J. Necas, Les methodes directes en theorie des equations elliptiques, Masson, 1967.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....element theory can be found for example in Nitsche s duality argument, multigrid convergence theorems, convergence of mortar finite element methods, etc. The shift estimates for the Laplace operator with Dirichlet boundary conditions on nonsmooth domains are well known (see, e. g, 21] 23] [27]) For the second order elliptic boundary value problems with mixed boundary conditions on nonsmooth domains, much less has been done. Even though the shift estimates concerning mixed boundary conditions are often used in the finite element theory, a proof of the regularity estimates is not given, ....

J. Necas . Les Methodes Directes en Theorie des Equations Elliptiques. Academia, Prague, 1967.

....properties are defined in terms of Sobolev spaces. The Sobolev spaces fH g for non negative integers s are defined to be the distributions which along with their partial derivatives of order s are in L A complete development and discussion of these spaces can be found in, e.g. 13] 16] [18]. The norm on H will be denoted k Deltak . For negative s, the space H is defined by duality and is the set of linear functionals on H ( Omega Gamma for which the norm kvk s = sup OE2C 1 ( Omega Gamma (v; OE) is finite. Here ( Delta; Delta) denotes the L inner product. We ....

J. Necas, Les M'ethodes Directes en Th'eorie des ' Equations Elliptiques, Academia, Prague, 1967.

....definite and bounded. We further assume that b i 2 L for i = 1; d. 4 BRAMBLE, et al. To describe and analyze the least squares method, we shall use Sobolev spaces. For non negative integers s, let H denote the Sobolev space of order s defined on Omega (see, e.g. 19] 26] [29]) The norm in H will be denoted by k Deltak s . For s = 0, H ( Omega Gamma coincides with L In this case, the norm and inner product will be denoted by k Deltak and ( Delta; Delta) respectively. The space W is defined to be the closure v 2 C ( Omega Gamma j v = 0 on Gamma D ....

J. Necas, Les M'ethodes Directes en Th'eorie des ' Equations Elliptiques, Academia, Prague, 1967.

....C(h J =h l ) C(h J =h l ) 9 We have used the Poincar e inequality for the last estimate above to replace the norm by the semi norm. We next estimate the second term on the right hand side of (4. 14) The subdomains have Lipschitz continuous boundaries and hence it follows from [13] that fl fl fl fl Gamma1=2 ffl; c kvk 1=2 ffl; H holds for any ffl in [0; 1=2] Here = n denotes the outward pointing normal derivative on H . Thus, 4.16) jA i (v; OE H )j = fi fi fi n ; OE AE fi fi C kvk 1=2 ffl; kOEk 1=2 Gammaffl; H ....

J. Necas, "Les M'ethodes Directes en Th'eorie des ' Equations Elliptiques," Academia, Prague, 1967.

....element theory can be found for example in Nitsche s duality argument, multigrid convergence theorems, convergence of mortar finite element methods, etc. The shift estimates for the Laplace operator with Dirichlet boundary conditions on nonsmooth domains are well known (see, e. g, 21] 23] [27]) For the second order elliptic boundary value problems with mixed boundary conditions on nonsmooth domains, much less has been done. One technique for proving shift results is by using the real method of interpolation of Lions and Peetre [2] 24] and [25] The resulting interpolation problems ....

J. Necas . Les Methodes Directes en Theorie des Equations Elliptiques. Academia, Prague, 1967.

....(2.3) in # also have unique weak solutions given by the solutions of problems (5.1) and (5.2) respectively. We know that there exist an isomorphism and homeomorphism of H 1(## H 1 0(## onto H 1 2 (#) see Theorem 7.53, p. 216, in [1] or Theorem 5.5, p. 99, and Theorem 5.7, p. 103, in [30]) i.e. there are two constants k 1 , k 2 0 such that we have the following. For any y # H 1(## , there exists v # H 1 2 (#) such that y = v on # and v H 1 2 (#) # k 1 y H 1(## . For any v # H 1 2 (#) there exists y # H 1(## such that y = v on # and ....

J. Necas, Les methodes directes en theorie des equation elliptiques, Editions de l'Academie Tschecoslovaque des Sciences, Prague, 1967.

....exists a subsequence of fv j g 1 j=0 (denoted by the same symbol again) such that v j u in H 1( Omega Gamma v j u in L 2( Omega Gamma v j u a.e. in Omega : 3. 9) To prove the strong convergence (for a subsequence) v j u in L 2 ( Gamma) we apply the well known inequality (see Necas [4]) kwk 2 0; Gamma kwk 2 1; Omega C kwk 2 0; Omega ; C = C 1 ; for small positive . Therefore, for small but fixed we deduce that kv j Gamma uk 2 0; Gamma kv j Gamma uk 2 1; Omega C kv j Gamma uk 2 0; Omega C C kv j Gamma uk 2 0; Omega : 54 M. ....

J. Necas. Les m'ethodes directes en th'eorie des 'equations elliptiques. Academia, Prague, 1967.

....is taken to be H = H 1 (#) 2.14) and the cost function is given by J(v) 1 2 # # # # #y(v) #nA (#) h # # # # # 2 H 1 (#) 2.15) Remark 2.1. Since y(v) # H 1 2(## and Ay(u) f # L 2(##7 we have y(v) # H 2 (D) for any domain D which satisfies # # D # D ## (see [30], Chap. 4, 1.2, Theorem 1.3, for instance) Therefore, y(v) # H 3 2 (#) with the same values on both the sides of #. Also, #y(v) #nA (#) # H 1 2 (#) #y(v) #nA(# #) # H 1 2 (#) and #y(v) #nA (#) #y(v) #nA(# #) 0. Consequently, above two cost functions make sense. ....

