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

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Abstract

Assume that the elliptic operator L=div (A(x)∇) is L p-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 L r- resolutive in the optimal range p−ε<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 ω L .

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.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia—Complesso Universitario Monte S. Angelo, 80126, Napoli, Italy

    Carlo Sbordone & Gabriella Zecca

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  1. Carlo Sbordone
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  2. Gabriella Zecca
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Correspondence to Carlo Sbordone.

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Communicated by Carlos Kenig.

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Sbordone, C., Zecca, G. The L p-Solvability of the Dirichlet Problem for Planar Elliptic Equations, Sharp Results. J Fourier Anal Appl 15, 871–903 (2009). https://doi.org/10.1007/s00041-009-9075-z

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  • DOI: https://doi.org/10.1007/s00041-009-9075-z

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