Brezis H. Analyse Fonctionnelle (Theorie et Applications). Dunod, 1999. In French.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of their corresponding matrices of second order coefficients are strictly positive, they are maximal monotone operators (see e.g. 9] In this case, the Hille Yosida theorem asserts that the associated homogeneous problem (i.e. when F 0) has a unique classical solution (see e.g. Brezis [3]) A classical solution of (11) is a function h 2 C ( 0; T ] H) C( 0; T ] D(A) such that (11) is verified (D(A) is the domain of definition of the operator A in H) It may be shown that any such function satisfies h(t) S(t) h R t S(t s) F (h(s) ds (12) where fS(t)g t 0 is the ....

H. Brezis. Analyse fonctionnelle. Theorie et applications. Masson, 1983.

....(3) for radially symmetric functions and hence with replaced by r 1 n # n 1 # . They showed that the corresponding H radial (r, s) is maximal for (r, s) being extremal which means r = 0 and s = 1 or vice versa. The critical number that they find for this radial case is as follows: r,s#[0,1] H radial (r, s) 1 2n . In the one dimensional case they also considered 2 c without assuming symmetry. Maximal lifetime on the disk. Gri#n, McConnell and Verchota in [9] considered H for general simply connected 2dimensional domains# but fixed y . Two of their main results for ....

H. Brezis, Analyse fonctionnelle, Masson, Paris, 1983. 9

....that every bounded sequence in BV(#) admits a subsequence converging in BV w#. This sequence is also relatively compact in L for 1 p N (N 1, and relatively weakly compact in L (#) for 2 [20] 1] We also have an extension to BV functions of the Poincare Wirtinger inequality [9], 1] for u BV(#) let u : dx. Then there exists M 0 such that #u u# Du (#) for every p 1 and for p 1) if N 1. Then, for N 1, we can take p 2. We deduce that if u BV(#) then u (#) BV(#) is continuously embedded in L (#) For any function u ....

Brezis H (1992) Analyse fonctionnelle. Masson, Paris

....admits a subsequence converging in BV Gamma w . This sequence is also relatively compact in L for 1 p N 1, and relatively weakly compact in L for p = and N 2 (Giusti [35] Acart Vogel [1] We have also an extension to BV Gammafunctions of the Poincar e Wirtinger inequality ([10], 1] for u 2 BV( Omega Gamma6 let u : u(x)dx: Then there exists M 0 such that ku Gamma uk L p M jDuj( Omega Gamma ; where p = 1 for N = 1 and p = for N 1. But, for N = 1, we can take p = 2 instead of p = 1. We deduce that if u 2 BV( Omega Gamma1 then u 2 L (BV( Omega Gamma ....

.... we get: w ; w = p ; w w; p 0 ; Kw p ; Dw w; 0; 8w 2 D( Omega Gamma : Then, we have: Omega Gamma : For p satisfying this relation, we obtain that divp ( Omega Gamma and then, we can define (by a theorem of Lions Magenes [10]) the trace of p Delta on Gamma = where represents the unit normal to Gamma and integrating by parts, we get, for v 2 W Gamma p Delta vd Gamma = P N i=1 (D i p i v)dx P N i=1 (p i D i v)dx = K 0 ; v v; Gamma p 0 ; Kv Gamma v; 0: In this ....

H. BREZIS, Analyse fonctionnelle, Masson 1992.

....We use the following result: Lernrna 10. If C is a closed convex cone in a Banach space X, then C 3 X = C. Proof of Lemma 10. First the inclusion C C C [3 X: if u C, then v f c , f, u) 0. Therefore u C . C rq X c C: suppose that u C rq X, u C. By the Hahn Banach theorem (e.g. author ) [1]) there exists f C such that f, u) 0. But this contradicts the assumption that u C . Proof of Lemma 9. Set M= ATgy : g 7 . M C K: let ATg M, implying that g P. Then V u K, ATg, u : g, Au k O. Therefore AT g K . K C M: we first show that M C K. If u M , then by definition f, u) 0 ....

