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Existence, Uniqueness, Optimization and Stability for Low Eigenvalues of Some Nonlinear Operators

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Existence, Uniqueness, Optimization and Stability for Low Eigenvalues of Some Nonlinear Operators UNIVERSITA` DEGLI STUDI DI TRENTO Dipartimento di Matematica DOTTORATO DI RICERCA IN MATEMATICA XXV CICLO A thesis submitted for the degree of Doctor of Philosophy Giovanni Franzina Existence, Uniqueness, Optimization and Stability for low Eigenvalues of some Nonlinear Operators Supervisor: Prof. Peter Lindqvist Existence, uniqueness, optimization and stability for low eigenvalues of some nonlinear operators Ph.D. Thesis Giovanni Franzina November 12th, 2012 Members of the jury: Lorenzo Brasco – Universit´eAix-Marseille Filippo Gazzola – Politecnico di Milano Peter Lindqvist – Norwegian Institute of Technology Francesco Serra Cassano – Universit`adi Trento Creative Commons License Attribution 2.5 Generic (CC BY 2.5x) Contents Introduction v Chapter 1. Basic preliminaries on nonlinear eigenvalues 1 1. Differentiable functions 1 2. Constrained critical levels and eigenvalues 4 3. Existence of eigenvalues: minimization and global analysis 6 4. Existence of eigenvalues for variational integrals 19 Chapter 2. Classical elliptic regularity for eigenfunctions 25 1. L∞ bounds 25 2. H¨older continuity of eigenfunctions 30 Chapter 3. Hidden convexity for eigenfunctions and applications 35 1. Hidden convexity Lemma 35 2. Uniqueness of positive eigenfunctions 39 3. Uniqueness of ground states 43 4. Uniqueness of positive fractional eigenfunctions 45 Chapter 4. Spectral gap 49 1. A Mountain Pass lemma 49 2. Low variational eigenvalues on disconnected domains 54 3. The Spectral gap theorem 56 Chapter 5. Optimization of low Dirichlet p-eigenvalues 61 1. Faber-Krahn inequality 63 2. The Hong-Krahn-Szego inequality 65 3. The stability issue 66 4. Extremal cases: p = 1 and p = 70 5. Sharpness of the estimates: examples∞ and open problems 73 Chapter 6. Optimization of a nonlinear anisotropic Stekloff p-eigenvalue 76 1. The Stekloff spectrum of the pseudo p-Laplacian 76 2. Existence of an unbounded sequence 79 3. The first nontrivial eigenvalue 81 4. Halving pairs 86 iii iv CONTENTS 5. An upper bound for σ2,p 89 Chapter 7. Anisotropic weighted Wulff inequalities 93 1. Basics on convex bodies 93 2. Differentiation of norms 93 3. Weighted Wulff Inequalities 94 4. Stability issues 98 Chapter 8. An eigenvalue problem with variable exponents 101 1. Preliminaries 101 2. The Euler Lagrange equation 104 3. Passage to infinity 111 4. Local uniqueness 115 5. Discussion about the one-dimensional case 121 Appendix A. Elementary inequalities in RN 125 1. The case p 2 125 2. The case 1 <p<≥ 2 125 3. A useful compactness criterion 128 Bibliography 131 Introduction This thesis is devoted to introduce and discuss the most noteworthy features of what it will be referred to as a nonlinear eigenvalue. A ultimate definition of such a mathematical object lacks in the literature, and may be missing also in the future. In fact, despite looking as a contradictio in terminis (“eigentheories” are all-linear theories), the vague idiomatic ex- pression of nonlinear eigenvalue does however arise naturally from the variational viewpoint. During the last decades, non-linear eigenvalue problems captured the interest of several re- searchers from different areas of mathematical analysis. The model problem is driven by the so-called p-Laplacian, defined by ∆ u := div u p−2 u p |∇ | ∇ for all smooth functions u : RN R. Note that taking p = 2 one is back to the familiar Laplace operator. Almost all features→ of a nonlinear eigenvalue problem are encoded in this nonlinear operator, which is singular or degenerate depending on whether p < 2 or p > 2. Several existence, uniqueness and stability results about the p-Laplacian are easily extended to rather general nonliner partial differential equations. In the thesis, an account is given also of the eigenvalues of some non-local operators. After being studied for a long time in potential theory and harmonic analysis, fractional operators defined via singular integrals are riveting attention since equations involving the fractional Laplacian or similar nonlocal operators naturally surface in several applications. A great attention in the thesis is devoted to carefully define a suitable notion of eigen- values when the exponent itself p is replaced by a function p(x). The definition of p(x)- eigenvalues seems to be new. The viscosity theory for second order differential equations allows one to study the asymptotic behaviour of p(x)-eigenvalues as p(x) approaches a “vari- able infinity” (x). This passage to infinity is accomplished replacing the variable exponent by jp(x) and∞ sending j . The limit problem is identified, and it has a nice geometric interpretation. → ∞ Owing to the importance of superposition principles in nature, the eigenvalues of linear second order elliptic operators appear in areas