....(2.3) in # also have unique weak solutions given by the solutions of problems (5.1) and (5.2) respectively. We know that there exits an isomorphism and homeomorphism of H 1(## H 1 0(## onto H 1 2 (#) see Theorem 7.53, p. 216, in [1] or Theorem 5.5, p. 99, and Theorem 5.7, p. 103, in [30]) i.e. there are two constants k 1 , k 2 0 such that we have . For any y # H 1(## , there exists v # H 1 2 (#) such that y = v on # and v H 1 2 (#) # k 1 y H 1(## . For any v # H 1 2 (#) there exists y # H 1(## such that y = v on # and y H ....

J. Necas, Les methodes directes en theorie des equation elliptiques, Editions de l'Academie Tschecoslovaque des Sciences, Prague, 1967.

....If Gamma N = and fl = 0 then p is determined only up to an additive constant. Thus, it is unique if we require that R Omega p dx = 0: The existence, stability, and regularity properties of solutions of the above problem are most naturally described in terms of Sobolev spaces (see, e.g. 40] [41]) Let ( Delta; Delta) denote the L 2 ( Omega Gamma inner product and jj Delta jj denote the corresponding norm. We will use the same inner product and norm notation for vector valued functions in the product space (L 2( Omega Gamma4 d . For positive values of s, let H s( Omega Gamma ....

....the corresponding norm. We will use the same inner product and norm notation for vector valued functions in the product space (L 2( Omega Gamma4 d . For positive values of s, let H s( Omega Gamma denote the Sobolev space of order s and jj Delta jj s denote the corresponding norm (cf. 34] [41]) We denote by H 1( Omega Gamma the space (H 1( Omega Gamma7 d : Let H 1 D( Omega Gamma be the set of functions in H 1( Omega Gamma with vanishing trace on Gamma D . In the case that Gamma D is all of the boundary, we denote this space as H 1 0( Omega Gamma9 Its dual will be called H ....

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J. Necas, Les M'ethodes Directes en Th'eorie des ' Equations Elliptiques, Masson, 1967.

....collect some well known results about these spaces that are used throughout this thesis. A more thorough introduction to the most important tools, specifically those used in domain decomposition theory, may be found in [102, Section 1. 2] For a more complete development of the theory, we refer to [85] and [64] Let Omega be a bounded Lipschitz region in R d . Let x represent a point in Omega or on its boundary let u, v, f , g be scalar valued functions, let u, v be vector valued functions, and let u i be the i th component of u, a vector with d components. The space L 2 ( Omega Gamma is ....

...., endowed with the norm jjvjj H 1=2 ( Omega Gamma . This space is isomorphic to the interpolation space [L 2 ( H 1 0 ( 1=2 ; see [71, Chapter 1] An equivalent norm for H 1=2 00 ( is given by: jjvjj 2 H 1=2 00 ( jvj 2 H 1=2 ( Z v 2 (x) d(x; dS(x) 1. 2) see [85]. 1.3 Ellipticity and the Babuska Brezzi Theory We start with the following well known lemma, which establishes existence, uniqueness and well posedness for the elliptic problems to be considered. Lemma 1.3.1 (Lax Milgram Lemma) Let B be a bilinear form on a Hilbert space H. Assume that B is ....

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Jindrich Necas. Les m'ethodes directes en th'eorie des 'equations elliptiques. Academia, Prague, 1967.

....2 (R n ) via the Fourier transform) 2,4 POSITIVITY IN L 2( Omega Gamma 23 It can be useful to know in what sense f 2 D( Delta) satisfies the boundary condition. We are further assuming some knowledge of the fractional Sobolev spaces on Omega which will be denoted by H s ( see e.g. [N] or [Tr] The Green formula (see (2.8) 2.9) Z Omega ( Deltaf )gdx Z Omega rf:rgdx = Z Omega f gdS holds for f 2 C 2 ( Omega Gamma ; g 2 C 1 ( Omega Gamma and, by continuity, also for f 2 X : ff 2 H 1( Omega Gamma4 Delta w f 2 L 2 g and g 2 H 1 ( Omega Gamma1 ....

.... Z Omega rf:rgdx = Z Omega f gdS holds for f 2 C 2 ( Omega Gamma ; g 2 C 1 ( Omega Gamma and, by continuity, also for f 2 X : ff 2 H 1( Omega Gamma4 Delta w f 2 L 2 g and g 2 H 1 ( Omega Gamma1 The following trace theorem can be deduced from (2. 9) for details see [N]) Trace theorem. There exists a uniquely determined operator T 2 L(H 1 ; H 1=2 ( Omega Gamma5 such that (i) Ker T = H 1 0 ( Omega Gamma1 (ii) R(T ) H 1=2 ( Omega Gamma1 (iii) Tf = f ffi Omega for any f 2 C 1 ( Omega Gamma2 For f 2 X denote by Phi f (g) the left hand side of ....

