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Boundary Value Problems for Elliptic Partial Differential Equations Mourad Choulli

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Boundary Value Problems for Elliptic Partial Differential Equations Mourad Choulli Boundary value problems for elliptic partial differential equations Mourad Choulli To cite this version: Mourad Choulli. Boundary value problems for elliptic partial differential equations. Master. France. 2019. hal-02406106 HAL Id: hal-02406106 https://hal.archives-ouvertes.fr/hal-02406106 Submitted on 12 Dec 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Mourad Choulli Boundary value problems for elliptic partial differential equations Preface This course is intended as an introduction to the analysis of elliptic partial differen- tial equations. The objective is to provide a large overview of the different aspects of elliptic partial differential equations and their modern treatment. Besides vari- ational and Schauder methods we study the unique continuation property and the stability for Cauchy problems. The derivation of the unique continuation property and the stability for Cauchy problems relies on a Carleman inequality. This inequal- ity is efficient to establish three-ball type inequalities which are the main tool in the continuation argument. We know that historically a central role in the analysis of partial differential equa- tions is played by their fundamental solutions. We added an appendix dealing with the construction of a fundamental solution by the so-called Levi parametrix method. We tried as much as possible to render this course self-contained. Moreover each chapter contains many exercices and problems. We have provided detailed solutions of these exercises and problems. The most parts of this course consist in an enhanced version of courses given by the author in both undergraduate and graduate levels during several years. Remarks and comments that can help to improve this course are welcome. 3 Contents 1 Sobolev spaces ................................................. 7 1.1 Lp spaces . .7 1.2 Approximation by smooth functions . 10 1.3 Weak derivatives . 15 1.4 W k;p spaces . 17 1.5 Extension and trace operators . 23 1.6 Imbedding Theorems . 31 1.7 Exercises and problems . 38 References . 44 2 Variational solutions ............................................ 45 2.1 Stampacchia’s theorem and Lax-Milgram lemma . 45 2.2 Elements of the spectral theory of compact operators . 48 2.3 Variational solutions for model problems . 60 2.3.1 Variational solutions . 60 2.3.2 H2-regularity of variational solutions . 64 2.3.3 Maximum principle . 72 2.3.4 Uniqueness of continuation across a non characteristic hypersurface . 74 2.4 General elliptic operators in divergence form . 81 2.4.1 Weak maximum principle . 82 2.4.2 The Dirichlet problem . 84 2.4.3 Harnack inequalities . 87 2.5 Exercises and problems . 98 References . 104 3 Classical solutions ..............................................105 3.1 Holder¨ spaces . 105 3.2 Equivalent semi-norms on Holder¨ spaces . 111 3.3 Maximum principle . 114 3.4 Some estimates for Harmonic functions . 121 5 6 Contents 3.5 Interior Schauder estimates . 126 3.6 The Dirichlet problem in a ball . 130 3.7 Dirichlet problem on a bounded domain . 133 3.8 Exercises and problems . 136 References . 145 4 Classical inequalities, Cauchy problems and unique continuation . 147 4.1 Classical inequalities for harmonic functions . 147 4.2 Two-dimensional Cauchy problems . 155 4.3 A Carleman inequality for a family of operators . 160 4.4 Three-ball inequalities . 163 4.5 Stability of the Cauchy problem . 168 4.6 Exercises and problems . 178 References . 184 Solutions of Exercises and Problems ..................................185 A Building a fundamental solution by the parametrix method . 237 A.1 Functions defined by singular integrals . 237 A.2 Weakly singular integral operators . 245 A.3 Canonical parametrix . 251 A.4 Fundamental solution . 258 References . 