Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Friedland-Hayman inequality and Caffarelli's contraction theorem

Thomas Beck, David Jerison

Abstract

The Friedland-Hayman inequality is a sharp inequality concerning the growth rates of homogeneous, harmonic functions with Dirichlet boundary conditions on complementary cones dividing Euclidean space into two parts. In this paper, we prove an analogous inequality in which one divides a convex cone into two parts, placing Neumann conditions on the boundary of the convex cone, and Dirichlet conditions on the interface. This analogous inequality was already proved by us jointly with Sarah Raynor. Here we present a new proof that permits us to characterize the case of equality. In keeping with the two-phase free boundary theory introduced by Alt, Caffarelli, and Friedman, such an improvement can be expected to yield further regularity in free boundary problems.

The Friedland-Hayman inequality and Caffarelli's contraction theorem

Abstract

The Friedland-Hayman inequality is a sharp inequality concerning the growth rates of homogeneous, harmonic functions with Dirichlet boundary conditions on complementary cones dividing Euclidean space into two parts. In this paper, we prove an analogous inequality in which one divides a convex cone into two parts, placing Neumann conditions on the boundary of the convex cone, and Dirichlet conditions on the interface. This analogous inequality was already proved by us jointly with Sarah Raynor. Here we present a new proof that permits us to characterize the case of equality. In keeping with the two-phase free boundary theory introduced by Alt, Caffarelli, and Friedman, such an improvement can be expected to yield further regularity in free boundary problems.

Paper Structure

This paper contains 6 sections, 9 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Proof of the Caffarelli contraction theorem
  3. Consequences of the contraction theorem
  4. A Poincaré-Wirtinger inequality
  5. The Friedland-Hayman inequality
  6. Remarks on transport maps on the sphere $\mathbb{S}^{n-1}$

Key Result

Theorem 1.1

FH Let $u_1$ and $u_2$ be non-negative, Hölder continuous functions defined on $\mathbb{R}^n$, with $u_1u_2\equiv0$, and harmonic where they are positive, that is, $\Delta u_i(x) = 0$ whenever $u_i(x)>0$. If $u_i$ is homogeneous of degree $\alpha_i$, then

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4: Brenier Br
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 3.1
  • Theorem 3.2: Erhard, Proposition 2.3 in Er and Carlen, Kerce, Theorem 3 in CK
  • Theorem 3.3: Beckner, Kenig, Pipher BKP; in Section 12.2 of CaS