Journal of Fourier Analysis and Applications

, Volume 15, Issue 6, pp 871–903

The Lp-Solvability of the Dirichlet Problem for Planar Elliptic Equations, Sharp Results

Article

DOI: 10.1007/s00041-009-9075-z

Cite this article as:
Sbordone, C. & Zecca, G. J Fourier Anal Appl (2009) 15: 871. doi:10.1007/s00041-009-9075-z

Abstract

Assume that the elliptic operator L=div (A(x)) is Lp-resolutive, p>1, on the unit disc \(\mathbb{D}\subset \mathbb {R}^{2}\) . This means that the Dirichlet problem
$$\left\{\begin{array}{l@{\quad}l}Lu=0&\mbox{in }\mathbb{D},\\[3pt]u=g&\mbox{on }\partial\mathbb{D}\end{array}\right.$$
is uniquely solvable for any \(g\in L^{p}(\partial\mathbb{D})\) . Then, there exists ε>0 such that L is Lr- resolutive in the optimal range pε<r≤∞ (Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, Conference Board of the Mathematical Sciences, vol. 83, Am. Math. Soc., Providence, 1991). Here we determine the precise value of ε in terms of p and of a natural “norm” of the harmonic measure ωL.

Simultaneous solvability for couples of operators which are pull-back of the Laplacian under a quasiconformal mapping F and its inverse F−1 is also studied.

Finally we consider sequences of operators and study the weak convergence of their harmonic measures.

Keywords

Divergence elliptic equationsDirichlet problemHarmonic-measureQuasiconformal mappings

Mathematics Subject Classification (2000)

35J2535B6535R05

Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”Via Cintia—Complesso Universitario Monte S. AngeloNapoliItaly