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J.Necas, Les m'ethodes directes en th'eorie des 'equations elliptiques, Academia Prague 1967.

....(2. 2) where (f; v) n X i=1 (f i ; v i ) Z Omega n X i=1 f i (x)v i (x) dx; a(u; v) Z Omega n X i;j=1 u i x j v i x j dx; b(u; p) Gamma(div u; p) The problem consists in finding a velocity vector u 2 V and a pressure p 2 L2;0( Omega Gamma1 As is well known (see [11], 9] this problem is solvable and its solution in a stable way depends on the data since the so called inf sup condition is satisfied: sup u6=0 b 2 (u; p) a(u; u) fi 2 0 kpk 2 for all 0 6= p 2 L2;0 ; fi 0 = const 0: 3 In both the two and three dimensional case the following ....

Necas, J.: Les M'ethodes Directes en Th'eorie des ' Equations Elliptiques, Masson, Paris 1967.

....to the constraints on the state y F i (y) 0; i = 1; m; 2.3) E(y) 2 K; 2.4) and to the constraints on the control u u a (x) u(x) u b (x) a. e. on Gamma: 2. 5) In this setting, Omega ae R n is a bounded domain with a Lipschitz boundary Gamma according to the definition by Necas [17]. Moreover, sufficiently smooth functions f : Omega Theta R R and g; b : Gamma Theta R 2 R are given. The symbol is used for the derivative in the direction of the unit outward normal on Gamma. The functionals F i : C( Omega Gamma R, i = 1; m, are supposed to be twice ....

....y Gamma b( Delta; y; u) dS m X j=1 j F j (y) hz ; E(y)i; 3.8) L : Y q;p Theta U Theta W 1;oe Theta R m Theta Z R. The regularity of y and fits together, as 2 W 1;oe for all oe n= n Gamma 1) ensures 2 L s( Omega Gamma for all s n= n Gamma 2) Necas [17], Thm. 3.4, p. 69) and j Gamma 2 L r ( Gamma) holds for all r 1 1= n Gamma 2) 17] Thm. 4.2, p.84) Therefore, this definition makes sense. In (3.8) h Delta; Deltai denotes the duality pairing between Z and its dual space Z . The Lagrange function L is of class C 2 with respect ....

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Necas, J.: Les M'ethodes Directes en Th'eorie des Equations Elliptiques. Editeurs Academia, Prague, 1967.

....developed in the present paper the above interaction problems can also be investigated (with corresponding obvious modification) when the interface is a non smooth polyhedral (or, in general, Lipschitz) surface. In this case we have to apply results obtained in [39] 63] 8] 14] 42] [49], 22] 50] 1 Classical Formulation of Problem 1.1. Elastic field. Let Omega 1 = Omega ae IR 3 be a bounded domain (diam Omega 1 1) with a smooth, connected, nonselfintersecting boundary S = Omega 1 ; Omega 2 = Omega Gamma = IR 3 n Omega 1 ; Omega 1 = Omega 1 [S. ....

....to the equation (2.8) we have to study some properties of the sesquilinear form B and the anti linear functional F . In particular, we will establish the boundedness of B and F , and the Garding type inequality for B, in appropriate function spaces (see, e.g. 36] Chapter 2, Section 9, 20] [49]. To this end, we recall the well known Ehrling inequalities (see, e.g. 36] Chapter 1, x16, Theorem 16.3; 41] Chapter 3, x7, Theorem 3.16; 38] Chapter 14, x3) Let u 2 W 1 2( Omega ) Then for arbitrary 0 there exist positive constants c 0 1 ( and c 0 2 ( such that jju ; ....

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Necas, J. Les M'ethodes Directes en Th'eorie des ' Equations ' Elliptique, Academia, Prague, 1967.

....2.6 Case m # n In what follows, we will need the following results: Theorem 2.6.1 (Vainikko and Kunisch [35] Suppose that n # 2. Let a function # # L #(## and a function v # C # 0(## satisfy the equation div ##v = 0 in H 1(## . Then # = 0 on x ## #v (x) #= 0 . Theorem 2.6. 2 (Necas [22], cf. 34] Let D be a bounded open set in R n with Lipschitz boundary. Then there exists c = c (D) 0 such that for every g # L 2 (D) #g# L 2 (D) R # c #grad g# [H 1 (D) n where L 2 (D) R is the factor space. 2.6.1 Case m = n = 2, First Counter Example Let A 1 and A 2 be ....

J. Necas, Les methodes directes en theorie des equations elliptiques, Academia, Prague, 1967.

....defined by interpolation between H 1 and L 2 ; see [39] Only the case of C 1 boundary is treated there, but many results carry over to the case of polyhedral domains. In all of R n , the Sobolev spaces, of integer or fractional order, can also be defined via the Fourier transform; see [47, 39]. Extension theorems for Sobolev spaces are used to extend this equivalent definition to bounded regions; see [47, 39] Any function that belongs to a Sobolev space H m( Omega Gamma7 with m large enough, is continuous; the higher the dimension n of R n , the larger m is required. This ....

