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On the Dirichlet-Neumann operator for nearly spherical domains

Pietro Baldi, Vesa Julin, Domenico Angelo La Manna

Abstract

We consider the Dirichlet-Neumann operator for a nearly spherical domain in R^n, and prove sharp analytic and tame estimates in Sobolev class. The novelty of this paper concerns technical improvements, the most important of which are the independence of the analyticity radius on the high norms and the regularity loss of one in the elevation function. These properties are expectable but nontrivial to prove. The result is obtained by introducing local charts and a convenient class of non-isotropic Sobolev spaces of high, possibly fractional tangential regularity and integer, limited regularity in the normal direction.

On the Dirichlet-Neumann operator for nearly spherical domains

Abstract

We consider the Dirichlet-Neumann operator for a nearly spherical domain in R^n, and prove sharp analytic and tame estimates in Sobolev class. The novelty of this paper concerns technical improvements, the most important of which are the independence of the analyticity radius on the high norms and the regularity loss of one in the elevation function. These properties are expectable but nontrivial to prove. The result is obtained by introducing local charts and a convenient class of non-isotropic Sobolev spaces of high, possibly fractional tangential regularity and integer, limited regularity in the normal direction.

Paper Structure

This paper contains 5 sections, 28 theorems, 248 equations.

Table of Contents

  1. Introduction and main results
  2. Function spaces
  3. Higher tangential Sobolev regularity
  4. Analysis of the Dirichlet-Neumann operator
  5. Appendix

Key Result

Theorem 1.1

Let $s_0, s \in {\mathbb R}$, with $s \geq 1/2$, $s_0 > (n-1)/2$. There exist $\delta, C_0, C_s > 0$ such that, for $h$ in the set for $\psi \in H^{s+1}({\mathbb S}^{n-1})$, the function $G(h)\psi$ satisfies where $(\chi_{s>s_0}) = 1$ if $s>s_0$, $(\chi_{s>s_0}) = 0$ if $s \leq s_0$. The constants $\delta, C_0$ are independent of $s$, the constant $C_s$ is increasing in $s$, and all $\delta, C_0

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 48 more