[Paper Review] The Boundary value problems for second order elliptic operators satisfying a Carleson condition

Martin Dindoš, Jill Pipher+1 more|arXiv (Cornell University)|Jan 3, 2013

Advanced Harmonic Analysis Research17 references4 citations

TL;DR

This paper establishes the solvability of the Dirichlet regularity problem with data in $H^{1,p}(ar{\partial}\Omega)$ and the Neumann problem with $L^p(\partial\Omega)$ data for second-order elliptic operators in divergence form on Lipschitz domains with small Lipschitz constant. The coefficients satisfy a Carleson condition with small norm, allowing rough but controlled oscillations, and the results hold for all $1 < p < \infty$, completing earlier work on $L^p$ Dirichlet problems and extending solvability to non-symmetric, non-divergence form operators in higher dimensions.

ABSTRACT

Let $Ω$ be a Lipschitz domain in $\mathbb R^n$ $n\geq 2,$ and $L=\mbox{div} (A abla\cdot)$ be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in $H^{1,p}(\partialΩ)$ and of the Neumann problem with $L^p(\partialΩ)$ data for the operator $L$ on Lipschitz domains with small Lipschitz constant. We allow the coefficients of the operator $L$ to be rough obeying a certain Carleson condition with small norm. These results complete the results of [5] where $L^p(\partialΩ)$ Dirichlet problem was considered under the same assumptions and [6] where the regularity and Neumann problems were considered on two dimensional domains.

Motivation & Objective

• To extend the solvability of boundary value problems for non-symmetric, divergence-form elliptic operators with minimal regularity assumptions on coefficients.
• To establish solvability of the Dirichlet regularity problem with data in $H^{1,p}(ar{\partial}\Omega)$ for $1 < p < \infty$ on Lipschitz domains with small Lipschitz constant.
• To prove solvability of the Neumann problem with $L^p(\partial\Omega)$ data under the same conditions, completing the full range of $p$ for both problems.
• To unify and generalize prior results on $L^p$ Dirichlet problems and Neumann problems in two dimensions, extending them to higher dimensions.

Proposed method

• The authors use a Carleson condition on the oscillation of coefficients, defined via $ d\mu = \delta(X)^{-1} (\text{osc}_{B(X,\delta(X)/2)} a_{ij})^2 dX $, with small Carleson norm.
• They apply layer potential methods and maximal function estimates, relying on the smallness of the Lipschitz constant of the domain and the Carleson norm of the coefficient oscillation.
• A key technique involves integration by parts in tangential directions to handle second derivatives of the coefficients, reducing higher-order terms to lower-order ones.
• The proof uses a bootstrap argument based on pointwise estimates for $|\nabla u|$ away from the boundary, derived from the Carleson condition on $|\nabla A|$.
• The Neumann problem is analyzed via the Lax-Milgram theorem in $H^1$ spaces, followed by non-tangential maximal function estimates and decay control in the interior.
• The argument is adapted from smooth to Lipschitz domains using a localization and flattening technique, with the small Lipschitz constant ensuring uniform control.

Experimental results

Research questions

• RQ1 Can the Dirichlet regularity problem with $H^{1,p}$ data be solved for non-symmetric, divergence-form elliptic operators with rough coefficients satisfying a Carleson condition on the domain boundary?
• RQ2 Is the Neumann problem with $L^p$ data solvable for the same class of operators and domains, for all $1 < p < \infty$?
• RQ3 Does the solvability of the Neumann problem extend to higher dimensions when the coefficients satisfy a Carleson condition with small norm, even without symmetry?
• RQ4 How does the smallness of the Lipschitz constant of the domain interact with the Carleson norm of the coefficient oscillation to ensure solvability?
• RQ5 Can the results from two-dimensional settings be generalized to arbitrary dimensions under minimal regularity assumptions on the coefficients?

Key findings

• The Dirichlet regularity problem with $H^{1,p}(ar{\partial}\Omega)$ data is solvable for all $1 < p < \infty$ on Lipschitz domains with sufficiently small Lipschitz constant.
• The Neumann problem with $L^p(\partial\Omega)$ data is solvable for all $1 < p < \infty$ under the same assumptions, with a uniform estimate $\|N(\nabla u)\|_{L^p(\partial\Omega)} \leq C\|f\|_{L^p(\partial\Omega)}$.
• The solvability holds under the assumption that the measure $ d\mu = \delta(X)^{-1} (\text{osc}_{B(X,\delta(X)/2)} a_{ij})^2 dX $ is a Carleson measure with small norm.
• The results complete the $L^p$ solvability program for divergence-form operators by extending the full range of $p$ to both regularity and Neumann problems.
• The proof establishes that the Neumann problem solvability is robust under small perturbations of the domain and coefficients, even in non-symmetric settings.
• The method applies uniformly to both smooth and Lipschitz domains with small constant, and the estimates are stable under the smallness assumptions on $\ell$ and $\|\mu\|_{\text{Carl},r_0}$.