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Nontangential Maximal Function estimates for the elliptic Mixed Boundary Value Problem with variable coefficients

Hongjie Dong, Martin Ulmer

Abstract

We consider an elliptic operator $L$ with variable, merely bounded, and measurable coefficients on a Lipschitz domain, and study solutions to $Lu=0$ that attain given Neumann and Dirichlet-regularity data on different parts of the boundary. The boundary data lies in $L^p$ or $W^{1,p}$ respectively, and we show nontangential maximal function estimates of the gradient of the solution. This mixed boundary value problem generalizes the pure Dirichlet, regularity, and Neumann problem with rough boundary data in $L^p$, and the already established mixed boundary value problem for the Laplacian.

Nontangential Maximal Function estimates for the elliptic Mixed Boundary Value Problem with variable coefficients

Abstract

We consider an elliptic operator with variable, merely bounded, and measurable coefficients on a Lipschitz domain, and study solutions to that attain given Neumann and Dirichlet-regularity data on different parts of the boundary. The boundary data lies in or respectively, and we show nontangential maximal function estimates of the gradient of the solution. This mixed boundary value problem generalizes the pure Dirichlet, regularity, and Neumann problem with rough boundary data in , and the already established mixed boundary value problem for the Laplacian.
Paper Structure (11 sections, 14 theorems, 126 equations)

This paper contains 11 sections, 14 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Geometry of the domain and interface
  3. Further definitions
  4. The pure Dirichlet, regularity, and Neumann boundary value problem
  5. Preliminaries: Properties of solutions
  6. Existence of solutions for the mixed boundary value problem
  7. Local estimates for the nontangential maximal function
  8. Existence of solutions to the and mixed boundary value problem
  9. Decay estimates for the nontangential maximal function
  10. Proof of
  11. Uniqueness - Proof of

Key Result

Theorem 1.7

Let $\Omega$ be a bounded Lipschitz domain with Lipschitz constant $l$, $L$ be an elliptic operator as in eq:DefEllipticOperator, and $p_0>2$ the reverse Hölder exponent from Lemma lemma:RevHolderBoundary. Assume that the geometric condition Assumption ass:corkscrew on $\mathcal{D}$ holds and let $

Theorems & Definitions (33)

  • Theorem 1.7
  • Corollary 1.13
  • Definition 2.1
  • Definition 2.2: Lipschitz domain
  • Remark 2.5
  • Proposition 3.3: mourgoglou_solvability_2025
  • Definition 3.8: $(D)_p^L$
  • Definition 3.9: $(N)_p^L$
  • Definition 3.11: $(R)_p^L$
  • Proposition 3.13: dindos_lp_2007 for Dirichlet, dindos_boundary_2017 for Regularity and Neumann
  • ...and 23 more