H. Brezis. Analyse fonctionnelle. Masson, Paris, 1983.

....and u be the unique solution of Poisson s equation: AU = f ulo = 0 We have the following equality: x, dX=fo Vu. Nda(x) where N is the inside pointing unit normal to O and da(x) its area element. Proof. Because of our assumptions, Poisson s equation has a unique classical, i.e. solution in [2, 20] and we have: ff(x, dx = Audx = foVU. Nda(x) the last equality being a consequence of the Green Riemann theorem. A region functional can always be expressed as a boundary functional, via the solution of Poisson s equation with Dirichlet conditions. 3.2. Tranformation of boundary ....

H. Brezis. Analyse fonctionnelle. Th4orie et applications. Masson, 1983.

....from below. We are now ready to introduce the phase space E of the model. Let k k 2 denote the usual norm on L ; dxdy) On C ) k k = kr y k 2 de nes a norm. Let E be the completion of C R ) with this norm. Actually, as a consequence of the Sobolev imbedding theorems ([B],chapter 9) E is the space L ; D; dx) where D = f 2 L 2n ; dy)jr y 2 L ; dy)g: We then de ne E = E R L ) R with the norm: jY j E = k k 2 jpj for Y = q; p) With this norm, E is a Hilbert space. We now write the problem (1:2) in a more ....

Brezis H., Analyse fonctionnelle. Thorie et applications, Masson (1993).

....theorem 2.1 are analytic in . On , the solutions (f; Q) have at least the regularity C 1 m; if 2 C 2 m; m 0. Proof of proposition 2. 2 If f; Q 2 L 1( rf; curl Q 2 L 2( then for nite : 1 2 f = 1 2 G 0 f 2 L 1 in D 0( n f = 0 in H 1 2 ( Then (see [2] for the regularity of weak solutions) f 2 H 2( and by the standard L p elliptic theory, f 2 W 2;p and therefore f 2 C 1; Schauder theory) Let us remark that the equation on Q is elliptic in dimension 1, but is not elliptic in dimension n 2. So it is interesting to obtain an ....

....= 1 2 G 0 Q as curl H = f 2 Q where curl H = 2 H 1 H , we see that H = curl Q = curl ( curl H f 2 ) r ( rH f 2 ) Thus ( r ( rH f 2 ) H = 0 in D 0( H = H 0 in H 1 2 ( 3 (2.3) 8 Q may be found by the inverse formula Q = curl H f 2 . Then H 2 C 2; see [2]) Consider the elliptic system: 8 : 1 2 f = 1 2 G 0 f (f; curl H f 2 ) n f = 0 r( rH f 2 ) H = 0 H = H 0 (2.4) A classical bootstrap argument permits us to see that if (f; H) 2 C m; C m 1; with m 1, then (f; H) 2 C m 1; C m 2; and then f; ....

H. Brezis, Analyse fonctionnelle, theorie et applications, (1993).

....(4) Let H # # f # H : #f#H 0 and S(f) be the functional S : # # # # # H # # R f # ## f) 1 #f# 2 # n## #f, # n # 2 (22) This functional is bounded on H # , hence it is continuous and the restriction of S to the unit ball in span # n n=1. N reach its infimum (Brezis 1983): there is g # span # n n=1. N with #g# = 1 such that 1 #g# 2 # n## #g, # n # 2 = inf # 1 #f# 2 # n## #f, # n # 2 , f # H # # let A be # n## #g, # n # 2 . Hence A 0, and as #g# = 1, one has : A#f# 2 # N # n=1 #f, # n # 2 9 Step 3 Now ....

Brezis, H. (1983). Analyse fonctionnelle, Theorie et applications, Masson.

....2 1 d 2 min j4u(t)j 2 2 j xf j 2 1 j uf j 2 1 sup st Gamma1 jru(s)j 2 2 j vf j 2 1 sup st Gamma1 jrv(s)j 2 2 2dmin R 2 d min mes( Omega Gamma jf j 2 1 d 2 min t 2 7 with R defined in ( 2.9) This proves the first estimate of proposition 2.5. According to [2], application u Gamma r fi fi fiu fi fi fi 2 2 fi fi firu fi fi fi 2 2 fi fi fi 4 u fi fi fi 2 2 is a norm on H 2 n( Omega Gamma7 So we have shown a uniform H 2( Omega Gamma estimate in time of u in order to ensure a time uniform W 1;6 ( Omega Gamma estimate. For v, ....

