Abstract. For a general subcritical second-order elliptic operator P in a domain Ω ⊂ R n (or noncompact manifold), we construct Hardyweight W which is optimal in the following sense. The operator P − λW is subcritical in Ω for all λ < 1, null-critical in Ω for λ = 1, and supercritical near any neighborhood of infinity in Ω for any λ > 1. Moreover, if P is symmetric and W > 0, then the spectrum and the essential spectrum of W −1 P are equal to [1, ∞), and the corresponding Agmon metric is complete.Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation P u = 0, the existence of which depends on the subcriticality of P in Ω.

Abstract. Let Ω be a domain in R n and p ∈ (1, ∞). We consider the (generalized) Hardy inequality Ω |∇u| p ≥ K Ω |u/δ| p , where δ(x) = dist (x, ∂Ω). The inequality is valid for a large family of domains, including all bounded domains with Lipschitz boundary. We here explore the connection between the value of the Hardy constantΩ |∇u| p / Ω |u/δ| p and the existence of a minimizer for this Rayleigh quotient. It is shown that for all smooth n-dimensional domains, µp(Ω) ≤ cp, where cp = (1 − 1 p ) p is the one-dimensional Hardy constant. Moreover it is shown that µp(Ω) = cp for all those domains not possessing a minimizer for the above Rayleigh quotient. Finally, for p = 2, it is proved that µ 2 (Ω) < c 2 = 1/4 if and only if the Rayleigh quotient possesses a minimizer. Examples show that strict inequality may occur even for bounded smooth domains, but µp = cp for convex domains.

Let Ω be a domain in R d , d ≥ 2, and 1 < p < ∞. Fix V ∈ L ∞ loc (Ω). Consider the functional Q and its Gâteaux derivative Q ′ given byIn the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q ′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every ψ ∈ C ∞ 0 (Ω) satisfying ψv dx = 0 there exists a constant C > 0 such that C −1 W |u| p dx ≤ Q(u) + C uψ dx p for all u ∈ C ∞ 0 (Ω). As a consequence, we prove positivity properties for the quasilinear operator Q ′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.

Preface page xi 1 Introduction 1.1 Preliminaries 1.2 Classification 1.3 Differential operators and the superposition principle 1.4 Differential equations as mathematical models 1.5 Associated conditions 1.6 Simple examples 1.7 Exercises 2 First-order equations 2.1 Introduction 2.2 Quasilinear equations 2.3 The method of characteristics 2.4 Examples of the characteristics method 2.5 The existence and uniqueness theorem 2.6 The Lagrange method 2.7 Conservation laws and shock waves 2.8 The eikonal equation 2.9 General nonlinear equations 2.10 Exercises 3 Second-order linear equations in two indenpendent variables 3.1 Introduction 3.2 Classification 3.3 Canonical form of hyperbolic equations 3.4 Canonical form of parabolic equations 3.5 Canonical form of elliptic equations 3.6 Exercises vii viii Contents 4 The one-dimensional wave equation 4.1 Introduction 4.2 Canonical form and general solution 4.3 The Cauchy problem and d'Alembert's formula 4.4 Domain of dependence and region of influence 4.5 The Cauchy problem for the nonhomogeneous wave equation 4.6 Exercises 5 The method of separation of variables 5.1 Introduction 5.2 Heat equation: homogeneous boundary condition 5.3 Separation of variables for the wave equation 5.4 Separation of variables for nonhomogeneous equations 5.5 The energy method and uniqueness 5.6 Further applications of the heat equation 5.7 Exercises 6 Sturm-Liouville problems and eigenfunction expansions 6.1 Introduction 6.2 The Sturm-Liouville problem 6.3 Inner product spaces and orthonormal systems 6.4 The basic properties of Sturm-Liouville eigenfunctions and eigenvalues 6.5 Nonhomogeneous equations 6.6 Nonhomogeneous boundary conditions 6.7 Exercises 7 Elliptic equations 7.1 Introduction 7.2 Basic properties of elliptic problems 7.3 The maximum principle 7.4 Applications of the maximum principle 7.5 Green's identities 7.6 The maximum principle for the heat equation 7.7 Separation of variables for elliptic problems 7.8 Poisson's formula 7.9 Exercises 8 Green's functions and integral representations 8.1 Introduction 8.2 Green's function for Dirichlet problem in the plane 8.3 Neumann's function in the plane 8.4 The heat kernel 8.5 Exercises Contents ix 9 Equations in high dimensions 9.1 Introduction 9.2 First-order equations 9.3 Classification of second-order equations 9.4 The wave equation in R 2 and R 9.5 The eigenvalue problem for the Laplace equation 9.6 Separation of variables for the heat equation 9.7 Separation of variables for the wave equation 9.8 Separation of variables for the Laplace equation 9.9 Schrödinger equation for the hydrogen atom 9.10 Musical instruments 9.11 Green's functions in higher dimensions 9.12 Heat kernel in higher dimensions 9.13 Exercises 10 Variational methods 10.1 Calculus of variations 10.2 Function spaces and weak formulation 10.3 Exercises 11 Numerical methods 11.1 Introduction 11.2 Finite differences 11.3 The heat equation: explicit and implicit schemes, stability, consistency and convergence 11.4 Laplace equation 11.5 The wave equation 11.6 Numerical solution...

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