Extrapolation for the Lᵖ Dirichlet Problem in Lipschitz domains

الأصل · EN

Let L be a second-order linear elliptic operator with complex coefficients. We show that if the Lᵖ Dirichlet problem for the elliptic system L(u)=0 in a fixed Lipschitz domain Ω in Rᵈ is solvable for some 1<p=p₀< 2(d-1)/d-2, then it is solvable for all p satisfying p₀<p< 2(d-1)/d-2 +ε. The proof is based on a real-variable argument. It only requires that local solutions of L(u)=0 satisfy a boundary Cacciopoli inequality.