H. Brezis, Analyse fonctionnelle, Th'eorie et applications, (MASSON).

....in L 2 DEFINITION 2. A path (X(t, t#[0, 1] is said to be admissible in L 2 (X(t, # L 2 for all t) if there exists a path, denoted ( X t (t, t#[0, 1] such that . for all # #L 2 , the scalar function t # Z 1 0 X(t, s)#(s)ds is di#erentiable in the generalized sense [6], and its derivative is t # Z 1 0 X t (t, s)#(s)ds. The total energy is finite: Z 1 0 Z 1 0 X t (t, s) 2 dtds #. X(t, t#[0, 1] is admissible in G if it is admissible in L 2 and (s # X(t, s) # G for all t. The length of an admissible path in L 2 is ....

H. BREZIS, Analyse fonctionnelle, theorie et applications, Masson, Springer Verlag, Paris, 1983. (English translation).

....compact topological space with a countable basis, and m is a positive Radon measure on X with Supp[m] X and m(X) 1. By (u; v) R X uvm(dx) we denote the inner product of H; and by jj:jj the related norm. If D 2 H, we denote by D the subset of m a.e. positive functions. Definition 1. 1 ([5]) A symmetric linear operator A on H with a dense domain D(A) is said to be monotone if, for all u 2 D(A) Au; u) 0. A is maximal if morover it satisfies R(I A) H that is, for all f 2 H there exists u 2 D(A) such that u Au = f: Lemma 1.1 ( 7] The product of two commuting self adjoint ....

....)W 1 f; W 1 f) I Gamma P ) I Gamma W 1 )f; W 1 f) W 1 (I Gamma P ) I Gamma W 1 )f; f) 0: Hence GammaL(I Gamma P ) is a self adjoint operator by the hypothesis (H 3 ) Using ( 4] proposition 1.1.2) we have that GammaL(I Gamma P ) is a maximal monotone operator. Corollary 2. 1 ([5]) We have i) GammaL(I Gamma P ) is closed. ii) For all ff 0; ffI Gamma L(I Gamma P ) is one to one from D(L) into H; ffI Gamma L(I Gamma P ) Gamma1 is a bounded operator and jjff(ffI Gamma L(I Gamma P ) Gamma1 jj 1. Theorem 2.1 Let V ff = ffI Gamma L(I Gamma P ) Gamma1 ....

BREZIS, H. : Analyse fonctionnelle, th'eorie et applications. Masson (1983).

....= lim m 1 PE ( i i p (l m ) j : 4. 10) Indeed, notice that Gamma (x Gamma y) is bounded, smooth, and nonnegative, and that for any x 2 R 2n ; Z R 2n Gamma (x Gamma y) i p (l m) y; u) Gamma p (y; u) j dydu Gamma 0 as m 1: Hence, we may appeal to Egorov s Lemma [4] to conclude that the convergence is almost uniform in x. More precisely, for any r; ffi 0; there exists a subset Omega r;ffi ae B r = n jxj 2 r 2 o with jB r n Omega r;ffi j ffi and such that Z R 2n Gamma (x Gamma y) i p (l m ) y; u) Gamma p (y; u) j dydu ....

Brezis, H., Analyse fonctionnelle, Masson, Paris, 1983.

....C 0 is a constant depending only on M in (11) Inserting these estimates into (13) and using Cauchy s inequality we have RHS of (13) C(jru j 2 1) in R n Theta R (14) where C 0 is a constant depending only on M in (11) hence C depends only on I . Applying the maximum principle [3] to (14) yields for all t 2 [0; T ] for any T 1) kru ( Delta; t)k L 1 (R n ) e ct krI k L 1 (R n ) e ct krIk L 1 (R n ) C T ; 15) where C T 0 depends only on T , and I . This implies that ju (x; t) Gamma u (y; t)j C T jx Gamma yj; for 8x; y 2 R n and 8t 2 [0; T ....

H. Brezis. Analyse Fonctionnelle, Th'eorie et Applicatons. Masson, Paris, 1987.

.... u i (s 0 ) i (s 0 ) i (s) Gamma i (s 0 ) Gamma (s Gamma s 0 ) 0: 53) The vector u i is a row vector and corresponds to the dual element to u i such that u i (s 0 ) u i (s 0 ) k u i (s 0 )k 2 0 : The existence of such an element is given by the Hahnn Banach theorem [46]. The constraints (52) and (53) are called the pseudo arclength normalizations [33] For our theoretical and numerical analysis, we will use the constraints (53) 3.5 Arclength continuation about regular and limit points At first, we will justify the arclength continuation procedure, applied to ....

