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

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 e>0 such that L is Lr- resolutive in the optimal range p−e<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 e in terms of p and of a natural “norm” of the harmonic measure ωL.