The main goal of the book is to provide a comprehensive and self-contained proof of the, relatively recent, theorem of characterization of the strong maximum principle due to Molina-Meyer and the author, published in Diff. Int. Eqns. in 1994, which was later refined by Amann and the author in a paper published in J. of Diff. Eqns. in 1998. Besides this characterization has been shown to be a pivotal result for the development of the modern theory of spatially heterogeneous nonlinear elliptic and parabolic problems; it has allowed us to update the classical theory on the maximum and minimum principles by providing with some extremely sharp refinements of the classical results of Hopf and Protter-Weinberger. By a celebrated result of Berestycki, Nirenberg and Varadhan, Comm. Pure Appl. Maths. in 1994, the characterization theorem is partially true under no regularity constraints on the support domain for Dirichlet boundary conditions.
Instead of encyclopedic generality, this book pays special attention to completeness, clarity and transparency of its exposition so that it can be taught even at an advanced undergraduate level. Adopting this perspective, it is a textbook; however, it is simultaneously a research monograph about the maximum principle, as it brings together for the first time in the form of a book, the most paradigmatic classical results together with a series of recent fundamental results scattered in a number of independent papers by the author of this book and his collaborators.
Chapters 3, 4, and 5 can be delivered as a classical undergraduate, or graduate, course in Hilbert space techniques for linear second order elliptic operators, and Chaps. 1 and 2 complete the classical results on the minimum principle covered by the paradigmatic textbook of Protter and Weinberger by incorporating some recent classification theorems of supersolutions by Walter, 1989, and the author, 2003. Consequently, these five chapters can be taught at an undergraduate, or graduate, level. Chapters 6 and 7 study the celebrated theorem of Krein–Rutman and infer from it the characterizations of the strong maximum principle of Molina-Meyer and Amann, in collaboration with the author, which have been incorporated to a textbook by the first time here, as well as the results of Chaps. 8 and 9, polishing some recent joint work of Cano-Casanova with the author. Consequently, the second half of the book consists of a more specialized monograph on the maximum principle and the underlying principal eigenvalues.

Preface vii

1 The minimum principle 1

1.1Concept of ellipticity.First consequences 2

1.2Minimum principle of E.Hopf 5

1.3Interior sphere properties 12

1.4Boundary lemma of E.Hopf 19

1.5Positivity properties of super-harmonic functions 23

1.6Uniform decay property of E.Hopf 25

1.7The generalized minimum principle of M.H.Pr otter and H.F.Weinberger 30

1.8Appendix:Smooth domains 32

1.9Comments on Chapter 1 38

2.Classifying super solutions 41

2.1First classification theorem 42

2.2Existence of positive strict super solutions 47

2.3Positivity of there solvent operator 52

2.4Behavior of the positive super solutions on To 52

2.5Second classification theorem 53

2.6Appendix:Partitions of the unity 58

2.7Comments on Chapter 2 60

3.Representation theorems63

3.1The projection on a closed convex set 65

3.2The orthogonal projection on a closed subspace 69

3.3The representation theorem ofF.Riesz 71

3.4Continuity and coercivity of bilinear forms 75

3.5The theorem of G.Stampacchia 76

3.6The theorem of P.D.Lax and A.N.Milgram 78

3.7Projecting on a closed convex set of au.c.B-space78

      3.7.1Basic concepts and preliminaries79

      3.7.2The projection the oren 82

      3.7.3The projection on a closed linear subspace 85

      3.7.4The projection on a closed hyperplane 87

3.8Comments on Chapter 3 88

4.Existence of weak solutions 91

4.1Preliminaries.Sobolev spaces 93

      4.1.1Test functions 93

      4.1.2Weak derivatives.Sobolev spaces 94

      4.1.3Holder spaces of continuous functions 97

      4.1.4Sobolev'sim beddings 98

      4.1.5Compact imbeddings 101

4.2Trace operators 102

4.3Weak solutionsm 114

4.4Continuity of the associated bilinear form 117

4.5Invertibility of(4.4) when β≥ 0 118

      4.5.1Coercivity of the associated bilinear form 118

      4.5.2Existence of weak solutions.There solvent operator 120

4.6Invertibility of(4.4) for arbitraryβ 122

4.7Comments on Chapter 4 126

5.Regularity of weak solutions 129

5.1LP(RN) -estimates for the Laplacian 131

5.2LP(Q) -estimates for the Laplacian 135

5.3General elliptic LP(Q) -estimates when T 1=0 138

5.4The method of continuity 139

5.5Regularity of weak solutions when T 1=0 141

5.6A first glance to the general case when T 10 147

5.7Comments on Chapter 5 152

6.The Krein-Rutman theorem 155

6.1Orderings.Ordered Banach spaces 155

6.2Spectral theory of linear compact operators 161

6.3The Krein-Rutman theorem 164

6.4Preliminaries of the proof of Theorem 6.3 166

6.5Proof of Theorem 6.3 171

6.6Comments on Chapter 6 184

7.The strong maximum principle 187

7.1Minimum principle of J.M.Bony 189

7.2The existence of the principal eigenvalue 195

7.3Two equivalent weak eigenvalue problems 206

7.4Simplicity and dominance of б[ε,β,Ω] 208

      7.4.1Proof of the strict dominance in case To = 0 210

      7.4.2Proof of the strict dominance in case Ti= 0 212

      7.4.3Proof of the strict dominance in the general case 215

7.5The strong maximum principle 215

7.6The classical minimum principles revisited217

7.7Comments on Chapter 7 220

8.Properties of the principal eigenvalue225

8.1Monotonicity properties 226

8.2Point-wise min-max characterizations 229

8.3Concavity with respect to the potential 232

8.4Stability of e along the Dirichlet components of an 234

      8.4.1Proof of Proposition 8.5 236

      8.4.2Proof ofΩ Theorem 8.4 240

8.5Continuous dependence with respect toΩ 240

8.6Continuous dependence with respect to β(χ) 254

8.7Asymptotic behavior of σ(β) as minβ↑∞ 260

8.8Lower estimates of σ[Σ,D,Ω] in terms of|Ω|264

8.9Comments on Chapter 8 267

9.Principal eigenvalues of linear weighted boundary value problems 273

9.1General properties of the mapZ(X) 274

9.2Characterizing the existence of a principal eigenvalue 278

9.3Ascertaining lim x-o.o[S+XV, B,] when V≥0 284

      9.3.1The simplest case 285

      9.3.2The admissible V's satisfying the main theorem 287

      9.3.3The main theorem 290

9.4Characterizing the existence of principal eigenvalues for admissible potentials 307

9.5Comments on Chapter 9 311

Bibliography 319

Index 331

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