....since the operator A 1 is bounded, the operator C is coercive : hCw; wi fi 1 kwk 2 0 8w 2 W h ; INRIA Arclength continuation for semiconductor equations 93 where fi 1 = A 0 1 L fi 0 . Thanks to Lax Milgram theorem [45] C is invertible on W h . Besides, the application of the Open Map [46], 47] implies that C Gamma1 is continuous on W h . The problem (127) is now equivalent to the following 8 : A B t C Gamma1 B]D 1 = 0; OE 1 = C Gamma1 BD 1 : 128) It is sufficient now to prove that the operator A B t C Gamma1 B is invertible. We have hAv; vi ....

H. BREZIS, Analyse fonctionnelle, Th'eorie et Application. Masson, Paris 1988.

....that i 6= j there exists a test function OE ij 2 H 1 0 (B) such that D 1; OE ij (k 1 ) E = 0; 3.26) k i (k 1 ) OE ij (k 1 ) AE = 1; 3.27) k j (k 1 ) OE ij (k 1 ) AE = 0; 3. 28) where h; i is the H Gamma1 ; H 1 0 duality product (cf. for example [8], p. 41, lemma 3.2) Taking the duality product of (3.25) with OE ij (k 1 ) for i 6= j) we obtain the convergence in H Gamma1 (B) of A ij n (k) s n k j (k) a ij n k i (k) Gamma b ij n k j (k) Gamma c ij n (3.29) where a ij n = s n k j (k 1 ) ....

.... GammaE g (f) C 3 C 2;fi d 2 (H(f) M) C 1;fi d 2 (H(f) A) 3. 37) where d denotes the distance associated to L 2 (B) Note now that since M is finite dimensional and since A is closed (in L 2 (B) A M is also closed (in L 2 (B) Then, according to the open mapping theorem (see [8] for example) we get a constant C fi 0 such that for any f verifying the estimate fi f(k) 1 Gamma fi a.e. GammaE g (f) C fi d 2 (H(f) M A) 3.38) Since M A is the space of functions spanned by 1 and , then, according to the estimate 8x; y 2 IR; fi fi fi fi exp x 1 exp ....

H. Brezis, Analyse fonctionnelle, th'eorie et applications, Masson, Paris, (1983).

....Remark 2.6 This proof can be extended easily to the more general case of a curvilinear Lipschitz polyhedron. For that, two results have to be generalized. The first one is the decomposition y = y R rs. This is carried out by simply resuming the proof of Proposition 5. 1 in [6] It uses results of [2], 13] 19] and [10] which are valid in the case of curvilinear Lipschitz polyhedra. The second one states that X H 1( Omega Gamma is closed in X and that jj Delta jj 1 and jj Delta jj 0;curl;div are equivalent norms on X H 1 ( Omega Gamma : Corollary 2.5 of [9] can still be applied. ....

....3 Integration by parts formulae 3. 1 General case First of all, let us set: H 0 (div ; fu 2 L 2( Omega Gamma : div u 2 L 2( Omega Gamma u Delta n j Gamma = 0g H 0 (div 0; fu 2 H 0 (div ; div u = 0g 18 The following Hodge decomposition holds for every function u 2 H(curl ; [2]: u = Phi rp with Phi 2 H 0 (div 0; Omega Gamma and curl Phi = curl u : 23) where p is uniquely defined up to a constant as the solution of the following variational problem: Find p 2 H 1 ( Omega Gamma =R such that Z Omega rp Delta rq d Omega = Z Omega u Delta rq d Omega 8q ....

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H. Brezis (1983), Analyse fonctionnelle. Th'eorie et applications, Masson, Paris.

....(x; t) 2 Q o j q(x; t) 0 g. Relation (5.10) shows that u = b on the first set and u = a on the second set. Let us call Q the last set and suppose it has positive measure (otherwise the proof is finished) We have to prove that y = z d on this subset. We use a result found in Brezis [7] p.195 : Lemma 5.2: Let z be in W 1;ff ( with 1 ff 1 and any open subset of IR n . Then rz = 0 a.e. on the set fx 2 j z(x) k g, where k is a real number. As q 2 W 2;1;r loc (Q o ) for r 1, we first apply this result to any compact subset ae Q o and z = q; so q t and ....