....many results carry over to the case of polyhedral domains. In all of R n , the Sobolev spaces, of integer or fractional order, can also be defined via the Fourier transform; see [47, 39] Extension theorems for Sobolev spaces are used to extend this equivalent definition to bounded regions; see [47, 39]. Any function that belongs to a Sobolev space H m( Omega Gamma7 with m large enough, is continuous; the higher the dimension n of R n , the larger m is required. This well known result follows from Sobolev s lemma. 4 Theorem 1.1 If 2m n, then, for any u 2 H m( Omega Gamma max x2 Omega ....

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Jindrich Necas. Les m'ethodes directes en th'eorie des 'equations elliptiques. Academia, Prague, 1967.

.... and elliptic and that it is bounded in V Theta V: In the case of Poisson s equation, the bilinear form is defined by a(u; v) Z Omega ru Delta rv dx : 2) We assume that Omega is a Lipschitz region in R n ; n = 2; 3; and that its diameter is on the order of 1: We will follow Necas [45] when defining Lipschitz regions and Sobolev spaces on Omega : The bilinear form a(u; v) is directly related to the Sobolev space H 1( Omega Gamma that is defined by the semi norm and norm juj 2 H 1 ( Omega Gamma = a(u; u) and kuk 2 H 1( Omega Gamma = juj 2 H 1( Omega Gamma kuk ....

....Il in [38] for some similar inequalities. As before, Omega ae R n ; n =2 or 3, is a bounded, polygonal region and f Omega i g a nonoverlapping decomposition of Omega into substructures. To simplify our considerations, we now assume that the substructures are squares or a cubes; cf. e.g. Necas [45] where simple maps and partions of unity are used to derive bounds for Lipschitz regions from bounds for such special regions; if we can handle a corner of a square or cube, then we can analyze the general polygonal case. Our estimates, given in the next two sections, are developed for one ....

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Jindrich Necas. Les m'ethodes directes en th'eorie des 'equations elliptiques. Academia, Prague, 1967.

....we recall a classical result used essentially in finite element analysis. A generalization of the Bramble Hilbert lemma: For a proof of the following lemma, we refer to Lesaint [24] chap. 0. Lemma A1 Let Omega be a bounded open subset of IR n with a continuous boundary (in the sense of [28]) let p be given with 1 p 1, let k and r be two non negative fixed integers, and let f 2 i W k r 1;p j 0 be such that (f; u) 0 for all u 2 P k : Then there exists a constant C = C(n; k; r; p; Omega Gamma 27 such that j(f; u)j Ckfk k r 1;p; Omega k r 1 X l=k 1 juj ....

J. NECAS. Les M'ethodes Directes en Th'eorie des Equations Elliptiques. Masson, Paris, 1967.

.... The currents are then applied on Gamma ae Omega ff n Gamma ff and the voltages are also measured on Gamma . We observe that Gamma ae Omega Gamma By H s ( Omega Gamma ; s 2 R, we mean the L 2 based Sobolev space over Omega Gamma whose norm we denote by k : k s; Omega , see [10] for a treatment of these spaces. When considering the model problem we assume that by Omega Gamma Omega ff , Gamma and Gamma ff ; ff 0 we mean Omega = f(x 1 ; x 2 ) 2 R 2 ; 0 x 1 L; 0 x 2 Hg; Omega ff = f(x 1 ; x 2 ) 2 R 2 ; 0 x 1 L; 0 x 2 ffg; Gamma = f(x 1 ; x ....

J. Necas, Les m'ethodes directes en th'eorie des 'equations elliptiques, Masson, Paris, 1967. 19

.... the case where an essential boundary condition is imposed only on a subset Omega D ae Omega of positive measure with a natural boundary condition on its complement Omega N = Omega n Omega D : By Friedrichs inequality, a( Delta; Delta) is still positive definite; cf. Necas [35]. The region Omega is a bounded, polyhedral region in three dimensions. A coarse triangulation is introduced by dividing Omega into nonoverlapping simplices Omega i , i = 1; N , also called substructures. We assume that Omega D is the union of the closure of faces of some, or all, of ....

Jindrich Necas. Les m'ethodes directes en th'eorie des 'equations elliptiques. Academia, Prague, 1967.

....H s (G) 0 s 1, is defined as the space of all u 2 L 2 (G) such that juj 2 H s (G) Z G Z G ju(x) Gamma u(y)j 2 jx Gamma yj d 2s dx dy 1 (1. 1) with norm kuk 2 H s (G) juj 2 H s (G) 1 H 2s G kuk 2 L 2 (G) For a bounded Lipschitz domain G, it can be shown [49, 62] that the space H s (G) is the completion of the space C 1 (G) or C 1 ( G) with respect to k Delta k H s (G) see also [1] The 6 space C 1 (G) consists of the infinitely continuously differentiable functions defined in G. The space C 1 ( G) ae C 1 (G) is the restriction of ....

....or a smooth region to a bounded Lipschitz regions. Lemma 1.1 Let G 1 and G 2 be bounded open regions, and let be a bi Lipschitz mapping from G 1 to G 2 . Then, for u 2 H s (G 2 ) 0 s 1, c ju ffi j H s (G 1 ) juj H s (G 2 ) C ju ffi j H s (G 1 ) This lemma is proved in Necas [62] for s = 0; 1. For intermediate s, we prove it by working straightforwardly with formula (1.1) see also Grisvald [49] 7 1.2.1 Traces spaces We shall need Sobolev spaces on manifolds such as G, or an open subset Gamma ae G. Let us assume that G is a bounded Lipschitz region in d with ....