of mathematical physics ranging from classical to quantum mechanics. For example, the normal modes in the small oscillations near stable equilibria are determined by eigenvalues. Furthermore, in a quantum system the eigenvalues of Schr¨odinger operator represent the possible energy levels. Linear eigenvalues also play a v vi INTRODUCTION crucial role for a better understanding of qualitative properties and long time behaviour of solutions to several partial differential equations governing many physical phaenomena. A model case of elliptic linear eigenvalue problem is given by the celebrated Helmholtz equation ∆u = λu, in Ω, − with the Dirichlet conditions u = 0 on the boundary ∂Ω of the open set Ω RN . The eigenvalues, i.e. the numbers λ such that the above problem is solvable, are⊂ the critical values of the Dirichlet integral u 2 dx |∇ | ZΩ subject to the constraint u2 dx =1. ZΩ Reading the other way round, by computing the first variation of the Rayleigh quotient u 2 dx |∇ | ZΩ u2 dx ZΩ one ends up with an eigenvalue problem for the linear Laplace operator ∆. Note that eigenfunctions can be multiplied by constants. In addition to that, the− equation is also additive. The linearity is due to the quadratic growth in the integrals. If the square is replaced by a different power the linearity is destroyed. Nonetheless the homogeneity is preserved. The fact that the eigenfunctions may be multiplied by constants is an expedient feature of the eigenvalue problems. Despite being non-linear, the problems considered in this thesis do however satisfy this property. Let H(x, z) be convex, even and positively homogeneous of degree p> 1 in the variable z RN . Then, the critical values λ of the variational integrals ∈ H(x, u) dx ∇ ZΩ subject to the constraint u p dx =1 | | ZΩ are the numbers λ such that the Euler-Lagrange equation 1 div H(x u) = λ u p−2u, in Ω, −p ∇z ∇ | | admits a non-trivial weak solution attaining zero Dirichlet conditions on the boundary of Ω. The equation fails to be linear unless p = 2. Nevertheless, if u is solves the equation, then INTRODUCTION vii so does cu. Thus the critical values λ of the quotient H(x, u(x)) dx ∇ ZΩ u(x) p dx | | ZΩ will be called eigenvalues. The corresponding critical points solve the Euler-Lagrange equa- tion and they are said to be eigenfunctions. The same names are used for critical values and critical points of the quotient if the Dirichlet integral is replaced by the double integral u(y) u(x) p K(x, y) dxdy | − | RNZZ×RN where K(x, y) is some convolution kernel that makes the integral meaningful. In that case the (weak) Euler-Lagrange equation reads u(y) u(x) p−2(u(y) u(x))(ϕ(y) ϕ(x))K(x, y) dxdy = λ u(x) p−2u(x)ϕ(x) dx. | − | − − Ω | | RNZZRN Z When p = 2, a suitable choice of the kernel leads to the eigenvalue problem for the linear operator formally definded by s u(x + y)+ u(x y) 2u(y) ( ∆) u(x)= Cs,N N+2−s − dy, − − RN y Z | | where CN,s is some normalization constant. This is called the fractional s-Laplacian. For some accounts about the integro-differential equation involving this non-local operator, the interested reader is referred to the survey [F3] written with Enrico Valdinoci. Basic preliminaries on nonlinear eigenvalues. It is well known that the Dirichlet Laplace operator admits infinitely many eigenvalues 0 <λ λ ... λ . 1 ≤ 2 ≤ ≤ n →∞ This basically relies on the Rellich compactness Theorem for the embedding of the Sobolev 1 2 space H0 (Ω) into L (Ω), which makes compact the “resolvent”, i.e. the self-adjoint operator mapping each right hand side f L2(Ω) to the solution u H1(Ω) of equation ∈ ∈ 0 ∆u = f, in Ω, − with Dirichlet boundary conditions on ∂Ω. Hence by spectral theorem the resolvent admits a sequence of eigenvalues µn converging to zero, and the corresponding un’s are in fact −1 eigenfunctions of ∆ with eigenvalues λn = µn . Moreover, these eigenfunctions give an Hilbert basis of L2−(Ω). To prove the existence of (nonlinear) eigenvalues obtained by minimizing non-quadratic quotients, the lack of linearity makes a bit inefficient the standard methods used in the linear viii INTRODUCTION case, and tools of nonlinear analysis may be of help. There is no reliable spectral theory for producing a “basis” of eigenfunctions. However, there is a plenty of classical procedures to produce eigenvalues. The first chapter is focused on a well established formula for defining a non-decreasing unbounded sequence of critical values of a convex p-homogeneous functional , defined on some Banach space X, along the one-codimensional manifold M = −1( 1 ), whereF is another convex and p-homogeneous functional on X. Namely, one setsG { } G λn = inf max (fω) f ω F for all n N. Here ω ranges among all unit vectors in Rn and the infimum is performed on the class∈ of all odd continuous mappings ω f from the unit sphere of Rn to M. 7→ ω According to the main existence result of Chapter 1, the λn’s are critical values of along M provided that the Palais-Smale condition holds, see Theorem 1.3.3.
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