Point estimates for Green s matrix to boundary value problems for second order elliptic systems in a polyhedral cone

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1 Maz ya, V. G., Roßmann, J.: Estimates for Green s matrix 1 ZAMM Z. angew. Math. Mech , 1 30 Maz ya, V. G.; Roßmann, J. Point estimates for Green s matrix to boundary value problems for second order elliptic systems in a polyhedral cone In this paper we are concerned with boundary value problems for general second order elliptic equations and systems in a polyhedral cone. We obtain point estimates of Green s matrix in different areas of the cone. The proof of these estimates is essentially based on weighted L 2 estimates for weak solutions and their derivatives. As examples, we consider the Neumann problem to the Laplace equation and the Lamé system. Keywords: second order elliptic systems, Green s matrix, nonsmooth domains MSC 1991: 35J25, 35J55, 35C15, 35Q72 1 Introduction We deal with the Dirichlet, Neumann and mixed problems for elliptic systems of second order equations in a polyhedral cone K. Our main goal is to obtain point estimates for Green s matrix. In a forthcoming work we will prove, by means of such estimates, solvability theorems and regularity assertions in weighted L p Sobolev and Hölder spaces. As is well-known, the nonsmoothness of the boundary causes singularities of the solutions at the edges even if the right-hand side of the differential equation and the boundary data are smooth. Therefore, Green s matrix Gx, ξ is singular not only at the diagonal but also for x or ξ near the vertex or an edge. For a cone without edges these singularities were described by Maz ya and Plamenevskiĭ [12] in terms of eigenvalues, eigenfunctions and generalized eigenfunctions of a certain operator pencil. The presence of edges on the boundary makes the investigation of Green s functions more difficult. In [10] Maz ya and Plamenevskiĭ obtained estimates for Green s functions of boundary value problems in a dihedral angle. The results in [10] are applicable, e.g., to the Dirichlet problem for elliptic equations but not to the Neumann problem. Green s functions for the Dirichlet problem in polyhedral domains were studied in papers by Maz ya and Plamenevskiĭ [13] Lamé and Stokes systems, Maz ya and Roßmann [15] strongly elliptic 2m order equations. Concerning the Neumann problem for the Laplace equation in domains with edges, we refer to the preprints of Solonnikov [22], Grachev and Maz ya [5]. We outline the main results of our paper. Let K = {x R 3 : ω = x/ x Ω} be a polyhedral cone with faces Γ j = {x : x/ x γ j } and edges M j, j = 1,..., n. Here Ω is a curvilinear polygon on the unit sphere bounded by the sides γ 1,..., γ n. Suppose that K coincides with a dihedral angle D j in a neighborhood of an arbitrary edge point x M j. By S we denote the set M 1 M n {0}. We consider the boundary value problem L x u = 3 A i,j xi xj u = f in K, 1.1 i, u = g j on Γ j for j J 0, B x u = A i,j n j xi u = g k on Γ k for k J i, where A i,j are constant l l matrices such that A i,j = A j,i, J 0 J 1 = {1,..., n}, J 0 J 1 =, u, f, g are vector-valued functions, and n 1, n 2, n 3 denotes the exterior normal to K\S. Weak solutions of problem can be defined by means of the sesquilinear form 3 b K u, v = A i,j xi u xj v dx, 1.4 K i, where u v is the scalar product in C l of the vectors u and v. We denote by H the closure of the set {u C 0 K l : u = 0 on Γ j for j J 0 } with respect to the norm u H = K 3 1/2 xj u 2 C dx. 1.5 l

2 2 ZAMM Z. angew. Math. Mech Here C 0 K is the set of all infinitely differentiable functions on K with compact supports. From the above assumptions on the coefficients A i,j it follows that b K u, v = b K v, u for u, v H. Throughout this paper, it will be assumed that the form b K is H-coercive, i.e., b K u, u c u 2 H for all u H. 1.6 By Lax-Milgram s lemma, this implies that the variational problem b K u, v = F, v K for all v H 1.7 is uniquely solvable in H for arbitrary F H. Here, K denotes the scalar product in L 2 K l or its extension to H H. In Section 2 we consider the boundary value problem in a dihedron D = K R, where K is an infinite angle in the x 1, x 2 -plane with opening θ. The main goal of this section is the estimation of Green s matrix. We give here the estimates in the case of the Neumann problem to the Laplace equation, which was also considered in [22]. Let α = α 1, α 2, α 3 and γ = γ 1, γ 2, γ 3 be arbitrary multi-indices. Then x α γ ξ Gx, ξ c x ξ 1 α γ x min0,π/θ α1 α 2 ε ξ min0,π/θ γ1 γ 2 ε x ξ x ξ for x ξ min x, ξ, where x = x 1, x 2, ξ = ξ 1, ξ 2, and ε is an arbitrarily small positive number. For x ξ < min x, ξ there is the estimate α x γ ξ Gx, ξ c x ξ 1 α γ. The same inequalities hold for Green s matrix of the Neumann problem to the Lamé system if θ < π, while in the case θ > π the number π/θ in the exponent has to be replaced by ξ + θ/θ, where ξ + θ is the smallest positive root of the equation sin ξ ξ + sin θ θ = 0. For the proof of these inequalities, we use weighted L 2 estimates for weak solutions and their derivatives. Section 3 concerns the parameter-dependent boundary value problems 1.8 Lλ u = f in Ω, u = g j on γ j, j J 0, Bλ u = g k on γ k, k J generated by problem on the intersection Ω of the cone K with the unit sphere S 2. Here Lλu = ρ 2 λ L x ρ λ uω, Bλu = ρ 1 λ B x ρ λ uω, 1.10 ρ = x, and ω = x/ x. Let Aλ be the operator of problem 1.9. We prove that problem 1.9 is uniquely solvable in a certain class of weighted Sobolev spaces for all λ, except finitely many, in a double angle of the complex plane containing the imaginary axis. Furthermore, we obtain an a priori estimate of the solution. In Section 4, by means of these results, solvability theorems for the boundary value problem in weighted Sobolev spaces are obtained. In particular, we prove the existence of weak solutions u Vβ 1Kl, where Vβ 1 K is the weighted Sobolev space with the norm u V 1 β K = x 2β u 2 + x 2 u 2 1/2. dx 1.11 K Here, for example, by a weak solution of the Neumann problem we mean a vector function u V 1 β Kl satisfying b K u, v = F, v K for all v V 1 βk l, where F is a given continuous functional on V β 1 Kl. We prove that the absence of eigenvalues of the pencil A on the line Re λ = β 1/2 ensures the unique existence of a weak solution u Vβ 1Kl. Furthermore, we prove regularity assertions for the solution. For example, we conclude from our results that the second derivatives of the solution u H of the Dirichlet and Neumann problems for the Laplace equation and other second order differential equations, including the Lamé system are square summable if the angles at the edges are less than π and there are no eigenvalues of the pencil A with positive real part 1/2. In particular, the W 2 regularity holds for the Dirichlet problem to the Laplace equation and to the Lamé system if K is convex. This follows from the monotonicity of real eigenvalues of the pencil A in the interval [0, 1] see, e.g., the monograph by Kozlov, Maz ya and Roßmann [8, Ch.2,3]. For the Neumann problem to the Laplace equation the W 2 regularity was proved by Dauge [3, 4].

