ [0 , t ) , for some 0 < t < min { a, b } / p ( s ) = w x ◦ β ( s ) β ′ ( s ) for s ∈ [0 , t ] ,q ′ ( t ) = w x ( t, − t ) − w x ( t, − t ) , −√ q ( t ) = w x ( t, − t ) + w x ( t, − t ) (4.53)for t ∈ (0 , t ) . where w ( x ) = w ◦ ψ − ( x ) . Moreover, we have D w ( T α t , T α t ) = D w ( ∂x , ∂x ) = w x x ( t, − t ) − D ∂x ∂x ( w )( t, − t ) = ϕ + φ , and D w ( T α t , T α t ) = ϕ + φ . By differential the equations in (4.53) in t ∈ (0 , t ) and using the formulas (4.1) and (4.18),we obtain ϕ = 12 [ q ′′ ( t ) − √ q ′ ( t )] , ϕ = 12 [ q ′′ ( t ) + √ q ′ ( t )] for t ∈ (0 , t ) , = some frist order terms of w, φ = some frist order terms of w. By Theorem 4.3 p ∈ W , (0 , t ) , q ∈ W , (0 , t ) , q , ϕ t / , ϕ , φ , φ ∈ L (0 , t ) . Thus q ′′ t / = ( ϕ + ϕ ) t / ∈ L (0 , t ) , q ′ t / = 1 √ ϕ − ϕ ) t / ∈ L (0 , t ) . We also need the following.
Lemma 4.8
Let z ( t ) = 12 [ q ′ ( t ) + √ q ( t )] for t ∈ (0 , t ) . Then z ∈ C [0 , t ] and p (0) + z (0) β ′ (0) = 0 . (4.54) Proof of Lemma 4.8
It follows from z ′ = ϕ ∈ L (0 , t ) that z ∈ C [0 , t ] . We have w x ◦ β ◦ β − ( t ) − w x ( t, − t ) = Z β ◦ β − ( t ) − t w x x ( t, s ) ds, from which we obtain | w x ◦ β ◦ β − ( t ) − w x ( t, − t ) | ≤ [ β ◦ β − ( t ) + t ]] Z β ◦ β − ( t ) − t | w x x ( t, s ) | ds. For ε > ϑ ∈ [ ε/ , ε ] be fixed such that | w x ◦ β ◦ β − ( ϑ ) − w x ( ϑ, − ϑ ) | = inf t ∈ [ ε/ ,ε ] | w x ◦ β ◦ β − ( t ) − w x ( t, − t ) | . Then | w x ◦ β ◦ β − ( ϑ ) − w x ( ϑ, − ϑ ) | ≤ ε [ β ◦ β − ( ε ) + ε ] Z εε/ Z β ◦ β − ( t ) − t | w x x ( t, s ) | ds ≤ σ Z ε Z β ◦ β − ( t ) − t | w x x ( t, s ) | ds for t ∈ [ ε/ , ε ] . Thus, w ∈ W , (Ω) implies, by (4.53), that (4.54) holds.Let 0 < ε < t given small. We shall construct ˆ q and ˆ q to satisfy the following.(1) ˆ q ( t ) = q ( t ) , ˆ q ( t ) = q ( t ) for t ∈ [ ε, a );(2) ˆ q ∈ W , (0 , a ) and ˆ q ∈ W , (0 , a );(3) The following 1th order compatibility conditions hold at the point α (0 , , p (0) + [ˆ q ′ (0) + √ q (0)] β ′ (0) = 0;484) If ˆ w ∈ Υ(Ω) is such thatˆ w ◦ α ( t,
0) = ˆ q ( t ) , h D ˆ w, ( T − T ) α t i ◦ α ( t,
0) = ˆ q ( t ) for t ∈ (0 , a ) , h D ˆ w, T α s i ◦ α (0 , s ) = p ( s ) , h D ˆ w, T α s i ◦ α (0 , s ) = p ( s ) for s ∈ (0 , b ) , then Γ(Ω , ˆ w − w ) → ε → . For the above purposes, we defineˆ q ( t ) = ( σ ( ε ) + [ q ′ ( t ) − R t ε ϕ ds − R t ϕ ds ] t + R t ( t − s ) ϕ ( s ) ds, t ∈ [0 , ε ) ,q ( t ) t ∈ [ ε, a ] , and ˆ q ( t ) = q ( t ) + √ R t ε ϕ ds − √ R t t ϕ ds, t ∈ (0 , ε ) ,q ( t ) , t ∈ [ ε, a ] , where σ ( ε ) = q ( ε ) − q ′ ( ε ) ε + Z ε sϕ ( s ) ds. Clearly, (1) and (2) hold for the above ˆ q and ˆ q . Since q ′ ( t ) − Z t ε ϕ ds − Z t ϕ ds = q ′ ( ε ) − Z ε ϕ ( s ) ds = q ′ ( ε ) − z ( ε )+ z (0) , for t ∈ (0 , ε ) , √ Z t ε ϕ ds − √ Z t t ϕ ds = q ( ε ) − q ( t ) + 1 √ z ( t ) − z ( ε )] , using (4.54), we have2 p (0) + [ˆ q ′ (0) + √ q (0)] β ′ (0) = q ′ ( ε ) + √ q ( ε ) − z ( ε ) = 0 . Next, we check (4). It follows that | q ( t ) − ˆ q ( t ) | = | Z εt Z εs q ′′ ( τ ) dτ ds + Z εt ( t − s ) ϕ ( s ) ds | ≤ ε − t + t ln tε ) Z ε | q ′′ ( τ ) | τ dτ + 23 ε Z ε | ϕ ( s ) | ds for t ∈ (0 , ε ) . In addition, | q ′ ( t ) − ˆ q ′ ( t ) | = | Z εt ϕ ( s ) ds | ≤ (ln εt ) Z ε | ϕ ( s ) | sds for t ∈ (0 , ε ) . Similarly, we have | q ( t ) − ˆ q ( t ) | ≤ (ln εt ) Z ε | ϕ ( s ) | sds for t ∈ (0 , ε ) . α ( · , , w − ˆ w ) = X j =0 k D ( w − ˆ w ) ◦ α ( · , k L (0 ,ε ) + Z ε [ | D ( w − ˆ w )( T α t , T α t ) | t + | D ( w − ˆ w )( T α t , T α t ) | ( a − t )] dt ≤ C Z ε ( | q ( s ) − ˆ q ( s ) | + | q ′ ( t ) − ˆ q ′ ( t ) | + | q ( t ) − ˆ q ( t ) | + | ϕ − ˆ ϕ | t + | ϕ − ˆ ϕ | ) dt ≤ C Z ε [( | q ′′ ( t ) | + | ϕ ( t ) | ) t + | ϕ ( t ) | ] dt, (4.55)where ˆ ϕ = 12 [ˆ q ′′ ( t ) − √ q ′ ( t )] = 0 , ˆ ϕ = 12 [ˆ q ′′ ( t ) + √ q ′ ( t )] = ϕ . Thus (4) follows.
Step 2
As in Step 1, we change the values of q and q near the point α ( a,
0) to get ˆ q and ˆ q in W , (0 , a ) and in W , (0 , a ) , respectively, such that the 1th order compatibilityconditions at α ( a,
0) hold to approximate q and q . Then we change the values of p and p near the points α (0 , b ) and α ( a, b ) , respectively, such that ˆ p , ˆ p ∈ W , (0 , b ) approximate p and p , respectively. Thus the proof completes from Theorem 4.2. ✷ Proof of Theorem 1.1
Let Ω ⊂ M be a noncharacterisic region of class C , . For U ∈ C , (Ω , T ) given, we consider problemsym ∇ y = U on Ω . (5.1)(1) Consider problem h D v, Q ∗ Π i = P ( U ) − vκ tr g Π + X ( v ) for x ∈ Ω , (5.2)where P ( U ) and X are given in (2.26) and (2.27), respectively, with the boundary data h Dv, T α s i ◦ α (0 , s ) = h Dv, T α s i ◦ α ( a, s ) = 0 for s ∈ (0 , b ) , (5.3) v ◦ α ( t,
0) = 1 √ h Dv, ( T − T ) α t i ◦ α ( t,
0) = 0 for t ∈ (0 , a ) , (5.4)where T and T are given in (4.2).Since P ( U ) ∈ L ∞ (Ω) , X ∈ L ∞ (Ω) , it follows from Theorem 4.1 that problem (5.2) with the data (5.3) and (5.4) has a uniquesolution v ∈ C , (Ω) with the bounds k v k C , (Ω) ≤ C k U k C , (Ω ,T ) . (5.5)50rom Theorem 2.1, there is a solution y ∈ C , (Ω , IR ) to (5.1). Let w = h y, ~n i , W = y − w~n. Then w ∈ C , (Ω) . It follows from [12, lemma 4.3] that W ∈ C , (Ω , T ) and (1.2) holds.