Inequalities for the norms of vector functions in a spherical layer
aa r X i v : . [ m a t h - ph ] O c t Inequalities for the norms of vector functionsin a spherical layerValery V. Denisenko , Semen A. Nesterov Institute of Computational Modelling, Russian Academy of Sciences, Siberian Branch660036 Krasnoyarsk, Russia, [email protected]
Abstract
We consider the vector functions in a domain homeomorphic to a spherical layer boundedby twice continuously differentiable surfaces. Additional restrictions are imposed on the domain,which allow to conduct proofs using simple methods. On the outer and inner boundaries, thenormal and the tangential components of the vector are zero, respectively. For such functions,the integral over the domain of the squared vector is estimated from above via the integral ofthe sum of squared gradients of its Cartesian components. The last integral is estimated throughthe integral of the sum of the squared divergence and rotor. These inequalities allow to definetwo norms equivalent to the sum of the norms of the Cartesian components of vector functionsas the elements of the space W (1)2 (Ω). The integrals over the boundaries of the squared vectorare also estimated. The constants in all proved inequalities are determined only by the shape ofthe domain and do not depend on a specific vector function. The inequalities are necessary forinvestigating the operator of a mixed elliptic boundary value problem. Key words: space of the vector functions, equivalent norms, mixed elliptic boundary valueproblem, spherical layer
Introduction
We consider the set of smooth vector functions P in the domain Ω that is bounded and homeomor-phic to the spherical layer. Its outer Γ and inner γ boundaries are twice continuously differentiablesurfaces and homeomorphic to the sphere. Normal (positive outward direction) and tangent tothe boundary components of vectors are marked with the indices n and τ .The purpose of this work is to prove two inequalities for the vector function P , satisfying theconditions: P n | Γ = 0 , P τ | γ = 0 . (1)The first inequality Z | P | d Ω ≤ C Z | grad P | d Ω , (2)where | grad P | is the sum of squared modulus of the gradients of all Cartesian components of P ,and the constant C is determined only by the shape of the domain Ω and does not depend onthe specific function P .This inequality allows to use its right-hand side as the norm P , which is equivalent to the sumof the norms of the Cartesian components P as the elements of the space W (1)2 (Ω).This inequality is similar to the Friedrichs inequality for scalar functions which are equalto zero at the boundary. If the entire vector P is equal to zero at the boundary, for each of itsCartesian components one can use Friedrichs inequality to obtain the required inequality. However,of interest are the functions with only normal or only tangent components equal to zero at theboundary. 1he second required inequality Z ((rot P ) + (div P ) ) d Ω ≥ C Z | grad P | d Ω , (3)where the constant C is also determined only by the shape of the domain.Similar inequalities are used in the study of the orthogonal decomposition of the space of thevector functions [2] and allow investigating the properties of operators of the elliptic boundaryvalue problems [3].It is easy to check the inequality(rot P ) + (div P ) ≤ | grad P | . Together with (2, 3) this inequality allows to use the left-hand side of (3) as one more normof P , that is equivalent to the sum of the norms of the Cartesian component of P as the elementsof the space W (1)2 (Ω).Both inequalities (2, 3) for the functions with one of the conditions (1), set on the entireboundary of an arbitrary multiply connected domain are proved in the paper [2]. In appliedresearch, mixed boundary value problems sometimes arise when conditions at different sections ofthe boundary differ, for example, have the form (1), and for this case new proofs are required.We also get estimates of the traces of the vector function on the boundary of the domain. To limit ourselves to simple means, we make additional assumptions: both surfaces are smooth,the surface Γ is convex, the curvature of γ is bounded, and we also consider the differences of bothsurfaces to be limited from two spheres with a common center and not too different radii. Let’sgive these constraints a specific look.We consider these surfaces to be defined in spherical coordinates using the functions R Γ ( θ, ϕ )and R γ ( θ, ϕ ), such that0 < R ≤ R γ ( θ, ϕ ) ≤ R , < R Γ ( θ, ϕ ) − R γ ( θ, ϕ ) ≤ δR, R Γ ( θ, ϕ ) ≤ R . (4)It is convenient to write down the convexity condition for the surface Γ in local form. At anarbitrary point of Γ, construct a tangent plane with Cartesian coordinates x, y and direct the z axis along the outward normal. The surface equation can be written as z = ˜ f ( x, y ) (5)with a twice continuously differentiable function ˜ f ( x, y ) which is obtained by passing to localcoordinates from given function R Γ ( θ, ϕ ). At the point x = y = z = 0 the function itself ˜ f ( x, y )and its first derivatives by construction are equal to zero.The next terms of the expansion in its Fourier series form a quadratic form ∂ ˜ f∂x x + 2 ∂ ˜ f∂x∂y xy + ∂ ˜ f∂y y . (6)If this quadratic form is not negative definite, then in a sufficiently small neighborhood, wherethe terms of the series of the higher orders can be neglected, the function on some rays willhave positive values. This means finding the points of the region above the tangent plane, which2ontradicts the convexity of the domain. For the quadratic form to be negative definite, thewell-known inequalities for the coefficients must be satisfied. For (6) they take form: ∂ ˜ f∂x ≤ , ∂ ˜ f∂y ≤ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˜ f∂x∂y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∂ ˜ f∂x ∂ ˜ f∂y . (7)On the boundary γ , we denote a function similar to ˜ f ( x, y ) by f ( x, y ) and require the fulfillmentof the inequality: ∂ f∂x + ∂ f∂y ≤ R , (8)where R is a given positive constant. This condition is satisfied, for example, if the surface γ bounds a convex region and R is the minimum radius of curvature.Boundedness condition for the angles between the normals to the surfaces Γ and γ and theradial direction we write down in a form convenient for use. The radial components of the unitnormal vectors are considered to be bounded from below: n Γ ,r ≥ ξ , − n γ,r ≥ ξ , (9)where ξ is a strictly positive constant.We also require that the scalar product of the normals n Γ and n γ calculated on one ray ispositive: n Γ ( θ, ϕ ) · n γ ( θ, ϕ ) ≥ ξ > . (10)For ξ > / √ q − ξ ≥
0. Therefore, all scalar product > ξ − >
0, and the last expression cansimply be denoted by ξ .We also require that the differences between both surfaces and two spheres with a commoncenter and not too different radii, is limited. Namely, let the constants introduced above satisfythe inequality: 2 R δRξ ξ R R < . (11)In mathematical modeling of the D-layer of the ionosphere, this condition is met with a largemargin, the fraction is less than 0 .
02. If the boundaries γ and Γ are the concentric spheres withradii R and R , then ξ = ξ = 1, R = R , and (11) becomes the condition R /R < . L norm of the trace of a vector functionon the inner part of the boundary In this section, we prove the first of the inequalities I γ | P | dγ ≤ C Z Ω | grad P | d Ω , I Γ | P | d Γ ≤ C Z Ω | grad P | d Ω , (12)where the constants C , C are determined only by the shape of the region, and in the next section- the second one. 3hese inequalities mean the estimates of the L norms of the traces of the vector function P on two sections of the boundary. Without concrete values of the constants, they follow from thetheorems of embedding of the space W (1)2 (Ω) to the space L (Γ) [4], if we additionally require thatthe values of the Cartesian components P averaged over Ω are equal to zero.Consider the ray segment θ = const , ϕ = const from γ to Γ. We fix the normal n Γ at thepoint of intersection of the ray with the surface Γ and denote α ( r ) = n Γ · P . (13)Since the first factor does not vary along the ray, ∂α ( r ) ∂r = n Γ · ∂∂r P . (14)Taking into account that α equals to zero at Γ by the second condition (1), by the Newton-Leibniz formula we obtain at a point on γ : α = Z r = R γ ( θ,ϕ ) r = R Γ ( θ,ϕ ) ∂α ( r ) ∂r dr. We use the Cauchy-Bunyakovsky inequality taking into account the constraint (4), before thelast one: α ≤ δR Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂α ( r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dr. (15)Since n Γ is a unit vector (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂α ( r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂∂r P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | grad P | . This allows to continue the evaluation (15): α ≤ δR Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P | dr. At γ , by the last condition (1), the tangential components of the vector P are equal to zero,therefore P = | P | n γ . Consequently α = | P | n γ · n Γ , and due to (10) | P | ≤ α/ξ . The normal component of the vector also satisfies this inequality, especially since the tangentialones are equal to zero. We strengthen the inequality by introducing a factor that is greater than1 due to (4): P n ≤ δRξ Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P | r R dr. (16)We integrate this inequality over γ and reduce the integration over surface to integration overthe spherical angles using the equality: I γ P n dγ = I γ P n dγR γ ( θ, ϕ ) sin θ dθdϕ R γ ( θ, ϕ ) sin θ dθdϕ, (17)4here the fraction equals to the ratio of the area at γ to the area of the projection of this surfaceelement onto the sphere. By virtue of (9) it does not exceed 1 /ξ .Substituting (16) in (17) and replacing R γ ( θ, ϕ ) with its maximum value R , we obtain anintegral over the domain Ω: I γ P n dγ ≤ R δRR ξ ξ Z Ω | grad P | r sin θ dr dθ dϕ. Since the tangential components