3 Maz ya, V. G., Roßmann, J.: Estimates for Green s matrix 3 The absence of eigenvalues of the pencil A on the line Re λ = β 1/2 guarantees also the existence of a unique solution Gx, ξ of the problem L x Gx, ξ = x ξ I l, x, ξ K, 1.12 Gx, ξ = 0, x Γ j, ξ K, j J 0, 1.13 B x Gx, ξ = 0, x Γ j, ξ K, j J Gx, ξ belongs to the space Vβ 1Kl l I l denotes the l l identity matrix such that the function x ζ x ξ rξ for every fixed ξ K and for every smooth function ζ on 0, equal to one in 1, and to zero in 0, 1 2. We obtain point estimates for the derivatives of Gx, ξ in different areas of K K. For example, Green s function of the Neumann problem to the Laplace equation satisfies the following estimate for x < ξ /2: α x γ ξ Gx, ξ c x Λ α +ε j ξ 1 Λ γ ε rj x min0,π/θj α ε x rj ξ min0,π/θj γ ε. ξ j Here Λ < Re λ < Λ + is the widest strip in the complex plane containing the line Re λ = β 1/2 which is free of eigenvalues of the pencil A, θ j is the angle at the edge M j, r j is the distance to M j, and ε is an arbitrarily small positive number. The same estimate holds for the Lamé system if θ j < π for j = 1,..., n. If θ j > π, then the number π in the exponent has to be replaced by ξ + θ j. In the case β = 0, when Λ + = 0 and Λ = 1, these estimates can be improved. 2 The boundary value problem in a dihedron Let D be the dihedron {x = x, x 3 : x = x 1, x 2 K, x 3 R}, where K is the angle {x = x 1, x 2 : 0 < r <, 0 < ϕ < θ}. Here r, ϕ are the polar coordinates in the x 1, x 2 -plane. Furthermore, let Γ = {x : ϕ = 0} and Γ + = {x : ϕ = θ} be the sides of D, and let d ± {0, 1}. We consider the boundary value problem L x u = f in D, d ± u + 1 d ± B x u = g ± on Γ ±. 2.1 This means, for d + = d = 1 we are concerned with the Dirichlet problem, for d + = d = 0 with the Neumann problem, and for d + d with the mixed problem. We denote by H D the closure of the set {u C 0 D l : d ± u = 0 on Γ ± } with respect to the norm 1.5, where K is replaced by D, and by b D the sesquilinear form b D u, v = Suppose again that b D u, u c D i, Then the variational problem D 3 A i,j xi u xj v dx. 2.2 u 2 C l dx for all u H D. 2.3 b D u, v = F, v D for all v H D 2.4 has a unique solution u H D for arbitrary F HD. A large part of this section deals with the regularity of weak solutions. For the Dirichlet and mixed problems, which are handled at the end of the section, we give only the formulation of a theorem which follows from results of Maz ya and Plamenevskiĭ [9], Nazarov and Plamenevskiĭ [18]. The more complicated case of the Neumann problem is studied in Sections The results here were partially obtained by Zajaczkowski and Solonnikov [23], Nazarov [16, 17], Roßmann [20], Nazarov and Plamenevskiĭ [18]. The proof of point estimates for Green s matrix in this section is essentially based on weighted L 2 estimates for weak solutions and their derivatives. As examples, we consider the Neumann problem for the Laplace equation and the Lamé system.

4 4 ZAMM Z. angew. Math. Mech Weighted Sobolev spaces in a dihedron and in an angle Let > 1. Then L k D denotes the closure of C 0 D with respect to the norm u L k D = D α =k r 2 α x u 2 dx 1/2, where r = x = x x 2 2 1/2. Furthermore, we set W k D = k L j D. For arbitrary real let V k D be the closure of C 0 D\M with respect to the norm u V k D = D α k r 2 k+ α α x u 2 dx 1/ Analogously, we define the spaces L k K, V kk and W k K for a plane angle K with vertex in the origin then in the above norms D has to be replaced by K. By Hardy s inequality, every function u C0 D satisfies the inequality D r 2 1 u 2 dx c D r 2 u 2 dx 2.6 for > 0 with a constant c depending only on. Consequently, the space L k D is continuously imbedded into L k 1 1 D if > 0. If > k 1, then Lk D = V k D. Furthermore, from Hardy s inequality it follows that D x x 0 2 ux 2 dx c u 2 L 2 D l 2.7 for every u H D and for an arbitrary point x 0 on the edge M of D. This means that any vector function u H D is square integrable on every bounded subset of D. From 2.6 and 2.7 we conclude that D r 2 φu 2 dx c u 2 L 2D l. 2.8 for > 1 if u H D and φ is a function in C 1 D with compact support. The spaces of the traces of functions from L k D, V k are denoted by L k 1/2 Γ ±, V k 1/2 Γ ± and W k 1/2 u L k 1/2 Γ ± = inf { v L k D : v L k D, v Γ ± = u }. Analogously, the norms in V k 1/2 by see [9, Le.1.4] u = + + γ ± R R γ ± Γ ± R Γ ± and W k 1/2 D and W kd, k 1, on the sides Γ± = γ ± R of D Γ ±, respectively. The norm in L k 1/2 Γ ± is defined as Γ ± are defined. An equivalent norm in V k 1/2 Γ ± is given r 2 x k 1 3 ur, x 3 y k 1 3 r, y 3 2 dx 3 dy 3 x 3 y 3 2 dr r1 r k 1 γ ± ur 1, x 3 k 1 r ur 2, x 3 2 dr 1 dr 2 r 1 r 2 2 dx 3 k 1 1/2. r 2 k+j+1 rur, j x 3 p dr dx For > k 1 this is also an equivalent norm in L k 1/2 Γ ±.

5 Maz ya, V. G., Roßmann, J.: Estimates for Green s matrix The operator pencil corresponding to the boundary value problem Let H 0,θ = {u W 1 0, θ l : d u0 = d + uθ = 0}, where W 1 denotes the usual Sobolev space and d ± are the numbers introduced in the beginning of this section. Furthermore, let a K u, v; λ = 1 log 2 K 1< x <2 2 A i,j xi U xj V dx, i, where U = r λ uϕ, V = r λ vϕ, u, v H 0,θ, λ C. Aλ : H 0,θ H0,θ by Aλu, v = ak u, v; λ, u, v H 0,θ. The form a K, ; λ generates a continuous operator Here, denotes the scalar product in L 2 0, θ l. As is known, the spectrum of the pencil A consists of isolated points, the eigenvalues. The line Re λ = 0 contains no eigenvalues if d + d or d + = d = 1. In the case d + = d = 0 the case of the Neumann problem, the line Re λ = 0 contains the single eigenvalue λ = 0. The eigenvectors corresponding to this eigenvalue are constant vectors. Every of these eigenvectors has exactly one generalized eigenvector see [8, Ch.12]. We set L x, 0 = 2 A i,j xi xj, B x, 0 = i, and denote by γ ± be sides of K. Remark 2.1. The vector function u = r λ 0 s k=0 2 A i,j n j xi i, L x, 0 u = 0 in K, d ± u + 1 d ± B x, 0 u = 0 on γ ± 1 k! log rk v s k ϕ is a solution of the problem if and only if λ 0 is an eigenvalue of the pencil Aλ and v 0, v 1,..., v s is a Jordan chain corresponding to this eigenvalue see [8, Le ]. We denote by λ 1 the eigenvalue of the pencil Aλ with smallest positive real part and by µ 1 its real part. 2.3 Regularity results for the solution of the Neumann problem Let d + = d = 0. We assume that F is a functional on H D which has the form F, v D = f v dx + g ± v dσ ±, v H D, 2.10 D ± Γ ± where f L 0 Dl, g ± L 1/2 Γ ± l, 0 < < 1. Then the solution of 2.4 belongs to the Sobolev space W 2 and satisfies the equations loc D L x u = f in D, B x u = g ± on Γ ± Note that the right-hand side of 2.10 always defines a functional on H D if f L 0 Dl, g ± L 1/2 Γ ± l, and the supports of f and g ± are compact. For the first term on the right of 2.10, this can be easily proved by means of 2.8. Furthermore, we have L 1/2 Γ ± = V 1/2 Γ ± for > 0 and, due to the equivalence of the norm in V k 1/2 Γ ± to 2.9, r 2 1 g ± 2 dσ ± c g ± 2. Γ ± V 1/2 Γ ± This implies for arbitrary φ, φ = 1 on supp g ± g ± v dσ ± 2 c r 2 1 g ± 2 dσ ± r 1 2 φv 2 dσ ± Γ ± Γ ± Γ ± The following lemma can be found in [9, Le.3.1]. c g ± 2 V 1/2 Γ ± l φv 2 V 1/2 1 Γ± l c g± 2 L 1/2 Γ ± l φv 2 H D.