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Jindrich Necas. Les M'ethodes Directes en Th'eorie des Equations Elliptiques. Academia, Prague, 1967.

....interfaces. Before studying the stability properties of this operator, we need two lemmas for the piecewise linear conforming finite element space. The following lemma is a Poincar e Friedrichs inequality. The idea of the proof can be found in Ciarlet (Theorem 6:1) 2] and in Necas (Chapter 2.7. 2) [10]. Lemma 5. Let Gamma be a subset of Omega i , such that Gamma and Omega i have measures of order H. Then, kuk 2 L 2( Omega i ) H 2 juj 2 W 1( Omega i ) Z Gamma u(x) dx) 2 ; 8u 2 W 1( Omega i ) 26) As a consequence, if R Gamma u(x) dx = 0, we have the ....

Jindrich Necas. Les m'ethodes directes en th'eorie des 'equations elliptiques. Academia, Prague, 1967.

....the space of observations will be H = H 1 (#) 2.14) and the cost function will be J(v) 1 2 #y(v) #nA (#) h # 2 H 1 (#) 2.15) Remark 2. 1 Since y(v) # H 1 2(## and Ay(u) f # L 2(##5 we have y(v) # H 2 (D) for any domain D which satisfies # # D # D ## (see [30], Chap. 4, 1.2, Theorem 1.3, for instance) Therefore, y(v) # H 3 2 (#) having the same values on both the sides of #. Also, #y(v) #nA (#) # H 1 2 (#) #y(v) #nA(# #) # H 1 2 (#) and #y(v) #nA (#) #y(v) #nA(# #) 0. Consequently, above two cost functions make sense. ....

....(2.3) in # also have unique weak solutions given by the solutions of problems (5.1) and (5.2) respectively. We know that there exits an isomorphism and homeomorphism of H 1(## H 1 0(## onto H 1 2 (#) see Theorem 7.53, p. 216, in [1] or Theorem 5.5, p. 99, and Theorem 5.7, p. 103, in [30]) i.e. there are two constants k 1 , k 2 0 such that we have . For any y # H 1(## , there exists v # H 1 2 (#) such that y = v on # and v H 1 2 (#) # k 1 y H 1(## . For any v # H 1 2 (#) there exists y # H 1(## such that y = v on # and y H 1(## ....

J. Necas, Les methodes directes en theorie des equation elliptiques, Editions de l'Academie Tschecoslovaque des Sciences, Prague, 1967.

.... argument, for any u 2 H(curl ; Omega Gamma3 curl u Delta n j Gamma 2 H Gamma1=2 00 ( Gamma ) Moreover, if u 2 H 0; Gamma Gamma (curl ; Omega Gamma1 fl (u) is replaced by fl ;0 (u) and it is not hard to check that curl u Delta n j Gamma 2 H Gamma1=2 ( Gamma ) see [12]. The corresponding theorem related to the tangential components trace mapping is the following: Theorem 2.12 Let us set: H Gamma1=2 ;00 (curl Gamma ; Gamma ) f 2 H Gamma1=2 ;00 ( Gamma ) curl Gamma 2 H Gamma1=2 00 ( Gamma )g: H Gamma1=2 (curl Gamma ; ....

....1 one simply has to resume the proof given in [6] for a Lipschitz polyhedron, and use in particular [5] which is valid in the more general case. 20 decomposition. Namely, let Ym = H 0; Gamma Gamma (curl ; Omega Gamma H 0; Gamma (div 0; Omega Gamma3 the following decomposition holds (see [12]) L 2 ( Omega Gamma = rH 1 0; Gamma Gamma( Omega Gamma Phi H 0; Gamma (div 0; Omega Gamma2 Therefore, 8v 2 H 0; Gamma Gamma (curl ; Omega Gamma ; 9 Psi 2 Ym ; q 2 H 1 0; Gamma Gamma( Omega Gamma such that v = Psi rq: 30) Assuming that Ym = Ym H 1( Omega Gamma2 Phi rSm holds ....

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J. Necas (1967) Les M'ethodes directes en Th'eorie des Equations Elliptiques, Masson, Paris.

....u; bv 2 H 1 0 ( Omega Gamma ; ja (bu; bv)j fl fl fl 1 p b u fl fl fl fl fl fl 1 p bv fl fl fl k p rbukk p rbvk C(1 )kbuk 1 kbvk 1 ; 2.3) where C only depends on Omega Gamma and : Thus, a ( Delta; Delta) is continuous. An application of the Lax Milgram lemma [8, 23] gives uniqueness and existence. Remark 2.1. The estimate (2.2) implies that the coercivity is independent of , and (2.3) implies jja jj C(1 ) 2.3. Stability and Regularity. We are now going to establish stability and regularity of the solution of the problem (1.2) Using coercivity of ....

J. Necas. Les M'ethodes Directes en Theorie des ' Equations Elliptiques. Masson, Paris, 1967.

....2 ( Omega Gamma g: Using Thom ee s explicit form of Ehrling s inequality cf. 8] Lemma 4. 1 for Omega smooth and provable for more general domains (with Lipschitz continuous boundary) via Green s theorem or the compactness of the natural imbedding H 1( Omega Gamma L 2 ( cf. 6 [25] Thm. 6.2 using Conseq. 6.2) and Lions compactness result Thm. 16.4 in [23] 9C 0 8 0: Z Omega j u j 2 ds Cf Gamma1 k u k 2 L 2( Omega Gamma k u k 2 H 1( Omega Gamma g and choosing = k v k 0 =k v k 1 , we see that Z Omega j v j 2 ds 2Ck v ....