(2) Let Ω ∈ C m +2 , and U ∈ C m +1 , (Ω , T ) be given for some m ≥ . Let q ( t ) = q ( t ) = 0 for t ∈ [0 , a ] . Let Q k (cid:16) , , P ( U ) (cid:17) ( t ) be given in the formula (4.8) for t ∈ [0 , a ] and 1 ≤ k ≤ m − . Wedefine φ j ( s ) = , m = 1 , P m − l =1 p ( l ) j ( t j ) l ! s l , m ≥ , for s ∈ [0 , b ] , j = 1 , , (5.6)where p ( l ) j ( t j ) are given by the right hand sides of (4.10) for 1 ≤ l ≤ m − ≤ j ≤ , where q = q = 0 and f = P ( U ) . Clearly, the m th compatibility conditions hold true forthe above q , q , φ , φ , and P ( U ) . From Theorem 4.1, there is a solution v ∈ C m, (Ω)to problem (5 .
2) with the data h Dv, T α s i ◦ α (0 , s ) = φ ( s ) , h Dv, T α s i ◦ α ( a, s ) = φ ( s ) for s ∈ (0 , b ) ,v ◦ α ( t,
0) = 1 √ h Dv, ( T − T ) α t i ◦ α ( t,
0) = 0 for t ∈ (0 , a ) . Moreover, it follows from (4.11) and (2.26) that k v k C m, (Ω) ≤ C k U k C m +1 , (Ω ,T ) , which implies the estimate (1.3) is true. ✷ Proof of Theorem 1.2
Let V = W + w~n, w = h V, ~n i . The regularity of sym DW = − w Π ∈ W , (Ω , IR )implies W ∈ W , (Ω , T ) . Let E , E be a frame field on Ω with the positive orientation and let v = 12 [ ∇ V ( E , E ) − ∇ V ( E , E )] . From Theorem 2.1 v is a solution to problem h D v, Q ∗ Π i = − vκ tr g Π + X ( v ) for x ∈ Ω , (5.7)51here κ tr g Π ∈ C m, (Ω) and X = ( ∇ ~n ) − Dκ ∈ C m − , (Ω , T ) , where C − , (Ω , T ) = L ∞ (Ω , T ) . It is easy to check that ∇ E i V = D E i W + w ∇ E i ~n + [ E i ( w ) − Π( W, E i )] ~n for i = 1 , . Thus v = DW ( E , E ) − DW ( E , E ) ∈ W , (Ω) . From Theorems 4.4, 4.1, and 4.2, there are solutions v n ∈ C m, (Ω) to problem (5.7) suchthat lim n →∞ k v n − v k W , (Ω) = 0 . Let u n = − Q ( ∇ ~n ) − Dv n , u = − Q ( ∇ ~n ) − Dv.
Then u n ∈ C m − , (Ω) . From Theorem 2.1 (see (2.11)), there exist ˆ V n ∈ C m, (Ω , IR ) such that ( ∇ E ˆ V n = v n E + h u n , E i ~n, ∇ E ˆ V n = − v n E + h u n , E i ~n, for n = 1 , , · · · . (5.8)Define V n ( α ( t, s )) = ˆ V n ( α (0 , s )) − ˆ V n ( α (0 , V ( α (0 , Z t ∇ α t ˆ V n dt for n = 1 , , · · · . Thus V n ∈ V (Ω , IR ) ∩ C m, (Ω , IR )) satisfy (1.4). ✷ Proof of Theorem 1.3
As in [5] we conduct in 2 ≤ i ≤ m. Let u ε = i − X j =0 ε j w j be an ( i − m − i +1)+1 , (Ω , IR ) , where w = id and w = V for some i ≥ . Then k X j =0 ∇ T w j ∇ w k − j = 0 for 0 ≤ k ≤ i − . Next, we shall find out w i ∈ C m − i )+1 , (Ω , IR ) such that φ ε = u ε + ε i w i is an i th order isometry. From Theorem 1 . w i ∈ C m − i )+1 , (Ω , IR )to problem sym ∇ w i = −
12 sym i − X j =1 sym ∇ T w j ∇ w i − j k w i k C m − i )+1 , (Ω ,IR ) ≤ C k i − X j =1 sym ∇ T w j ∇ w i − j k C m − i +1) , (Ω ,IR ) ≤ C i − X j =1 k w j k C m − i +1)+1 , (Ω ,IR ) k w i − j k C m − i +1)+1 , (Ω ,IR ) . The conduction completes. ✷ Theorem 1.4 will follow from the density of the Sobolev space and Proposition 5.1below.