6 6 ZAMM Z. angew. Math. Mech Lemma 2.1. Let g ± V l+d± 3/2 Γ ± l, where l 1 if d + = d = 1, l 2 else. Then there exists a vector function u V ldl such that d ± u + 1 d ± Bu = g ± on Γ ± and u V l D l c ± g ± l+d V ± 3/2 Γ ± l with a constant c independent of g + and g. Since V 1/2 Γ ± = L 1/2 Γ ± for > 0 and V 2D L2 D, we conclude that for all g± L 1/2 Γ ± l there exists a vector function v L 2 Dl such that B x v = g ± on Γ ±. For the proof of the following lemma we refer to [23] and [20] for general elliptic problems see also [17, 18]. Lemma 2.2. Let φ, ψ be infinitely differentiable functions on D with compact supports such that ψ = 1 in a neighborhood of supp φ. If u H D is a solution of 2.4 and F is a functional of the form 2.10, where ψf L 0 Dl and ψg ± L 1/2 Γ ± l, max1 µ 1, 0 < < 1, then φu L 2 Dl and φu L 2 D c ψf l L 0 D + ψg l 1/2 L + ψu Γ ± l HD 2.12 ± Corollary 2.1. Let max1 µ 1, 0 < < 1. Then for every u L 2 Dl the estimate u L 2 D c L l x u L 0 D + B l x u 1/2 L Γ ± l ± is valid. Here the constant c is independent of u. Proof: Due to Lemma 2.1, we may assume, without loss of generality, that B x u = 0. If the support of u is contained in the ball x 1, then by Lemma 2.2, we have u L 2 D l c L x u L 0 D l + u L 1 0 Dl Let supp u be contained in the ball x < N. Then the support of the function vx = unx is contained in the unit ball x 1. Furthermore, B x v = 0 on Γ ±. Therefore, v satisfies From this inequality, by means of the coordinate change x = y/n, one obtains u L 2 D c L l x u L 0 D + N 1 u l L 1 0 D l with the same constant c as in The result follows. The following theorem generalizes Lemma 2.2. Theorem 2.1. Let φ, ψ be the same functions as in Lemma 2.2. If u H D is a solution of 2.4 and the functional F has the form 2.10, where ψ x j 3 f L 0 Dl and ψ x j 3 g ± L 1/2 Γ ± l for j = 0,..., k, max1 µ 1, 0 < < 1, then φ x j 3 u L 2 Dl for j = 0,..., k and k k k φ x j 3 u L 2 D c ψ j l x 3 f L 0 D + ψ j l x 3 g ± 1/2 L + ψu Γ ± l L 1 D l ± 2.14 with a constant c independent of u. Proof: We prove the theorem by induction in k. For k = 0 the assertion follows from Lemma 2.2 and from the unique solvability of problem 2.4 in H D. Suppose the theorem is proved for k 1. Then, under our assumptions on F, we have χ x j 3 u L 2 Dl for j = 0,..., k 1. Let v = x k 1 3 u. Then φv L 2 Dl. We consider the vector function v h x = h 1 vx, x 3 + h vx, x 3, where h is a sufficiently small real number. Obviously, v h is a solution of the problem Lv h = Φ h in D, Bv h = Ψ ± h on Γ ±, where Φ = x k 1 3 f, Ψ ± = x k 1 3 g ±. Consequently, φv h L 2 D c χφ l h L 0 D + χψ ± l h + χv L 1/2 Γ ± l h L 1 D 2.15 ±

7 Maz ya, V. G., Roßmann, J.: Estimates for Green s matrix 7 with a constant c independent of h. Here χφ h = χφ h χ h Φ and, for sufficiently small h, χφ h 2 L = r 2 h 2 χφx, x h χφx, x 3 2 dx Dl = χ h Φ 2 L c ψ k 1 0 Dl x 3 f 2 L 0 Dl D 1 r 2 χφ x, x 3 + th dt 2 dx D 0 x 3 c ψ x k 1 3 f 2 L + ψ k 0 Dl x 3 f 2 L, 0 Dl Analogously, χψ ± h c ψ k 1 L 1/2 Γ ± l x 3 g ± 2 + L 1/2 Γ ± l ψ k x 3 g ± 2. L 1/2 Γ ± l D r 2 x3 χxφx 2 dx For the proof of the last inequality one can use the equivalence of the norm in L 1/2 Γ ± with the norm 2.9. Furthermore, χv h L 1 D c η x k 1 3 u L 1 D + l η k x 3 u L 1 D, 2.16 l where η is a smooth function such that η = 1 in a neighborhood of supp χ and ψ = 1 in a neighborhood of supp η. Since the theorem was assumed to be true for k 1, the right-hand side of 2.16 is majorized by the right-hand side of Consequently, the limit as h 0 of the left-hand side of 2.15 is majorized by the right-hand side of This proves the theorem. W l+k 3/2 +k Lemma 2.3. Let u be a solution of problem 2.11 such that ψu W ldl, ψf W l+k 2 +k D l and ψg ± Γ ± l, l 1, > 1. Here φ, ψ are the same functions as in Lemma 2.2. Then φu W l+k +k Dl and φu W l+k c ψf +k Dl W l+k 2 +k D + l ± ψg ± W l+k 3/2 +k Γ ± l + ψu W l D l Proof: By [21, Cor.2,Rem.2], the vector function φu W ldl admits the representation φu = v + w, where v V ldl and w W l+k +k Dl. Let first k = 1. Then Lv = φf + [L, φ]u Lw W l 1 +1 Dl V l 2 D l V l 1 +1 Dl here [L, φ] = Lφ φl denotes the commutator of L and φ and, analogously, Bv = φg ± + [B, φ]u Bw V l 1/2 +1 Γ ± l. Using [9, Th.10.2], we obtain v V l+1 +1 Dl and, therefore, φu W l+1 +1 Dl. This proves the lemma for k = 1. Repeating this argument, we obtain the assertion for k Higher regularity of the solution to the Neumann problem We improve the results of the previous subsection for the case µ 1 > 1. Let us consider first the Neumann problem in the plane angle K. Lemma 2.4. Let the integer k > 0 be not an eigenvalue of the pencil Aλ. Then for arbitrary homogeneous polynomials p k 2, q ± k 1 of degrees k 2 and k 1, respectively p k 2 = 0 if k = 1 there exists a homogeneous polynomial p k of degree k such that L x, 0 p k = p k 2 in K, B x, 0 p k = q ± k 1 on γ± Proof: Let p k 2 = k 2 b j x j 1 xk 2 j 2 and q ± k 1 γ ± = c ± r k 1 with b j, c ± C l be given. Inserting p k = k a j x j 1 xk j into 2.18 and comparing the coefficients of x j 1 xk 2 j 2 and r k 1, respectively, we get a linear system of k + 1 equations with k + 1 unknowns a 0, a 1,..., a k. Since k is not an eigenvalue of the pencil Aλ, the corresponding homogeneous system has only the trivial solution see Remark 2.1. Therefore, there exists a unique polynomial 2.19 satisfying 2.18.

