The Dirichlet problem for elliptic operators having a BMO anti-symmetric part. (English) Zbl 1496.35184
Authors’ abstract: The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equation \(\text{div}(A\nabla u)=0\) in the upper half-space \((x,t)\in{\mathbb{R}}^{n+1}_+\) is uniquely solvable when \(n\geq 2\) and the boundary data is in \(L^p({\mathbb{R}}^n,dx)\) for some \(p\in (1,\infty).\) This result is equivalent to saying that the elliptic measure associated to \(L\) belongs to the \(A_\infty\) class with respect to the Lebesgue measure \(dx,\) a quantitative version of absolute continuity.
Reviewer: Dian K. Palagachev (Bari)
MSC:
| 35J15 | Second-order elliptic equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42B37 | Harmonic analysis and PDEs |
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