J. Necas. Les M'ethodes Directes en Th'eorie des ' Equations Elliptiques. Masson, 1967.

.... analogous to the corresponding proof for the Dirichlet boundary condition as given in [4] However, we note that instead of the maximum minimum principle for the heat equation, for the Neumann problem we need to apply the divergence theorem in a domain with a piecewise C 2 boundary curve (see [15]) Theorem 1.1 Let D and D be two annuli with a common exterior boundary Gamma 2 and interior boundaries Gamma 1 and Gamma 1 , respectively. Denote by u and u classical solutions to the initial boundary value problem (1.1) 1.3) in the domains D and D, respectively, for 1 = 0 ....

Necas, J.: Les Methodes Directes en Theorie des Equations Elliptiques. Masson, Paris 1967.

....get (2.4) What remains is to give the characterization of U . This is, according to its definition (2. 11) the orthogonal complement of gradients of functions from H 1 per (Y ) into L 2 (Y ; R n ) We make three steps: a) U ae H div 0 (Y ; R n ) Recall the definition (see [12] or [24]) u 2 H div 0 (Y ; R n ) j Z Y hu; r idy = 0 8 2 H 1 0 (Y ) 2.12) As H 1 0 (Y ) ae H 1 per (Y ) then u 2 U must have zero divergence. Also recall the following Green s formula for u 2 H div 0 (Y ; R n ) and 2 H 1 (Y ) Z Y hu; r idy = hh T u; ii H Gamma1=2 ( Y ....

....Y hu; r idy = hh T u; ii H Gamma1=2 ( Y ) ThetaH 1=2 ( Y ) 2.13) b) U ae H div 0 per (Y ; R n ) For contradiction, take skew periodic u 2 H div 0 (Y ; R n ) i.e. its the normal fluxes over the opposite faces of Y are equal. Applying an arbitrary mollifier function (see [24]) 2 C 1 per (R n ) on T u 2 H Gamma1=2 ( Y ) we get = T u) 2 H 1=2 ( Y ) The skew periodicity of u implies periodicity of . But now, putting these u and into (2.13) and varying over all 2 C 1 per (R n ) 0, we do not get zero unless T u = 0. Thus, ....

J. Necas, Les m'ethodes directes en th'eorie des 'equations elliptiques, Masson, Paris, 1967.

.... is of positive measure then there exists a positive constant C( Omega ; Gamma 0 ) such that, for all u 2 H 1( Omega Gamma , jjujj 2 H 1( Omega Gamma C( Omega ; Gamma 0 ) juj 2 H 1( Omega Gamma 1 H Z Gamma 0 u 2 ) Proofs of Poincar e s inequality may be found in [38] [69], 71] Friedrichs inequality may be found in [71] and Poincar e Friedrichs inequality in [69] 1.3 Trace Spaces Certain trace spaces play an important role in the analysis of many domain decomposition algorithms. We consider the same Lipschitz region Omega as in the previous section. Let Gamma ....

.... that, for all u 2 H 1( Omega Gamma , jjujj 2 H 1( Omega Gamma C( Omega ; Gamma 0 ) juj 2 H 1( Omega Gamma 1 H Z Gamma 0 u 2 ) Proofs of Poincar e s inequality may be found in [38] 69] 71] Friedrichs inequality may be found in [71] and Poincar e Friedrichs inequality in [69]. 1.3 Trace Spaces Certain trace spaces play an important role in the analysis of many domain decomposition algorithms. We consider the same Lipschitz region Omega as in the previous section. Let Gamma be a simple, closed curve (surface) in Omega : We define juj 2 H 1=2 ( Gamma) min ....

Jindrich Necas. Les m'ethodes directes en th'eorie des 'equations elliptiques. Academia, Prague, 1967.

....fi fi fi fi fi f j x k fi fi fi fi fi 2 L 2 (D) 63) where c 0 1 = 1 2 1 (K) c 0 2 = 73 18 C u (64) are the constants from (56) Explicit L 1 estimate for rp requires calculating constants in the well known Stampacchia s truncation argument. We follow the construction from Necas [19] and calculate explicitly all constants. Proposition 4 Under the above assumptions we have fi fi fi fi fi p x k fi fi fi fi fi L 1 (D) R(D) c 0 2 c 0 1 fl(q) fi fi fi fi fi f x k fi fi fi fi fi L 1 (D) n (65) where R(D) diam D) n=2 1 2 Gamma 1 2n n jDj ....

.... i M 4 (D) 2 j f xm j L 2 (D) n 2M 2 (D)j f xm j L 1 (D) n j Finally j(p m ) 2 j H 1 (D) R(D) c 0 2 c 0 1 j f x k j L 1 (D) n 2 (69) where R(D) is given by (66) We now start the recurrence procedure from the proof of De Giorgi s theorem (see e.g. Necas [19]) Let q 2]6=5; 4=3[ q= q Gamma 1) and 1 = 2. Then we define the sequence k by k 1 = 1 k 2 2 Gamma q q Gamma 1 ; k 2 N : 70) It should be noted that k 1 2 k . Then using (p m ) 1 k 1 = p xm ) 1 k 1 as the test function in the weak form of (61) ....