8 8 ZAMM Z. angew. Math. Mech Lemma 2.5. Let u W l 1 K l be a solution of the problem L x, 0 u = f in K, B x, 0 u = g ± on γ ± 2.20 with f W l 2 K l, g ± W l 3/2 γ ± l, l 2, 0 < < l 1, not integer. Suppose that the strip l 2 Re λ l 1 does not contain eigenvalues of the pencil Aλ. Then u W lkl and u W l K l c u W l 1 K l + f W l K l + ± with a constant c independent of u. g ± l 3/2 W γ ± l Proof: Let k = l 1 be the greatest integer less than l 1. The vector function u has continuous derivatives up to order k 1 at the point x = 0 see [7, Le.7.1.3]. We denote by p k 1 the Taylor polynomial of degree k 1 of u and by ζ a smooth cut-off function equal to one near the origin and to zero outside the unit ball. Then v = u ζp k 1 belongs to V l 1 K l see [7, Th.7.1.1]. Consequently, L x, 0 v = f L x, 0 ζp k 1 W l 2 K l V l 3 K l, B x, 0 v γ ± = g ± B x, 0 ζp k 1 γ ± W l 3/2 K l V l 5/2 K l. By [7, Th.7.1.1], there are the representations L x, 0 v = ζp k 2 + F in K, B x, 0 v = ζq ± k 1 + G± on γ ±, where p k 2, q± l 2 k 1 are homogeneous polynomials of degrees k 2 and k 1, respectively, F V K l, G ± V l 3/2 γ ± l. By Lemma 2.4, there exists a homogeneous polynomial p k of degree k such that L x, 0p k = p k 2 in K and B x, 0p k = q± k 1 on γ±. Then v ζp k V l 1 K l, L x, 0 v ζp k V l 2 K l, B x, 0 v ζp k γ ± V l 3/2 γ ± l. Applying [7, Th.6.1.4], we obtain v ζp k V lkl and, therefore, u W lkl. Furthermore, the desired estimate holds. We prove an analogous result for the problem in the dihedron D. Lemma 2.6. Let u be a solution of problem 2.11, and let φ, ψ be smooth functions on D with compact supports such that φψ = φ. Suppose that ψu W l 1 D l, ψ x3 u W l 1 D l, ψf W l 2 D l, ψg ± W l 3/2 Γ ± l, 0 < < l 1, is not integer, and the strip l 2 Re λ l 1 does not contain eigenvalues of the pencil Aλ. Then φu W ldl and 1 φu W l D c ψ j l x 3 u W l 1 D + ψf l W l 2 D + l ± Here the constant c depends only on the C l norm of ζ. Proof: From the equation L x, x3 u = f it follows that L x, 0 φu = F, where F = φf + φl 1 x3 u + [L x, 0, φ] u. ψg ± l 3/2 W Γ ± l Here [L x, 0, φ] = L x, 0 φ φl x, 0 is the commutator of L x, 0 and φ, and L 1 is a first order differential operator with constant coefficients, L 1 x3 u = L x, 0 L x, x3 u. An analogous representation holds for G ± = B x, 0 φu Γ ±. For almost all x 3 we have φ, x 3 u, x 3 W l 1 K l. Furthermore, by the conditions of the lemma, F, x 3 W l 2 K l and G ±, x 3 W l 3/2 γ ± 3. Consequently, by Lemma 2.5, we obtain φ, x 3 u, x 3 W lkl and φ, x 3 u, x 3 2 W dx l 3 c φ, x 3 u, x F, x R W l 1 K R 3 2 W l 2 K l + G ±, x 3 2 dx W l 3/2 γ ± l 3. ± Here the right-hand side of the last inequality can be estimated by the right-hand side of This together with the assumption that ψ x3 u W l 1 D l implies the assertion of the lemma.

9 Maz ya, V. G., Roßmann, J.: Estimates for Green s matrix 9 Theorem 2.2. Let u H D be a solution of problem 2.11, and let φ, ψ be smooth functions on D with compact supports such that ψ = 1 in a neighborhood of supp φ. We suppose that ψf W l 2 D l, ψg ± W l 3/2 Γ ±, l 2, is not integer, and maxl 1 µ 1, 0 < < l 1. Then φu W ldl. Proof: We prove the theorem by induction in l 1. Here s denotes the greatest integer less than s. 1 If l 1 = 0, then max1 µ 1, 0 < l+2 < 1, ψf W l+2 0 Dl, ψg ± W 1/2 l+2 Γ± l. Consequently, according to Theorem 2.1, we have χu W l+2 2 Dl, where χ is a smooth function equal to one near supp φ such that ψ = 1 near supp χ. Applying Lemma 2.3, we obtain φu W ldl. 2 Let l 1 = 1. Then max2 µ 1, 0 < l + 3 < 1 < 1 and, by means of Theorem 2.1, we obtain χ x j 3 u W l+3 2 Dl for j = 0, 1. Consequently, it follows from Lemmas 2.6 and 2.3 that φu W ldl. 3 Let k < l 1 < k + 1, where k is an integer, k 2. We assume that the theorem is proved for l 1 < k. Then, by the induction hypothesis, χu W l 1 D l, l 4, and χ x3 u W l 2 D l H D. Since ψ x3 f W l 3 D and ψ x3 g ± W l 5/2 Γ ± l, we obtain, by the induction hypothesis, that χ x3 u W l 1 D l. By the assumptions of the lemma, there are no eigenvalues of the pencil Aλ in the strip 0 < Re λ l 1. Thus, Lemma 2.6 implies φu W ldl. The proof is complete. W l 3/2 Corollary 2.2. Let u H D be a solution of problem 2.11, where ψ x j 3 f W l 2 D l and ψ x j 3 g ± Γ ± l for j = 0,..., k, is not integer, maxl 1 µ 1, 0 < < l 1. Then φ x j 3 u W ldl and k k φ x j 3 u W l D c ψ j l x 3 f W l 2 D + l k ± ψ j x 3 g ± W l 3/2 Γ ± l + ψu L 1 D l. Proof: Let first l 1 < 1. Then max1 µ 1, 0 < l + 2 < 1, W l 2 K W l+2 0 l 3/2 K and W Γ ± W 1/2 l+2 Γ±. Consequently, by Theorem 2.1, we have χ x j 3 u W l+2 2 Kl, where χ is a smooth function such that χ = 1 in a neighborhood of supp φ and ψ = 1 in a neighborhood of supp χ. Applying Lemma 2.3, we get φ x j 3 u W lkl for j = 0,..., k. Now let l 1 > 1. Then l 3 and, by Theorem 2.2, we obtain χu W lkl, χ x3 u W l 1 K l H D. Since ψ x3 f W l 2 D l and ψ x3 g ± W l 3/2 Γ ± l, we conclude again from Theorem 2.2 that φ x3 u W lkl. Repeating this argument, we get φ j x 3 u W l Kl for j=2,...,k. Example. We consider the Neumann problem u = f in D, u ν = g± on Γ ± Here the eigenvalues of the corresponding operator pencil Aλ are the numbers λ j = jπ/θ, j = 0, ±1, ±2,.... Consequently, the assertion of Theorem 2.2 with µ 1 = π/θ holds. 2.5 The Neumann problem to the Lamé system We consider a special case, where λ = 1 is an eigenvalue of the pencil A and the eigenfunctions corresponding to this eigenvalue are restrictions of linear functions to the unit circle. A necessary and sufficient condition for this case is given in the following lemma. Lemma 2.7. Let θ π, θ 2π. Then the homogeneous boundary value problem L x, 0 u = 0 in K, B x, 0 u = 0 on γ ± has a solution of the form u = c x 1 + d x 2, c, d C l, if and only if the 2l 2l matrix A = is not invertible. A1,1 A 2,1 A 1,2 A 2,2 Proof: The linear function u = c x 1 + d x 2 satisfies the homogeneous boundary conditions B x, 0 u = 0 on γ ± if and only if n + 1 n + 2 n 1 n 2 A1,1 A 2,1 A 1,2 A 2,2 c d = 0.

