Differentiability of Solutions to Second-Order Elliptic Equations via Dynamical Systems

For a second-order elliptic equation in divergence form we investigate conditions on the coefficients which imply that all solutions are Lipschitz continuous or differentiable at a given point. We assume the coefficients have modulus of continuity satisfying the square-Dini condition, and obtain additional conditions that examples show are sharp. Our results extend those of previous authors who assume the modulus of continuity satisfies the Dini condition. Our method involves the study of asymptotic properties of solutions to a dynamical system that is derived from the coefficients of the elliptic equation.Comment: 25 page

LRC and Student Support

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Singular Sturm–Liouville Theory on Manifolds

AbstractIn this paper we investigate Schrödinger operators L=−Δg+a(x) on a compact Riemannian manifold (M, g), where the potential function a(x) is assumed to be continuous, but not necessarily bounded, outside of some closed set Σ⊂M of measure zero. Under certain geometric hypotheses on Σ and growth conditions on a(x) as x→Σ, we prove that the Dirichlet extension of L is bounded from below with discrete spectrum; in many cases, a(x) is allowed to approach −∞ as x→Σ. We also consider conditions on Σ and a(x) under which the Sturm–Liouville theory of L is “singular” in that no boundary conditions are needed to specify the eigenvalues and eigenfunctions of L; in particular, this occurs when the domain of L does not depend on boundary conditions, for example, when L is essentially self-adjoint or more generally “essentially Dirichlet” (a new property that we define). The behavior of L on weighted Sobolev spaces is also discussed. In most of the paper we assume that Σ is a k-dimensional submanifold without boundary, but in the last few sections we generalize our results to stratified sets