Journal of Fourier Analysis and Applications

, Volume 15, Issue 6, pp 871–903 | Cite as

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



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.


Divergence elliptic equations Dirichlet problem Harmonic-measure Quasiconformal mappings 

Mathematics Subject Classification (2000)

35J25 35B65 35R05 


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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

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