10 10 ZAMM Z. angew. Math. Mech Here the first matrix is invertible for θ π, θ 2π. This proves the lemma. Let r denote the rank of the matrix A. From the proof of the last lemma it follows that there are 2l r linearly independent eigenvectors of the form c cos ϕ + d sin ϕ corresponding to the eigenvalue λ = 1. Furthermore, the inhomogeneous boundary conditions B x, 0u = g ± on γ ± can be satisfied for a vector function u W 3 Kl only if g + and g satisfy 2l r compatibility conditions at x = 0. Such compatibility conditions must be also satisfied, in general, for the boundary data of the Neumann problem in the dihedron D. If u W 3 Dl, 0 < < 1, then the restriction of B x u to the edge M belongs to the space W 1 2 M l see, e.g., [14], [21], and we obtain A 1,1 n ± 1 + A 1,2n ± 2 x 1 u M + A 2,1 n ± 1 + A 2,2n ± 2 x 2 u M + A 3,1 n ± 1 + A 3,2n ± 2 x 3 u M = g ± M. The last system can be written in the form n + 1 n + 2 n 1 n 2 A1,1 A 2,1 A 3,1 A 1,2 A 2,2 A 3,2 x 1 u M x2 u M x3 u M g = + M g. M From this it follows that 2l r compatibility conditions must be satisfied for g + and g on the edge M, where r is the rank of the matrix A = A1,1 A 2,1 A 3,1. A 1,2 A 2,2 A 3,2 This means, there exist 2l r constant vectors c k such that c k g + M, g M = 0 for k = 1,..., 2l r We suppose that r = r. Then there are the same compatibility conditions for the Neumann problem 2.11 in the dihedron and the corresponding Neumann problem 2.20 in the angle K. This condition is satisfied, e.g., for the Neumann problem in isotropic and anisotropic elasticity. Furthermore, we assume that the geometric and algebraic multiplicity of the eigenvalue λ = 1 is equal to 2l r. This means that all eigenvectors corresponding to this eigenvalue have the form c cos ϕ + d sin ϕ and that there are no generalized eigenvectors corresponding to this eigenvalue. Lemma 2.8. Suppose that there are no eigenvalues of the pencil Aλ in the strip 0 < Re λ < 1 and the line Re λ = 1 contains the single eigenvalue λ = 1 having geometric and algebraic multiplicity 2l r = 2l r. Denote by λ 2 the eigenvalue with smallest real part greater than 1 and by µ 2 its real part. Furthermore, let φ, ψ be the same functions as in Theorem 2.1 and let u H D be a solution of problem 2.11, where ψf W 1Dl, ψg ± W 3/2 Γ ± l, max2 µ 2, 0 < < 1, and g + and g satisfy the compatibility condition Then φu W 3Dl and φu W 3 D l c ψf W 1 D l + ± with a constant c independent of u. ψg W 3/2 Γ ± l + ψu L 1 D l 2.24 Proof: Let χ be a smooth function on D such that χφ = φ and χψ = χ. From Theorem 2.1 it follows that χu W 2 Dl and χ x3 u W 2 Dl. Consequently, for almost all x 3 we have L x, 0 u, x 3 = f, x 3 L x, x3 L x, 0 u, x 3 = F, x 3, 2 B x, 0 u, x 3 = g ±, x 3 A 3,j n ± j x 3 u, x 3 = G ±, x 3, where χ, x 3 F, x 3 W 1Kl, χ, x 3 G ±, x 3 W 3/2 γ ± l. Since r = r and g +, g satisfy the compatibility condition 2.23, there exist vectors cx 3, dx 3 C l such that n + 1 n + 2 A1,1 A 2,1 A cx 3 3,1 n 1 n dx 2 A 1,2 A 2,2 A 3 g = + 0, x 3 3,2 g x3 u0, x 3 0, x 3

11 Maz ya, V. G., Roßmann, J.: Estimates for Green s matrix 11 for all x 3. From this it follows that px = cx 3 x 1 + dx 3 x 2 satisfies B x, 0w, x 3 = G ± 0, x 3 on γ ± for all x 3. Therefore, for v = u p we obtain L x, 0 v, x 3 = F, x 3 in K, B x, 0 v, x 3 = G ±, x 3 G ± 0, x 3 on γ ±. Here, according to [7, Th.7.1.1], χ, x 3 G ±, x 3 G ± 0, x 3 V 3/2 γ ±. By the assumptions of the theorem, λ = 1 is the only eigenvalues of the pencil A in the strip 0 < Re λ 2, all eigenfunctions are restrictions of linear functions to the unit circle, and generalized eigenfunctions corresponding to the eigenvalue λ = 1 do not exist. Thus, by [6, Th.1.2] see also [7, Th.6.1.4], φv admits the representation φxvx = c 0 x 3 + c 1 x 3 x 1 + c 2 x 3 x 2 + wx, where w, x 3 V 3 Kl and w, x 3 2 V 3 Kl c φ, x 3 F, x 3 2 V + 1Kl ± + χ, x 3 u, x 3 2 W 2Kl φ, x 3 G ±, x 3 2 V 3/2 γ ± l c χ, x 3 f, x 3 2 V + χ, x 1 3 g ±, x 3 2 Kl V 3/2 γ ± l ± + χ, x 3 u, x 3 2 W + χ, x 2 3 Kl x3 u, x 3 2 W 2Kl with a constant c independent of x 3. Since x α φu = α x w + α x φp for α = 3, the last estimate implies φ, x 3 u, x 3 2 L c χ, x 3 3 f, x 3 2 Kl V + χ, x 1 3 g ±, x 3 2 Kl V 3/2 γ ± l ± + χ, x 3 u, x 3 2 W 2 Kl + χ, x 3 x3 u, x 3 2 W 2 Kl. Integrating this inequality with respect to x 3 and using 2.14, we obtain The lemma is proved. Now, analogously to Theorem 2.2, the following statement holds. Theorem 2.3. Suppose that there are no eigenvalues of the pencil Aλ in the strip 0 < Re λ < 1 and the line Re λ = 1 contains the single eigenvalue λ = 1 having geometric and algebraic multiplicity 2l r = 2l r. Furthermore, we assume that u H D is a solution of problem 2.11, where ψf W l 2 D l, ψg ± W l 3/2 Γ ± l, l 2, is not integer, and maxl 1 µ 2, 0 < < l 1. In the case < l 2 we assume additionally that g +, g satisfy the compatibility condition Then φu W ldl and φu W l D l c ψf W l 2 D l + ± with a constant c independent of u. ψg L l 3/2 Γ ± l + ψu L 1 D l 2.25 Proof: If 0 < l 1 < 1, then the result follows directly from Theorem 2.2. Suppose that 1 < l 1 < 2. Then max2 µ 2, 0 < l+3 < 1, and Lemma 2.8 implies χu W 3 l+3 Dl, where χ is a smooth function such that χ = 1 in a neighborhood of supp φ and ψ = 1 in a neighborhood of supp χ. Applying Lemma 2.3, we obtain φu W l Dl. The proof for the case k < l 1 < k + 1, where k is an integer, k 2, proceeds analogously to the third step in the proof of Theorem 2.2. Moreover, the assertion of Corollary 2.2 with µ 2 instead of µ 1 is valid. Example. We consider the Neumann problem for the Lamé system u ν u = f in D, σu n = g± on Γ ±. 2.26