266 Index .............................................................267 Chapter 1 Sobolev spaces We define in this chapter the W k;p-spaces when k is non negative integer and 1 ≤ p < ¥ and we collect their main properties. We adopt the same approach as in the books by H. Brezis´ [2] and M. Willem [13]. That is we get around the theory of distributions. We precisely use the notion of weak derivative which in fact coincide with the derivative in the distributional sense. We refer the interested reader to the books by L. Hormander¨ [3] and C. Zuily [14] for a self-contained presentation of the theory of distributions. The most results and proofs of this chapter are borrowed from [2, 13]. 1.1 Lp spaces n In this chapter W denotes an open subset of R . As usual, we identify a measur- able function f on W by its equivalence class consisting of functions that are equal almost everywhere to f . For simplicity convenience the essential supremum and the essential minimum are simply denoted respectively by the supremum and the minimum. n Let dx be the Lebesgue measure on R . Define Z 1 L (W) = f : W ! C measurable and j f (x)jdx < ¥ : W Theorem 1.1. (Lebesgue’s dominated convergence theorem) Let ( fm) be a sequence in L1(W) satisfying (a) fm(x) ! f (x) a.e. x 2 W, 1 (b) there exists g 2 L (W) so that for any m j fm(x)j ≤ g(x) a.e. x 2 W. Then f 2 L1(W) and Z j fm − f jdx ! 0: W 7 8 1 Sobolev spaces 1 Theorem 1.2. (Fubini’s theorem) Let f 2 L (W1 ×W2), where Wi is an open subset n of R i , i = 1;2. Then Z 1 1 f (x;·) 2 L (W2) a:e: x 2 W1; f (·;y)dy 2 L (W1) W2 and Z 1 1 f (·;y) 2 L (W1) a:e: y 2 W2; f (x;·)dx 2 L (W2): W1 Moreover Z Z Z Z Z dx f (x;y)dy = dy f (x;y)dx = f (x;y)dxdy: W1 W2 W2 W1 W1×W2 Recall that Lp(W), p ≥ 1, denotes the Banach space of measurable functions f on W satisfying j f jp 2 L1(W). We endow Lp(W) with its natural norm Z 1=p p k f kp;W = k f kLp(W) = j f j dx : W Let L¥(W) denotes the Banach space of bounded measurable functions f on W. This space equipped with the norm k f k¥;W = k f kL¥(W) = supj f j: When there is no confusion k f kLp(W) is simply denoted by k f kp. The conjugate exponent of p, 1 < p < ¥, is given by the formula p p0 = : p − 1 Note that we have the identity 1 1 + = 1: p p0 If p = 1 (resp. p = ¥) we set p0 = ¥ (resp. p0 = 1). The following inequalities will be very useful in the rest of this text. 0 Proposition 1.1. (Holder’s¨ inequality) Let f 2 Lp(W) and g 2 Lp (W), where 1 ≤ p ≤ ¥. Then f g 2 L1(W) and Z j f gjdx ≤ k f kpkgkp0 : W Proposition 1.2. (Generalized Holder’s¨ inequality) Fix k ≥ 2 an integer. Let 1 < p j < ¥, 1 ≤ j ≤ k so that 1 1 + ::: + = 1: p1 pk p j k 1 If u j 2 L (W), 1 ≤ j ≤ k, then ∏ j=1 u j 2 L (W) and 1.1 Lp spaces 9 Z k k ∏ ju jjdx ≤ ∏ ku jkp j : W j=1 j=1 Proposition 1.3. (Interpolation inequality) Let 1 ≤ p < q < r < ¥ and 0 < l < 1 given by 1 1 − l l = + : q p r If u 2 Lp(W) \ Lr(W) then u 2 Lq(W) and 1−l l kukq ≤ kukp kukr : n As usual if E ⊂ R is a measurable set, its measure is denoted by jEj. Proposition 1.4. Let 1 ≤ p < q < ¥ and assume that jWj < ¥. If u 2 Lq(W) then u 2 Lp(W) and 1 − 1 kukp ≤ jWj p q kukq: We collect in the following theorem the main properties of the Lp spaces. Theorem 1.3. (i) Lp(W) is reflexive for any 1 < p < ¥. (ii) Lp(W) is separable for any 1 ≤ p < ¥. (iii) Let 1 ≤ p < ¥ and u 2 [Lp(W)]0, the dual of Lp(W). Then there exists a unique p0 gu 2 L (W) so Z p hu; f i = u( f ) = f gudx for any f 2 L (W): W Furthermore 0 kuk[Lp(W)]0 = kgukLp (W): Theorem 1.3 (iii) says that the mapping u 7! gu is an isometric isomorphism 0 between [Lp(W)]0 and Lp (W). Therefore, we always identify in the sequel the dual 0 of Lp(W) by Lp (W). The following theorem gives sufficient conditions for a subset of Lp(W) to be relatively compact. Theorem 1.4. (M. Riesz-Frechet-Kolmogorov)´ Let F ⊂ Lp(W), 1 ≤ p < ¥, admit- ting the following properties. (i) sup f 2F k f kLp(W) < ¥. (ii) For any w b W, we have lim kth f − f kLp(w) = 0; h!0 where th f (x) = f (x − h); x 2 w, provided that jhj sufficiently small. (iii) For any e > 0, there exists w b W so that, for every f 2 F , k f kLp(Wnw) < e. Then F is relatively compact in Lp(W). 10 1 Sobolev spaces We end the present section by a density result, where Cc(W) denotes the space of continuous functions with compact supported in W.
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