Revistes Catalanes amb Accés Obert (RACO)

Lp Regularity of the dirichlet problem for elliptic equations with singular drift

C. Rios

Resum


Let $\mathcal{L}_{0}$ and $\mathcal{L}_{1}$ be two elliptic operators in nondivergence form, with coefficients $\mathbf{A}_{\ell}$ and drift terms $\mathbf{b}_{\ell}$, $\ell =0,1$ satisfying
$$
\sup_{\left| Y-X\right| \le \frac{\delta \left( X\right) }{2}}\frac{\left| \mathbf{A}_{0}\left( Y\right) -\mathbf{A}_{1}\left( Y\right) \right| ^{2}+\delta \left( X\right) ^{2}\left| \mathbf{b}_{0}\left( Y\right) - \mathbf{b}_{1}\left( Y\right) \right| ^{2}}{\delta \left( X\right) }\,dX
$$
is a Carleson measure in a Lipschitz domain $\Omega \subset \mathbb{R}^{n+1}$, $n\ge 1$, (here $\delta \left( X\right) =\operatorname{dist}\left( X,\partial \Omega \right) $). If the harmonic measure $d\omega _{\mathcal{L}_{0}}\in A_{\infty }$, then $d\omega _{\mathcal{L}_{1}}\in A_{\infty }$. This is an analog to Theorem 2.17 in [8] for divergence form operators. As an application of this, a new approximation argument and known results we are able to extend the results in [10] for divergence form operators while obtaining totally new results for nondivergence form operators. The theorems are sharp in all cases.

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