12 12 ZAMM Z. angew. Math. Mech Here σu = {σ i,j u} is the stress tensor connected with the strain tensor { } { 1 } ε i,j u = 2 x j u i + xi u j by the Hooke law ν σ i,j u = 2µ 1 2ν ε 1,1 + ε 2,2 + ε 3,3 i,j + ε i,j µ is the shear modulus, ν is the Poisson ratio, ν < 1/2, and i,j denotes the Kronecker symbol. The corresponding problem 2.20 in the angle K is: u1 x + 1 u 2 1 2ν x x u1 f1 =, u 2 f x u 3 = f 3 in K, 2 g1 u 3 σu 1, u 2 n =, g 2 n = g 3 on γ ±. If the opening θ of the angle K is greater than π, then the eigenvalue with smallest positive real part of the pencil Aλ is ξ + θ/θ, where ξ + θ is the smallest positive root of the equation 1.8. This is shown, e.g., in [8, Sect.4.2]. Note that ξ + θ < π for π < θ < 2π. If θ < π, then the eigenvalues with smallest positive real parts are λ 1 = 1 and λ 2 = π/θ. The eigenvalue λ 1 is simple, the corresponding eigenvector is sin ϕ, cos ϕ. Let n ± be the exterior normal to Γ ±. If u W l Dl, < l 2, then it follows from the Neumann boundary conditions that σu n ± M = g ± M and consequently, n σ n + M = n g + M and n + σ n M = n + g M. Here a b denotes the scalar product in R 3. Since σ is symmetric, we have n σ n + = n + σ n. Consequently, g + and g must satisfy the compatibility condition n g + = n + g on M. Applying Theorem 2.3, we get the following result: 1 Let u H D be a solution of problem 2.26, where ψf W 0D3, ψg ± W 1/2 Γ ± 3, 0 < < 1 for θ < π, 1 ξ + θ/θ < < 1 for θ > π. Then φu W 2D3. 2 Let θ < π and let u H D be a solution of problem 2.26, where ψf W l 2 D 3, l 3, ψg ± W l 3/2 Γ ± 3, is not integer, and maxl 1 π/θ, 0 < < l 1. In the case < l 2 we assume additionally that n g + M = n + g M. Then φu W ld3. In particular, φu belongs to the Sobolev space W 2 D 3 if θ < π, f W 1 D3, g ± W 3/2 Γ ± 3, < 1, n g + M = n + g M. 2.6 Estimates for Green s matrix to the Neumann problem From the unique solvability of the Neumann problem in H D and from classical results on fundamental solutions of elliptic boundary value problems in a half-space we obtain the following assertions for the Laplace equation see [22]. Theorem There exists a unique solution Gx, ξ of the boundary value problem L x Gx, ξ = x ξ I l for x, ξ D, 2.27 B x Gx, ξ = 0 for x D\M, ξ D 2.28 such the function x ζ ξ 1 x Gx, ξ belongs to H l D for arbitrary fixed ξ = ξ, ξ 3 D. Here I l is the l l identity matrix and ζ is a smooth function on 0, equal to zero in the interval 3/4, 3/2 and to one outside the interval 1/2, 2. 2 The function Gx, ξ is infinitely differentiable with respect to x, ξ D\M, x ξ. For x ξ < min x, ξ there is the estimate α x β ξ Gx, ξ c x ξ 1 α β,

13 Maz ya, V. G., Roßmann, J.: Estimates for Green s matrix 13 where c is independent of x and ξ. 3 The function Gx, ξ is also the unique solution of the problem L ξ Gx, ξ = x ξ I l for x, ξ D, B ξ Gx, ξ = 0 for x D, ξ D\M such that the function ξ ζ ξ 1 x Gx, ξ belongs to H l D for arbitrary fixed x D. We establish now an estimate for the derivatives of Green s function Gx, ξ in the case x ξ min x, ξ. For this we need the following lemma analogous to Lemma 2.2 in [10]. Lemma 2.9. Let B be a ball with radius 1 and center x 0 such that distx 0, M 4. Furthermore, let φ, ψ be infinitely differentiable functions with supports in B such that ψ = 1 on supp φ. If ψu H D, Lu = 0 in D B and Bu = 0 on D\M B, then sup x max α µ 1+ε,0 φx α x x j 3 ux c ψu HD, 2.29 x D where ε is an arbitrarily small positive number. The constant c in 2.29 is independent of u and x 0. Proof: Let ε be such that µ 1 ε k, k +1. Then = k +1 µ 1 +ε 0, 1. Furthermore, let χ be a function from C0 B such that φχ = φ and χψ = χ. From Theorems 2.1 and 2.2 it follows that x j 3 χu W k+2 D l for j = 0, 1,... and χ j x 3 u W k+2 D l c ψu HD. Hence we have x α j x 3 χu W 2Dl for α k. Since W 2 K is continuously imbedded into CK, we have sup x α j x 3 χu c sup x α j x 3 χu, x 3 W 2 x K. l K, x 3 R x 3 R Using the continuity of the imbedding W 1 2 M CM, we obtain sup x α j x 3 χu, x 3 W 2 K c l x α j x 3 χu W 2 D + l α x j+1 x 3 χu W 2 D l x 3 R c ψu HD. This proves 2.29 for α k. Now let α k + 1. By Theorems 2.1 and 2.2, we have j x 3 χu W α +2 k+ α Dl and, therefore, α x j x 3 χu W 2 k+ α Dl V 2 k+ α Dl. Using Sobolev s lemma, it can be easily shown that sup x β k+1 vx c v V k β K for arbitrary v Vβ k K, k x K with a constant c independent of v and x. Applying this inequality to α x j x 3 χu, we obtain sup x k+ α 1 α x x j 3 χu c sup x α j x 3 χu, x 3 W 2 Kl. x k+ α K, x 3 R x 3 R Using again the continuity of the imbedding W 1 2 M CM, we arrive at Theorem 2.5. For x ξ min x, ξ there is the estimate x α j x 3 β ξ k ξ 3 Gx, ξ c x ξ 1 α β j k x min0,µ1 α ε ξ min0,µ1 β ε, 2.31 x ξ x ξ where ε is an arbitrarily small positive number. Proof: Since GT x, T ξ = T 1 Gx, ξ, we may assume, without loss of generality, that x ξ = 2. Then max x, ξ 4. Let B x and B ξ be balls with centers x and ξ, respectively, and radius 1. Furthermore, let η and ψ be infinitely differentiable functions with supports in B x and B ξ, respectively. Applying Lemma 2.9 to the function α x j x 3 Gx,, we obtain ξ max β µ1+ε,0 α x j x 3 β ξ k ξ 3 Gx, ξ c ψ α x j x 3 Gx, HD. 2.32