H. Brezis, Analyse fonctionnelle. Th'eorie et applications, Masson, Paris (1983).

....equation defined on a variable bounded domain of R N . We suppose in this paper that y is the solution in a Hilbert space V( to the variational equation (1) a( y ; p) l( p) 0 8p 2 V( Here a( ffl; ffl) is a continuous bilinear form satifying the V( ellipticity property [1], 3] and l( ffl) is a continuous linear form on V( Example (E) In many cases, the forms a( ffl; ffl) and l( ffl) can be written as follows: for y; p 2 H 1 0 ( a( y; p) Z X i;j a i;j i y j p dx l(F; p) Gamma Z f p dx ; where f 2 L 2 (R N ) the ....

Brezis, H. (1987): Analyse fonctionnelle, Masson

....) has a unique solution u 2 V . Proof: In this proof, as well as in the whole paper, c denotes various constants. The functional J is strictly convex. Moreover v G(h ffi oe jrvj) is continuous from W 1;q ( Sigma ) into L 1 ( Sigma ) use the Theorem IV: 9, p: 58 in [5], 2.3) and the Lebesgue dominated convergence Theorem) so that the integral on Sigma is a continuous function on W 1;q ( Sigma ) It is easy to check (using Poincar e inequality on Sigma ) that jjvjj V = jjvjj W 1;p jjrvjj L q ( Sigma ) is a norm on V which is ....

H. BREZIS, Analyse fonctionnelle, Th'eorie et Applications, Masson, Paris, 1983.

....(1:2) 1:3) Here Delta = 2 x 2 1 : 2 x 2 n , and u = u(x) u(x 1 ; xn ) A classical solution of (P ) is a function u 2 C 2 ( Omega Gamma C( Omega ) which satisfies (1.1) 1.3) pointwise. It will be more convenient here to work with so called weak solutions, see [Br] as a general reference. We recall that [A,Br] the Sobolev space H 1 is defined by H 1 ( Omega Gamma = fu 2 L 2 ( Omega Gamma : u x i 2 L 2 ; i = 1; ng where u x i are the distributional first order derivatives of u. The space H 1 is a Hilbert space with inner product (u; v) ....

....: 2 x 2 n , and u = u(x) u(x 1 ; xn ) A classical solution of (P ) is a function u 2 C 2 ( Omega Gamma C( Omega ) which satisfies (1.1) 1.3) pointwise. It will be more convenient here to work with so called weak solutions, see [Br] as a general reference. We recall that [A,Br] the Sobolev space H 1 is defined by H 1 ( Omega Gamma = fu 2 L 2 ( Omega Gamma : u x i 2 L 2 ; i = 1; ng where u x i are the distributional first order derivatives of u. The space H 1 is a Hilbert space with inner product (u; v) H 1 = Z Omega uv n X j=1 Z ....

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H. Brezis, Analyse fonctionnelle, Masson 1983.

....the laplacian. We shall need the following result Lemma 3.5. Consider an orthonormal basis of L 2( Omega Gamma formed by eigenvectors of A denoted by (w i ) i=1;1 . Then the set Z of points (x; y) 2 Omega such that Deltaw i (x; y) 6= 0; 8i 1 is dense in Omega Gamma Proof. According to [6] the functions w i are C 2( Omega Gamma3 so the set of points (x; y) 2 Omega such that Deltaw i (x; y) 6= 0; is open in Omega for all i 1. On the other hand if Deltaw i j 0 in an open set 0 ae Omega then w i (x; y) 1 i Delta 2 w i (x; y) 0 in 0 and by Holmgren s uniqueness ....

H. Brezis, Analyse fonctionnelle, Masson, Paris, 1988

....M) H3) f(X; Y ) 0 for jXj R for some fixed R 0 (H4) f 2 W 1;1 loc (R m ) where f(X) R M f(X; Y )jdY j. In (H1) and (H4) we use standard notation for Sobolev spaces (i.e. f 2 W 1;p loc means that f together with all its first order derivatives belong to L p loc ; cf. 1] or [5] for more details) The goal of averaging theory is to approximate the X components of solutions of (2.1) by solutions of the so called averaged system associated to (2.1) dW dt = fflf (W ) 2:2) Roughly speaking, if the vector field (fflf; fflg) is well behaved, this approximation method ....