Proposition 5.1
Let Ω ⊂ M be a noncharacteristic region of class C , . Then for U ∈ W , (Ω , T ) there exits a solution w ∈ W , (Ω , IR ) to problem sym ∇ w = U. Proof
Consider problem (5.2) with the data (5.3) and (5.4). By (4.9) the first or-der compatibility conditions hold. Since P ( U ) ∈ W , (Ω) , the proposition follows fromTheorems 4.2 and 2.1. ✷ Proof of Theorem 1.7
A recovery sequence can be constructed, based on Theorems1.2 and 1.3, as in the proof of [5, Theorem 6.2]. We present a skeleton of the proof. Forthe further details, see [5].From the density of Theorem 1.2 and the continuity of the functional I with respectto the strong topology of W , , we can assume V ∈ V (Ω , IR ) ∩ C m − , (Ω , IR ) . Step 1
Let ε = √ e h h so ε → , as h → , by assumption (1.6). Therefore, by Theorem1.3, there exists a sequence w ε : Ω → IR , equibounded in C , (Ω , IR ) , for all h > ,u ε = id + εV + ε w ε is a m th isometry of class C , . Then ε m +1 = o ( √ e h ) . Consider the sequence of deformations u h ∈ W , (Ω h , IR ) defined by u h ( x + t~n ) = u ε ( x ) + t~n ε ( x ) + t εd h ( x ) for x + t~n ( x ) ∈ Ω h , where ~n ε ( x ) denotes the unit normal to u ε (Ω) at u ε ( x ) and d h ∈ W , ∞ (Ω , IR ) is suchthat lim h → h / k d h k W , ∞ = 0 andlim h → d h ( x ) = 2 c ( x, sym ( ∇ ( A~n ) − A Π) tan ) for x ∈ Ω , c ( x, F tan ) denotes the unique vector satisfying Q ( x, F tan ) = Q ( F tan + c ⊗ ~n ( x ) + ~n ( x ) ⊗ c ) . We have ~n ε ( x ) = ~n ( x ) + εA~n + O ( ε ) . Step 2
We have E h ( u h ) e h = 1 e h h Z h / − h / Z Ω W ( ∇ h y h ( x + t~n ( x )))(1 + thh tr g Π + t h h κ ) dgdt, where ∇ h y h ( x + t~n ( x )) = ∇ u h ( x + thh ~n ) . Let K h ( x + t~n ( x )) = ( ∇ h y h ) T ∇ h y h − Id . Using the formulas ∇ T u ε ∇ u ε = Id + O ( ε m +1 ) = Id + o ( √ e h ) and hε = √ e h , we have K h tan = 2 t √ e h h ( Id + thh Π) − sym ( ∇ ( A~n ) − A Π)( Id + thh Π) − + o ( √ e h ) , h K h ~n, ~n i = 2 t √ e h h h ~n ε , d h i + o ( √ e h ) , h K h α, ~n i = t √ e h h h∇ u ε ( Id + thh Π) − α, d h i + o ( √ e h ) for α ∈ T x Ω . Then lim h → K h tan √ e h = th sym ( ∇ ( A~n ) − A Π) in L ∞ (Ω h ) , lim h → K h ~n √ e h = 2 th c ( x, sym ( ∇ ( A~n ) − A Π) tan ) in L ∞ (Ω h ) . Step 3
We have W ( ∇ y y h ) e h = 12 Q ( K h √ e h + 1 √ e h O ( | K h | ) + 1 e h o ( | K h | ) . Then the limit (1.8) follows from Step 2. ✷ Compliance with Ethical Standards
Conflict of Interest: The author declares that there is no conflict of interest.Ethical approval: This article does not contain any studies with human participantsor animals performed by the author.