14 14 ZAMM Z. angew. Math. Mech We consider the solution ux = ψ F, Gx, D of problem 2.4, where F HD. Since ψf vanishes in the ball B x, we conclude from Lemma 2.9 that x max α µ 1+ε,0 x α j x 3 ux c ηu HD. Consequently, the mapping HD F x max α µ1+ε,0 x α j x 3 ux = x max α µ1+ε,0 F, ψ x α j x 3 Gx, C D represents a linear and continuous functional on HD by a constant independent of x. This implies for arbitrary x D. The norm of this functional is bounded x max α µ1+ε,0 ψ α x j x 3 Gx, HD c what together with 2.32 yields the desired estimate. Using Theorem 2.3 instead of Theorem 2.2 in the proof of Lemma 2.9, we obtain the following result. Theorem 2.6. Suppose that there are no eigenvalues of the pencil Aλ in the strip 0 < Re λ < 1 and the line Re λ = 1 contains the single eigenvalue λ = 1 having geometric and algebraic multiplicity 2l r = 2l r r and r were defined in Section 2.5. Then Gx, ξ satisfies 2.31 with µ 2 instead of µ 1 = 1. Examples. 1 Green s matrix of the Neumann problem 2.22 for the Laplace equation satisfies 2.31 with µ 1 = π/θ. 2 For θ > π Green matrix of the Neumann problem 2.26 for the Lamé system satisfies 2.31 with µ 1 = ξ + θ/θ. In the case θ < π, the number µ 1 has to be replaced by π/θ. 2.7 Estimates for Green s matrices to the Dirichlet and mixed problems We consider problem 2.1 for the case when the Dirichlet condition is given on at least one of the sides Γ +, Γ, i.e., not both numbers d +, d equal zero. Then H D V0 1 D l. From Lax-Milgram s lemma and Lemma 2.1 it follows that the problem b D u, v = F, v D for all v H D, u = g ± on Γ ± for d ± = has a unique solution u V 1 0 D l for arbitrary F H D, g± V 1/2 0 Γ ± l. For the following theorem we refer to [9, Th.4.1,7.2] and [18, Ch.11,Prop.1.4]. Theorem 2.7. Let u V 1 0 D l be a solution of problem 2.33, where the functional F has the form F, v D = f, v D + ± 1 d ± g ±, v Γ ± with ψ x j 3 f V l 2 D l and ψ x j 3 g ± V l+d± 3/2 Γ ± l for j = 0, 1,..., k, l 1 µ 1 < < l 1. Here φ and ψ are the same cut-off functions as in Theorem 2.2. Then φ x j 3 u V ldl and k φ j x 3 u V l D l c k ψ j x 3 f V l 2 D l + k ψ x j 3 g ± l+ V ± + ψu 3/2 Γ ± l V0 1. Dl Analogously to Theorem 2.4, there exists a unique solution Gx, ξ of the problem L x Gx, ξ = x ξ I l for x, ξ D, d ± Gx, ξ + 1 d ± B x Gx, ξ = 0 for x Γ ±, ξ D such that the function x ζ ξ 1 x Gx, ξ belongs to HD l for arbitrary ξ D and for an arbitrary smooth function ζ on 0, equal to zero in the interval 3/4, 3/2 and to one outside the interval 1/2, 2. We call the matrix-valued function Gx, ξ Green s matrix of problem Using Theorem 2.7, one can prove the following estimates. ±

15 Maz ya, V. G., Roßmann, J.: Estimates for Green s matrix 15 Theorem 2.8. The matrix Gx, ξ satisfies the estimates x α β ξ Gx, ξ c x ξ 1 α β for x ξ < min x, ξ and x α j x 3 β ξ k ξ 3 Gx, ξ c x ξ 1 α β j k x µ1 α ε ξ µ1 β ε x ξ x ξ for x ξ > min x, ξ, where ε is an arbitrarily small positive number. 3 The parameter-dependent problem on a domain of the sphere In this section we study the parameter-dependent boundary value problem 1.9. We prove that this problem is uniquely solvable in a certain class of weighted Sobolev spaces for all λ, except finitely many, in a double angle of the complex plane containing the imaginary axis. This result is essentially known. For a smooth domain Ω on the sphere and Sobolev spaces without weight it was proved by Agranovich, M. S. and Vishik, M. I. [1]. 3.1 The parameter dependent Neumann problem in an angle Let again D be the dihedron K R, where K is the angle {x = x 1, x 2 : ϕ 0, θ}, and let b D be the sesquilinear form 2.2. We denote by ũ and ṽ the Fourier transforms with respect to x 3 of the vector-functions u and v. Then, by Parseval s equality, we have b D u, v = b K ũ, η, ṽ, η; η dη, where R b K u, v; η = K 2 A i,j xi u xj v + iη i, We consider the variational problem 2 i=1 A3,i u xi v A i,3 xi u v + η 2 A 3,3 u v dx. b K u, v; η = F, v K for all v W 1 2 K l, 3.1 which corresponds to the parameter-depending Neumann problem L x, iη u = B x, iη u = 2 A i,j xi xj u iη i, 2 A i,j xi u n j + iη i, where γ ± are the sides of K. 2 Ai,3 + A 3,i xi u + η 2 A 3,3 u = f in K, 3.2 i=1 2 A 3,j u n j = g ± on γ ±, 3.3 Theorem 3.1. The boundary value problem 3.2, 3.3 is uniquely solvable in W 2Kl for arbitrary f W 0Kl, g ± W 1/2 γ ± l, max1 µ 1, 0 < < 1, η R, η 0. The solution satisfies the inequality 2 η 2 j u L j c f Kl L 0 K + l ± with a constant c independent of f, g ± and η. g ± L 1/2 γ ± l + η 1/2 ± Proof: Let the functional F be given by F, v K = f v dx + g ± v dr, v W0 1 K l, K ± γ ± r g ± L2 γ ± l 3.4

16 16 ZAMM Z. angew. Math. Mech where f L 0 Kl, g ± W 1/2 γ ± l, 0 < < 1. It can be easily seen that this functional belongs to W0 1 K l. We set ux = N 1/2 e iηx3 φx 3 /N vx, where v W2 1 K l, and φ C0 R is a real-valued function such that + φt dt = 1. Then D 3 xj u 2 dx K x v 2 + η 2 2 v 2 dx c N 2 where c is independent of v and N. Analogously, b D u, u b K v, v; η + c N 2 v 2 dx. Consequently, 2.3 yields b K v, v; η c K K x v 2 + η 2 v 2 dx. K v 2 dx, Thus, by Lax-Milgram s lemma, for all real η 0 there exists a unique solution u W0 1 K l of problem 3.1 which is also a solution of problem 3.2, 3.3. We show that u L 2 Kl. Let χ be an arbitrary smooth cut-off function with compact support equal to one near the vertex of K. Then χu Vε 1 K l with an arbitrary positive ε and, therefore, also χu V1+εK 2 l see, e.g. [7, Le.6.3.1]. Furthermore, L x, iηχu V 0Kl, B x, iηχu V 1/2 γ ± l. Hence, according to [7, Th.6.4.1] and [8, Th ], the vector-function χu has the asymptotics χu = c + d log r + w, where c, d C l, w V 2 K l. Since u W2 1 K l, the vector d is equal to zero. This implies χu L 2 Kl. We consider the vector-function 1 χu. Obviously, 1 χu W 2 0 Kl, L x, iη 1 χu W 0Kl, while B x, iη 1 χu V 1/2 γ ± l W 1/2 γ ± l. Consequently, by [9, Th.4.1 ], we obtain 1 χu V 2Kl W 2Kl. Thus, we have shown that u L 2 K. Estimate 3.4 holds by applying the inequality of Corollary 2.1 to vx = N 1/2 e iηx 3 φn 1 x 3 ux, where φ C0 R and N is a large number. An analogous result holds for the parameter-dependent Dirichlet and mixed problems in the angle K. Here the spaces L j j can be replaced by V. 3.2 Solvability of problem 1.9 Let H Ω = {u W 1 Ω l : u = 0 on γ j for j J 0 }. We introduce the parameter-dependent sesquilinear form au, v; λ = 1 log 2 K 1< x <2 3 A i,j xi U xj V dx, i, where Ux = ρ λ ω, V x = ρ 1 λ vω, and define the operator Aλ : H Ω H Ω by Aλu, v Ω = au, v; λ, u, v H Ω. The pencil A has following properties see [8, Ch.10,12]. i The spectrum of the pencil A consists of isolated points, the eigenvalues of this pencil. All eigenvalues have finite algebraic multiplicity. ii If λ is an eigenvalue of the pencil A, then 1 λ is also an eigenvalue with the same geometric and algebraic multiplicity. s 1 iii The vector function u = r λ0 k! log rk u s k ω satisfies the equality b K u, v = 0 for all v H equal k=0 to zero in a neighborhood of the origin and infinity if and only if λ 0 is an eigenvalue of the pencil A and u 0,..., u s is a Jordan chain corresponding to this eigenvalue.