....in L 2 (B; uniformly in t 2 R. As a consequence 1 t (OE ffi X Gammat Gamma OE) b Delta rOE 2 L 1 (R; L 1 (B; and this expression, viewed as a family of functions of x parametrized by t, is bounded in L 2 (B; uniformly with respect to t 2 R. The Banach Alaoglu theorem (cf. [5]) then shows that the family 1 t (OE ffi X Gammat Gamma OE) b Delta rOE is relatively compact in L 2 (B; equipped with the weak topology. We are going to show that it converges to 0 as t 0 in a weaker topology, namely that of the distributions on B. But this is a standard ....

H. Brezis, Analyse fonctionnelle, th'eorie et applications, Masson, Paris, 1983.

....a measure of the distance between the function v and the data u. Usually R is the norm of a derivative of v or a combination of norms of different derivatives. The minimization is made on the space of functions for which R R(v) is well defined adding some constraints on the domain boundary (see [11] for a definition of Sobolev spaces) The first term is void if we impose the smoothness through a restriction on the shape. This is the case for deformable templates [8] see Sec. 6) or spline snakes [5] see Sec. 9) 2.2 Attraction Potential. Examples In [4] we give a survey of some ....

....the required hypotheses, alternate minimization of (27) will give a simple algorithm for minimization of (26) 4 Resolution using conjugate functions Under some assumptions on a given P , we find a potential P 1 to obtain Q = P . For this we recall the definition of the Conjugate function (see [11, 21, 22]) also called Legendre transform. 4.1 Conjugate functions Definition 1 If is a function from a Euclidean space E to IR, the conjugate function of is (u) sup v2E ( u; v) Gamma (v) 28) Remark 3 Here and may take infinite values but if we assume is not infinite ....

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H. Brezis. Analyse Fonctionnelle, Th'eorie et applications. Masson, Paris, 1983.

....dj Z fjjj Rg [fi( h 2 f(j) Gamma fi( h 1 f(j) dj and this integral tends to zero as R tends to infinity. Consequently fi(u) Gamma fi(f) has tails at infinity that are uniformly integrable in t, and the same holds for v Gammag. We can then apply for instance Corollary IV.26 of [7] or Theorem IV.8.20 of [11] to conclude that ffi(u(t) Gamma fi(f)g t 0 and fv(t) Gamma gg t 0 both are compact in L 1 (R) Let the limit set be defined as = fy 2 i L 1 (R) 2 : 9t n 1; z(t n ) Gamma y 0 in L 1 (R) 2 g: By Step 1, 6= CONVERGENCE TO TRAVELLING WAVES ....

H. Brezis. Analyse fonctionnelle. Masson, Paris, 1983.

....: L 2 ( Omega Gamma Gamma L 2 ( Omega Gamma are the adjoint operators of g : L 2( Omega Gamma Gamma L 2 ( Gamma Theta (0; T ) and S : L 2 ( Omega Gamma Gamma L 2 ( Omega Gamma by identifying the Hilbert spaces with themselves. By Lemma 5, S is also compact (e.g. Brezis [4]) Therefore by the Fredholm alternative, we have either (1) Gamma1 is an eigenvalue of S . 2) Gamma1 is not an eigenvalue of S . Case (1) We prove that this case corresponds to the first alternative in Theorem 1. By compactness of S , there exist N 1 and linearly independent OE k 2 ....

....eigenvalue of S . 2) Gamma1 is not an eigenvalue of S . Case (1) We prove that this case corresponds to the first alternative in Theorem 1. By compactness of S , there exist N 1 and linearly independent OE k 2 L 2( Omega Gamma (1 k N) such that (1 S )OE k = 0 (1 k N) e.g. [4]) Assuming that y(f) n (x; t) 0 (x 2 Gamma; 0 t T ) we have to prove that ( Delta; 0)f = P N k=1 ff k OE k with some ff k 2 R, that is, Delta; 0)f 2 Ker(1 S ) This is straightforward from (5.33) Moreover since 1 S is injective from L 2( Omega Gamma =X 0 to L ....

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Brezis, H., Analyses fonctionnelle, Th'eorie et Applicaitions, Masson, Paris, 1983.

