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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The $L^p$ Dirichlet problem and nondivergence harmonic measure


Author: Cristian Rios
Journal: Trans. Amer. Math. Soc. 355 (2003), 665-687
MSC (2000): Primary 35J25; Secondary 35B20, 31B35
DOI: https://doi.org/10.1090/S0002-9947-02-03145-8
Published electronically: October 1, 2002
MathSciNet review: 1932720
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Abstract: We consider the Dirichlet problem

\begin{displaymath}\left\{ \begin{array}{rcl} \mathcal{L} u & = & 0\quad\text{ in }D,\\ u & = & g\quad\text{ on }\partial D \end{array}\right.\end{displaymath}

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, $\Vert Nu\Vert _{L^p}\le C\,\Vert g\Vert _{L^p}$ 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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Additional Information

Cristian Rios
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Address at time of publication: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8R-B19 Canada
Email: riosc@math.mcmaster.ca

DOI: https://doi.org/10.1090/S0002-9947-02-03145-8
Keywords: Nondivergence elliptic equations, Dirichlet problem, harmonic measure
Received by editor(s): April 5, 2002
Received by editor(s) in revised form: May 17, 2002
Published electronically: October 1, 2002
Dedicated: In memory of E. Fabes
Article copyright: © Copyright 2002 American Mathematical Society