The 𝐿^{𝑝} Dirichlet problem and nondivergence harmonic measure

Rios, Cristian

doi:10.1090/S0002-9947-02-03145-8

The $L^p$ Dirichlet problem and nondivergence harmonic measure
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Trans. Amer. Math. Soc. 355 (2003), 665-687 Request permission

Abstract:

We consider the Dirichlet problem \[ \left \{ \begin {aligned} \mathcal {L} u & = 0 &&\text {in $D$},\\ u &= g &&\text {on $\partial D$} \end {aligned} \right .\] for two second-order elliptic operators $\mathcal {L}_k u=\sum _{i,j=1}^na_k^{i,j}(x) \partial _{ij} u(x)$, $k=0,1$, in a bounded Lipschitz domain $D\subset \mathbb {R}^n$. The coefficients $a_k^{i,j}$ belong to the space of bounded mean oscillation BMO with a suitable small BMO modulus. We assume that ${\mathcal {L}}_0$ is regular in $L^p(\partial D, d\sigma )$ for some $p$, $1<p<\infty$, that is, $\|Nu\|_{L^p}\le C \|g\|_{L^p}$ for all continuous boundary data $g$. Here $\sigma$ is the surface measure on $\partial D$ and $Nu$ is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients $\varepsilon ^{i,j}(x)=a^{i,j}_1(x)-a^{i,j}_0(x)$ that will assure the perturbed operator $\mathcal {L}_1$ to be regular in $L^q(\partial D,d\sigma )$ for some $q$, $1<q<\infty$.

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