....the laplacian. We shall need the following result Lemma 4.5. Consider an orthonormal basis of L 2( Omega Gamma formed by eigenvectors of the Laplace operator and denoted by (u i ) i=1;1 . Then the set Z of points x 2 Omega such that u i (x) 6= 0; 8i 1 is dense in Omega . Proof. According to [5] the functions u i are continous, so the set of points x 2 Omega such that u i (x) 6= 0; is open in Omega for all i 1. On the other hand, by Holmgren s uniqueness theorem the interior of the set of points x 2 Omega such that u i (x) 0; is void, for all i 1, so by applying Baire s lemma we ....

H. Brezis, Analyse fonctionnelle, Masson, Paris, 1988

....(44) if there exist p 2 C(Q) and OE 2 C(Q 0 ) with t OE 2 C(Q 0 ) and r x OE 2 C(Q 0 ) d such that F (t; x; a) t OE(t; x; a) p(t; x) 0; Phi(t; x; a) r x OE(t; x; a) 0 (45) for all (t; x; a) 2 Q 0 , and fi(F; Phi) 1 otherwise. The Legendre FenchelMoreau transforms (see [9], ch. I, for instance) of ff and fi are respectively given by ff (c; m) supf c; F m; Phi ; F 1 2 j Phij 2 0g (46) and fi (c; m) sup c Gamma c; F m Gamma m; Phi (47) where (F; Phi) 2 E satisfies (45) In (46) we recognize definition (25) of K(c; m) So K ....

....Functions ff and fi are convex with values in ] Gamma 1; 1] Moreover, there is at least one point (F; B) 2 E, namely F = Gamma1; Phi = 0; for which ff is continuous (for the sup norm) and fi is finite. Thus, by the Fenchel Rockafellar duality Theorem (an avatar of the Hahn Banach Theorem, see [9], ch. I, for instance) we have the duality relation inffff (c; m) fi (c; m) c; m) 2 E 0 g (49) supf Gammaff( GammaF; Gamma Phi) Gamma fi(F; Phi) F; Phi) 2 Eg and the infimum is achieved. More concretely, we get I (h) sup c; t OE p m;r x OE (50) with t OE ....

H.Brezis, Analyse fonctionnelle, Masson, Paris, 1974.

....) 2 E 0 g: 31) Both A and B are convex, lower semicontinuous functions from E into ] Gamma 1; 1] Moreover, there is a point (a ff ; b ff ) namely a ff = Gamma1; b ff = 0; where B is continuous (for the sup norm) and A is finite. Thus, we can use Rockafellar s duality theorem, as stated in [Brez], and infer I(ae 0 ; ae T ) minfA ( ae ff ; m ff ) B ( ae ff ; m ff ) ae ff ; m ff ) 2 E 0 g (32) supf GammaA(a; b) Gamma B( Gammaa; Gammab) a; b) 2 Eg: More concretely, we get I(ae 0 ; ae T ) sup X ff Z Q (ae ff ( t Phi ff p) m ff :r Phi ff ) 33) where p 2 C ....

H.Brezis, Analyse fonctionnelle, Masson, Paris, 1974.

.... H 1 0 ( 0; 1[ fv 2 L 2 ( 0; 1[ v 0 2 L 2 ( 0; 1[ and v(0) v(1) 0g; H 2 0 ( 0; 1[ fv 2 L 2 ( 0; 1[ v 0 ; v 00 2 L 2 ( 0; 1[ and v(0) v(1) v 0 (0) v 0 (1) 0g (by v and v we mean the first and second derivatives of v in the sense of distributions see [7]) For v = v 1 ; v 2 ) 2 V we denote: fl(v) v 0 1 cv 2 and ae(v) cv 1 Gamma v 0 2 ) 0 where c 2 C 1 ( 0; 1] is the curvature of the midcurve. Numerische Mathematik Electronic Edition page numbers may differ from the printed version page 434 of Numer. Math. 67: 427 440 ....

Brezis, H. (1983): Analyse fonctionnelle. Masson

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Brezis H. Analyse Fonctionnelle (Theorie et Applications). Dunod, 1999. In French.

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H. Brezis, Analyse fonctionnelle. Theorie et applications, Masson, Paris, 1983.

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H. Brezis, Analyse fonctionnelle, Masson 1987.

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