17 Maz ya, V. G., Roßmann, J.: Estimates for Green s matrix 17 We denote by J the set all j {1, 2,..., n} such that the Dirichlet condition in problem is given on at least one side adjacent to the edge M j, i.e. M j Γ k for at least one k J 0. Let = 1,..., n R n, j > 1 for j J. Then we define the norm in the weighted Sobolev space W l Ω; J by u W l Ω; J = K 1< x <2 α l j J r 2j l+ α j n j J r 2j j α x ux 2 dx 1/2, where u is extended by the equality ux = ux/ x to the cone K and r j x = dist x, M j. Furthermore, we set V l Ω = W l Ω; {1,..., n} and W l Ω = W l Ω;. From Hardy s inequality it follows that W l Ω; J = V l Ω if j > l 1 for j J. Furthermore, H Ω W 0 1Ω; J l. The trace spaces for V l Ω, W l Ω and W l Ω; J, l 1, on the arc γ j are denoted by V l 1/2 γ j, W l 1/2 γ j and W l 1/2 γ j ; J, respectively. In particular, W l 1/2 γ j ; J = V l 1/2 γ j for j J 0. Let D j be the dihedron which coincides with K near the point M j S 2. The boundary value problem for the system 1.1 in D j is connected with a pencil A j λ on an interval 0, θ j, where θ j is the interior angle at the edge M j see the definition of the pencil Aλ in Section 2. We denote by λ j 1 the eigenvalue with smallest positive real part and set µ j = Re λ j 1. Furthermore, let the operator A λ be defined as W 2 Ω; J l u Lλu, {u γj } j J0, {Bλu γj } j J1 W 0 Ω l γ j ; J l γ j ; J l, where L and B are given by W 3/2 j J 0 j J 1 W 1/2 Theorem 3.2. Let 1 µ j < j < 1 for j J and max1 µ j, 0 < j < 1 for j J. 1 Then the spectra of the pencils A and A coincide. 2 There exist positive real constants N and ɛ such that for all λ in the set { λ C : λ > N, Re λ < ɛ Im λ } 3.5 the operator A λ is an isomorphism. Furthermore, every solution u W 2 Ω l of problem 1.9 with λ in the set 3.5 satisfies the inequality 2 λ 2 j u W j Ω; J c f l V 0 Ω + l g V 3/2 γ j + λ 3/2 g l V 0 γ j l j J 0 where c is independent of u and λ. + j J 1 g V 1/2 γ j l + λ 1/2 g V 0 γ j l, 3.6 Proof: 1 We consider the differential operators Lλ and Bλ in a neighborhood of M 1 S 2. Without loss of generality, we may assume that M 1 coincides with the x 3 -axis and D 1 is the dihedron K R. By ω 1 = x 1 /ρ, ω 2 = x 2 /ρ we denote local coordinates on the unit sphere near the north pole N = M 1 S 2. Since the operator Lλ has the form x1 = ω 1 ρ + 1 ω2 1 ω1 ω 1ω 2 ω2, x2 = ω 2 ρ ω 1ω 2 ρ ρ ρ ω ω2 2 ω2, ρ x3 = 1 ω 2 1 ω 2 2 1/2 ρ ω 1 ρ ω 1 ω 2 ρ ω 2, Lλ = 2 A i,j ωi ωj λ 1 i, 2 Ai,3 + A 3,i ωi λ A 1,1 + A 2,2 λλ 1 A 3,3 i=1 +λ 2 P 0 ω + λ P 1 ω, ω + P 2 ω, ω, where P j are differential operators of order j with coefficients vanishing at the point ω 1, ω 2 = 0. Analogously, Bλ = 2 A i,j n j ωi + λ i, 2 A 3,j n j + λ Q 0 ω + Q 1 ω, ω

18 18 ZAMM Z. angew. Math. Mech near N, where Q j are differential operators of order j with coefficients vanishing at ω 1, ω 2 = 0. Furthermore Ω coincides with the wedge K in the coordinate system ω 1, ω 2 near M 1 S 2. Hence we conclude, analogously to the proof of Theorem 3.1, that every weak solution u W2 1 Ω l of problem 1.9 with support near N belongs to the space W 2 Ω; J l if f V 0 Ω l, g j V 3/2 γ j l for j J 0 and g j V 1/2 γ j l for j J 1. By means of a partition of unity on Ω, we obtain this result for arbitrary weak solutions. This implies, in particular, that every eigenfunction of the pencil A is an eigenfunction of A corresponding to the same eigenvalue. The same is true for generalized eigenfunctions. 2 We prove the second assertion first for purely imaginary λ = iη. Let ζ 0, ζ 1,..., ζ n be a partition of unity on Ω such that ζ j = 1 near M j S 2 and supp ζ j is sufficiently small for j = 1,..., n. We consider the vector-function ζ 1 u and assume, as above, that the edge M 1 coincides with the x 3 -axis. The difference of the operator Lλ in the coordinates ω 1, ω 2 introduced above and the operator 3.2 is small for large λ and small ω1 2 + ω2. 2 This means, there is the inequality Lλ L ω1, ω2, λ 2 ζ 1 u L 0 1 K ε λ 2 j ζ l 1 u L j K, l 1 where ε is small if supp ζ 1 is small and λ is large. The same is true for the difference of the operators Bλ and 3.3. Hence in the case of the Neumann problem it follows from Theorem 3.1 that 2 j λ 2 j ζ 1 u L j Ω c l Lλu W 0 Ω l + n Bλu W 1/2 γ j l + λ 1/2 Bλu W 0 γ j l for sufficiently large λ. The same inequality is true for the vector-functions ζ j u, j = 1,..., n. The validity of this inequality for ζ 0 u follows from a result of Agranovich and Vishik [1] see also [7, Th.3.6.1]. An analogous estimate holds for the Dirichlet and mixed problems. This implies 3.6 for purely imaginary λ, λ > N. For λ in the set 3.5 this estimate can be proved in the same way as in [1, 7]. 4 The boundary value problem in a polyhedral cone In the last section we consider problem in the cone K. We prove the existence of strong and weak solutions, obtain regularity assertions for the solutions and point estimates for Green s matrices. As in Section 2 we concentrate on the case of the Neumann problem. Analogous assertions for the Dirichlet and mixed problem are formulated at the end of the section and can by obtained by obvious modifications in the proofs. For the Dirichlet problem we refer also to the papers by Maz ya and Plamenevskiĭ [13] Lamé and Stokes systems, Maz ya and Roßmann [15] scalar 2m order elliptic equations which include solvability theorems in weighted Sobolev and Hölder spaces and estimates for Green s functions. The solvability of the Neumann problem for diagonalizable second order equations in Sobolev spaces without weight was studied by Dauge [3, 4]. 4.1 Weighted Sobolev spaces in K For an arbitrary point x K let ρx = x be the distance to the vertex of the cone and r j x the distance to the edge M j. Furthermore, we denote by rx the regularized distance to S, i.e., an infinitely differentiable function in K which coincides with distx, S in a neighborhood of S. Let l be a nonnegative integer, J the same subset of {1, 2,..., n} as in Section 3, β R, = 1,..., n R n, j > 1 for j J. By W l β, K; J we denote the weighted Sobolev space with the norm u W l β, K; J = ρ K 2β l+ α α l j J r j ρ 2j l+ α j J r j 2j α 1/2. ρ x u 2 dx Furthermore, we define V l β, K = Wl β, K; {1,..., n} and W l β, K = Wl β, K;. Passing to spherical coordinates ρ, ω, one obtains the following equivalent norm in W l β, K; J: u = 0 ρ 2β l+1 l k=0 ρ ρ k uρ, 2 W l k Ω; J dρ 1/2.

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