of the vector P at γ are equal to zero, this proves the firstinequality (12) with the constant C = R δRR ξ ξ . (18) L norm of the trace of a vector functionon the outer part of the boundary Consider the ray segment θ = const , ϕ = const from γ to Γ. At the point of intersection of theray with γ , we introduce Cartesian coordinates with the z axis along the outward normal to γ .We denote the projections of the vector P on the axis x , y , z as P x , P y , P z , respectively.Taking into account the equality to zero of P x , P y on γ due to the first condition (1), usingthe Newton-Leibniz formula, we obtain at a point on Γ: P x = Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) ∂P x ( r ) ∂r dr, P y = Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) ∂P y ( r ) ∂r dr. Applying the Cauchy-Bunyakovsky inequality and taking into account before the last constraint(4), we obtain: P x ≤ δR Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂P x ( r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dr, P y ≤ δR Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂P y ( r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dr. (19)Since ∂/∂r is one of the components of the vector grad (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂P x ( r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | grad P x | , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂P y ( r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | grad P y | . This allows to continue the evaluation (19). Let’s strengthen inequalities by introducing afactor that is greater than 1 due to (4): P x ≤ δR Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P x | r R dr, P y ≤ δR Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P y | r R dr. (20)We fix the normal n at the point of intersection of the ray with the boundary Γ. Due to thefirst condition (1), the normal component of the vector P is equal to zero at Γ. Therefore, at apoint at Γ: ( P · n ) = P x n x + P y n y + P z n z = 0 . (21)Hence, P z = (( P x n x + P y n y ) /n z ) . (22)5pplying the Cauchy-Bunyakovsky inequality for the numerator, we obtain( P x n x + P y n y ) ≤ (cid:16) P x + P y (cid:17) (cid:16) n x + n y (cid:17) . (23)Due to (10) n z ≥ ξ . So n x + n y = 1 − n z ≤ − ξ . (24)The (20, 22, 23, 24) result into P z ≤ δR (1 − ξ ) ξ Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) (cid:16) | grad P x | + | grad P y | (cid:17) r R dr. (25)Since all Cartesian components (20, 25) are estimated, the vector P satisfies the estimate | P | ≤ δRξ R Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P | r dr. (26)We integrate the inequality (26) over Γ and reduce the integration over the surface to integra-tion over the spherical angles using the equality: I Γ | P | d Γ = I | P | d Γ R ( θ, ϕ ) sin θ dθdϕ R ( θ, ϕ ) sin θ dθ dϕ, (27)where the fraction equals to the ratio of the area at Γ to the area of the projection of this surfaceelement onto the sphere. By virtue of (9) it does not exceed 1 /ξ .Substituting (26) in (27), and replacing R Γ ( θ, ϕ ) with its maximum value R , we obtain in theright-hand side the integral over the domain Ω: I Γ | P | d Γ ≤ δRR R ξ ξ Z Ω | grad P | r sin θ dr dθ dϕ. Thus, we have proved the second inequality (12) with the constant C = δRR R ξ ξ . L norm of a vector function In this section, we prove the inequality (2).Consider the ray segment θ = const , ϕ = const from γ to Γ, and divide it into two parts withsome point A inside the domain Ω. Let us fix the normal n at the point of intersection of the raywith the surface Γ. Arguments similar to those used in Section 2 give an estimate for α at thepoint A α ≤ δR Z r = R Γ ( θ,ϕ ) r = R A ( θ,ϕ ) | grad P | r R dr. (28)Now consider a segment of the same ray from the opposite boundary γ to the same point A .The point of intersection of the ray with γ is taken as the origin of the Cartesian coordinates.The z axis is directed along the outward normal to γ . Reasoning similar to those used in Section3 gives an estimate for P x , P y at the point AP x ≤ δR Z r = R A ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P x | r R dr, P y ≤ δR Z r = R A ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P y | r R dr. (29)6ince the scalar product does not exceed the product of the modulus of the vectors, | n | = 1,and the integration in (28, 29) is made over two different parts of the segment under consideration,at the point A | P · n | = | P x n x + P y n y + P z n z | ≤ s δR Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P | r R dr. This implies | P z | ≤ n z s δR Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P | r R dr + | P x n x + P y n y | ! . (30)Since the scalar product does not exceed the product of the absolute values of the vectors, | n | = 1, and due to (9) n z ≥ ξ , | P x n x + P y n y | ≤ q − ξ q P x + P y . Taking this inequality into account, we square both sides (30) and get P z ≤ ξ s δR Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P | r R dr + q − ξ q P x + P y ! . (31)For the right-hand side (31), we apply the inequality ( a + b ) ≤ a + b ), which is valid forany real numbers P z ≤ ξ δR Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P | r R dr + (1 − ξ )( P x + P y ) ! . (32)Due to (9) 1 − ξ ≤