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Necas J., Les m'ethodes directes en th'eorie des 'equations elliptiques, Masson, Paris, 1967.

....T (v #) v T # H 1 2 (#) H 1 2 (#) # # # K (v T , v T ) tr K) v # 2 # d# (3.2) for every v # [H 1(##0 n where # , # H 1 2 (#) H 1 2 (#) represents the dual pairing between elements in H 1 2 (#) and H 1 2 (#) Proof. For v # [C 2 (##2 n by Green s theorem [11] we have ## div v 2 dx = n # j,k=1 ## #v j #x j #v k #x k dx = n # j,k=1 ## v j # 2 v k #x j #x k dx n # j,k=1 # # v j #v k #x k # j d# = n # j,k=1 ## #v j #x k #v k #x j dx n # j,k=1 # # v j #v k #x k # j d# n # j,k=1 # # v j #v ....

J. Necas, Les methodes directes en theorie des equations elliptiques, Academia, Prague, 1967.

....should be small when k is large and the problem is more easily localized for large k. We remove the restriction that k is large by using the Fredholm theory. The ideas used here are borrowed from several sources. The use of the Rellich identity in boundary value problems has a long history, see [3, 9, 12]. However, the application to mixed problems presented here seems to be new. The estimates presented for Delta Gamma k 2 ; k 6= 0, are an adaptation of arguments in [1, 2] The observation in Lemma 2.2 is an adaptation of ideas in [13, 14] Finally, we note that much effort has been denoted to ....

Jindrich Ne¸cas, Les methodes directes en theorie des equations elliptique, Masson et Cie., Editeurs, Paris, 1967, Translated from Czech.

....0 uds: We shall have occasions to use the following elementary inequality (cf. Grisvard [43] kuk 0; Omega Gamma1 kuk 0; Omega juj 1; Omega 8u 2 H 1( Omega Gamma ; 2 (0; 1) 4. 8) The following theorem is an important case of the so called trace theorem (cf. Necas [57]) Theorem 4.1. The mapping u 7 uj Omega which is defined for u 2 C 1 ( Omega Gamma , has a unique continuous extension as an operator from H 1( Omega Gamma onto H 1=2 ( Omega ) namely kvk 1=2; Omega kvk 1; Omega ; jvj 1=2; Omega jvj 1; Omega : 4.9) This ....

J. Necas. Les m'ethodes directes en th'eorie des 'equations elliptiques. Academia, Prague, 1967.

....the mid surface of the plate, denotes a bounded domain with Lipschitz boundary. This implies that Omega is locally Lipschitz, too. Therefore the exterior unit normal n and the unit tangent vector t on fl (and hence also on Gamma) to (resp. to Omega Gamma are defined almost everywhere (cf. [13]) Points in Omega are denoted by (x; y) where x = x 1 ; x 2 ) 2 and jyj d. Analogously, points in Gamma are denoted by (s; y) with s denoting the arclength on fl. We further define the faces of the plate R Sigma = f(x; y) j x 2 ; y = Sigmadg : 2.3) Then the plate problem consists in ....

Necas, J., Les M'ethodes Directes en Th'eorie des Equations Elliptiques, Masson, Paris 1967.

....mp n: In this case, we have an obvious definition of boundary value, or trace on Omega Gamma of these smooth functions. This can be generalized to functions in H s( Omega Gamma and, more generally, to functions in W s;p under specific hypotheses on the boundary of Omega Gamma see Necas [67] or Lions and Magenes [52] We consider for simplicity 1=2 s 3=2, and we assume only that Omega is a Lipschitz region. For smooth domains, the result also holds for s 3=2. Theorem 1.1 (Trace) Let 1=2 s 3=2. The trace map fl : u uj Omega defined from C 1 0 ( Omega Gamma to L ....

....1. 1, the trace map fl has a continuous right inverse E E : H s Gamma1=2 ( Omega Gamma H s( Omega Gamma : Therefore, E satisfies flEg = g; 8g 2 H s Gamma1=2 ( Omega Gamma and kEgk H s( Omega Gamma C( Omega ; s)kgk H s Gamma1=2 ( Omega Gamma : For a proof and details, see Necas [67]. Trace and extension theorems are very important tools in the analysis of domain decomposition methods. In particular, when considering finite element spaces, it is important to obtain extension theorems with bounds independent of the discretization parameters. For the h version finite element ....

[Article contains additional citation context not shown here]

Jindrich Necas. Les M'ethodes Directes en Th'eorie des ' Equations Elliptiques. Academia, Prague, 1967.

....to be developed apply equally well when a source term is present on the right hand side of (1.1) 1.2. 2 The Method of Integral Equations The Helmholtz equation can of course be treated using the theory of boundary value problems for second order elliptic partial differential equations cf. [85, 112, 44]. However, in light of the close connection between the numerical techniques to be developed in this thesis and the theory of (boundary) integral equations for the Helmholtz equation, we will only give a short summary of the results we need coming from the latter. As in the classical potential ....