1. Therefore, using (29) for (32), we get P z ≤ δRξ R Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P | r dr, (33)Using (29, 33), estimate the whole vector P at the point A | P | ≤ δRξ R Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P | r dr + δRR Z r = R A ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P | r dr. (34)Due to (9) 2 + ξ ≤ | P | ≤ δRξ R Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P | r dr. We multiply this inequality by r and integrate it over r : Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | P | r dr ≤ δRξ R Z ˜ r = R Γ ( θ,ϕ )˜ r = R γ ( θ,ϕ ) | grad P | ˜ r d ˜ r Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) r dr. Here the right-hand side (34) is immediately removed from the integral, since it does notdepend on r . Due to (4) r ≤ R , and the integration interval does not exceed δR . Consequently Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | P | r dr ≤ δRR ξ R ! Z r = R Γ ( θ,ϕ ) r = R γ ( θ,ϕ ) | grad P | r dr. Multiplying both sides of this inequality by sin θ , and integrating over θ and ϕ , we prove theinequality (2) with the constant C = 3 ( δRR / ( ξ R )) . Estimate of the L norm of the gradient of a vector func-tion In the paper [1] an identity is given, which is easy to verify for multiply connected domains: Z ((rot P ) + (div P ) ) d Ω = Z (grad P ) d Ω ++ I Γ ((div P ) P n − ( P grad) P n ) d Γ + I γ ((div P ) P n − ( P grad) P n ) dγ. (35)Using the expression for the surface equation in the form (5), the integrand in the integral overthe boundary Γ with simple but cumbersome calculations can be converted to the form − ( ∂ ˜ f∂x + ∂ ˜ f∂y ) P z − ∂ ˜ f∂x P x − ∂ ˜ f∂x∂y P x P y − ∂ ˜ f∂y P y + ( ∂P x ∂x + ∂P y ∂y ) P z − ∂P z ∂x P x − ∂P z ∂y P y . (36)Because the normal compponent of the P is zero (1), P z = 0, ∂P z /∂x = 0, ∂P z /∂y = 0, andin (36) on Γ only the quadratic form remains − ∂ ˜ f∂x P x − ∂ ˜ f∂x∂y P x P y − ∂ ˜ f∂y P y . (37)It only differs in the sign of the coefficients from (6), and therefore is positive definite. There-fore, the integral over Γ in (35) is non-negative. On γ , the tangential components of P are equalto zero (1). Therefore P x = P y = 0, ∂P x /∂x = 0, ∂P y /∂y = 0, and in a similar (36) expression,only − ( ∂ f∂x + ∂ f∂y ) P z . (38)Using the resulting expression for the integrand (38) and the constraint (8), estimate theintegral over γ in (35) from above: (cid:12)(cid:12)(cid:12)(cid:12)I γ ((div P ) P n − ( P grad) P n ) dγ (cid:12)(cid:12)(cid:12)(cid:12) ≤ R I γ P n dγ. (39)Taking into account the first inequality (12), we obtain the estimate (cid:12)(cid:12)(cid:12)(cid:12)I γ ((div P ) P n − ( P grad) P n ) dγ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C R Z Ω | grad P | d Ω . (40)Since the integral over Γ, as already shown, is non-negative, and the integral over γ is estimatedfrom above in the inequality (40), taking into account the expression C (18), we obtain Z ((rot P ) + (div P ) ) d Ω ≥ (1 − R δRξ ξ R R ) Z (grad P ) d Ω . (41)For this estimate to be meaningful, the factor in front of the integral must be positive, i.e. thefraction must be less than 1. For this, the condition (11) was imposed.8 Conclusion
Thus, for vector functions P satisfying the mixed boundary conditions of the form (1) in a multiplyconnected domain, inequalities are proved which are necessary in the study of the operators ofelliptic boundary value problems. The inequality (2) estimates from above the L norm of a vectorfunction in terms of L the norm of its gradient. The inequality (3) estimates from above the L norm of the gradient of a vector function through the sum of the L norms of its rotor anddivergence. The inequalities (12) for the L norms of the traces of such vector functions on bothparts of the boundary are also obtained.Note that the condition of convexity of the outer part of the boundary Γ and the constraints(7) can be weakened, only slightly complicating the proof. Then the integral over Γ in (35) willno longer be non-negative, but it can be estimated in the same way as the integral over γ . Thiswill lead to the appearance of an additional negative term in the factor before the integral (41),and, therefore, will require a stronger constraint for the constants (11).It should also be said that the used restrictions on the shape of the domain are associated onlywith the possibility to obtain simple proofs, and the inequalities themselves, apparently, are validfor the same multiply connected domains of general form, for which these inequalities were provedin the work [2] under one of the conditions (1) on the whole boundary. Acknowledgments
The research is supported by Russian Foundation for Basic Research (project 18-05-00195).