....is usually taken to be W (R 2 n D) u 2 H 1 loc (R 2 n D) u r Gamma iku = o(r Gamma 1 2 ) as r 1 ) CHAPTER 1. INTRODUCTION 18 Since we are considering the Helmholtz equation with constant coefficients, the regularity theory for elliptic boundary value problems (cf. [85]) guarantees that solutions of the Helmholtz equation are analytic outside a disk containing the obstacle. Thus, the radiation condition makes sense. A regularity theorem for the boundary value problems of interest is given by Theorem 1.2.4 Let D ae R 2 be a bounded region with C 2 boundary ....

Jindrich Necas. Les M'ethodes Directes en Th'eorie des Equations Elliptiques. Masson et Cie., Prague, 1967.

....to the constraints on the state y F i (y) 0; i = 1; m; 2.3) E(y) 2 K; 2.4) and to the constraints on the control u u a (x) u(x) u b (x) a. e. on Gamma: 2. 5) In this setting, Omega ae R n is a bounded domain with a Lipschitz boundary Gamma according to the definition by Necas [17]. Moreover, sufficiently smooth functions f : Omega Theta R R and g; b : Gamma Theta R 2 R are given. The symbol is used for the derivative in the direction of the unit outward normal on Gamma. The functionals F i : C( Omega Gamma R, i = 1; m, are supposed to be twice ....

....y Gamma b( Delta; y; u) dS m X j=1 j F j (y) hz ; E(y)i; 3.8) L : Y q;p Theta U Theta W 1;oe Theta R m Theta Z R. The regularity of y and fits together, as 2 W 1;oe for all oe n= n Gamma 1) ensures 2 L s( Omega Gamma for all s n= n Gamma 2) Necas [17], Thm. 3.4, p. 69) and j Gamma 2 L r ( Gamma) holds for all r 1 1= n Gamma 2) 17] Thm. 4.2, p.84) Therefore, this definition makes sense. In (3.8) h Delta; Deltai denotes the duality pairing between Z and its dual space Z . The Lagrange function L is of class C 2 with respect ....

[Article contains additional citation context not shown here]

Necas, J.: Les M'ethodes Directes en Th'eorie des Equations Elliptiques. Editeurs Academia, Prague, 1967.

.... Dirichlet problem Our next goal is to establish the estimate (3.1) which is needed to solve the Dirichlet problem. Our argument has two steps. The first step is to consider solutions of (L i)u = 0 which satisfy D ff u = 0; jffj m Gamma 2. For such solutions, we may use an argument of Necas [20] to obtain the bound Z Omega ju(X)j 2 dX Z Omega jr m Gamma1 u(Q)j 2 dQ: Necas s arguments for fourth order equations are also used in [3] As we saw in section 3, such estimates are essential to extending Pipher and Verchota s argument to the equation (L i)u = 0. The second step ....

Jindrich Necas, Les methodes directes en theorie des equations elliptique, Masson et Cie., Editeurs, Paris, 1967, Translated from Czech.

....of the plate to be infinitesimally small. In Kirchhoff Love model, this yields a fourth order elliptic equation. In order to set up a well posed problem, the equations need to be complemented by boundary conditions. A boundary value problem may be reformulated in weak (variational) form [56]. Two basic kinds of boundary conditions are recognized. Essential boundary conditions, such as a Dirichlet boundary condition OE = 0 on Omega for (1.1) need to be explicitly imposed in the weak form of the problem. Natural boundary conditions, such as a Neumann boundary condition OE= n = 0 ....

....u 2 H 1 0( Omega Gamma such that a(u; v) f; v) 2. 2) for all v in V = H 1 0( Omega Gamma2 where H 1 0 is the completion of smooth functions with support in Omega with respect to the norm in H 1( Omega Gamma4 Since the bilinear form a(u; v) is symmetric, continuous and elliptic (coercive) [56], the problem has a unique solution by the Lax Milgram theorem. Let us now use the standard Galerkin approximation. We look for an approximate solution of (2.2) in a finite dimensional subspace V h( Omega Gamma of the space V . The Galerkin approximation is the solution of the following problem: ....

Jindrich Necas. Les m'ethodes directes en th'eorie des 'equations elliptiques. Academia, Prague, 1967.

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J. Necas, Les methodes directes en theorie des equations elliptiques, Masson, 1967.

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J. Necas, Les methodes directes en theorie des equations elliptiques, Masson, 1967.

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Necas, J. (1967). Les m'ethodes directes en th'eorie des 'equations elliptiques. Masson, Paris.

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Necas, J., Les m'ethodes directes en th'eorie des 'equations elliptiques. Academia, Prague, 1967.

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J. Necas,"les m'ethodes directes en th'eorie des 'equations elliptiques", Masson (1967).

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Necas, J., "Les m'ethodes directes en th'eorie des 'equations elliptiques," Academia, Prague, 1967.

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Jindrich Necas. Les m'ethodes directes en th'eorie des 'equations elliptiques. Academia, Prague, 1967.

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Necas J., Les m'ethodes directes en th'eorie des 'equations elliptiques, Academia, Praha, 1967.

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Jindrich Necas. Les m'ethodes directes en th'eorie des 'equations elliptiques. Academia, Prague, 1967.

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J. Necas. Les m'ethodes directes en th'eorie des 'equations elliptiques. Academia, Prague (1967)

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Necas, J. (1967): Les m'ethodes directes en th'eorie des 'equations elliptiques, Masson

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