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Optimal Sobolev Regularity for Second Order Divergence Elliptic Operators on Domains with Buried Boundary Parts

Joachim Rehberg, Elmar Schrohe

TL;DR

This work establishes sharp Sobolev regularity for solutions to second-order divergence-form elliptic problems with discontinuous coefficients on domains with buried boundary parts. By localizing the problem, applying bi-Lipschitz transformations to map the geometry to model constellations, and exploiting symmetry to reduce to mixed Dirichlet–Neumann problems, the authors show that near the buried boundary border the solution inherits the same singularity structure as in the corresponding mixed boundary case. Consequently, for data in $W^{-1,3+\varepsilon}$, the solution enjoys $W^{1,3+\varepsilon}$ regularity in a neighborhood of the buried boundary, with the precise $q$-range governed by local geometric and coefficient assumptions. These results provide a rigorous foundation for elliptic-operator analyses in semiconductor models with buried contacts and have potential implications for studying rigid inclusions in mechanics, while highlighting the intrinsic limits set by mixed-boundary singularities.

Abstract

We study the regularity of solutions of elliptic second order boundary value problems on a bounded domain $Ω$ in $\mathbb R^3$. The coefficients are not necessarily continuous and the boundary conditions may be mixed, i.e. Dirichlet on one part $D$ of the boundary and Neumann on the complementing part. The peculiarity is that $D$ is partly `buried' in $Ω$ in the sense that the topological interior of $Ω\cup D$ properly contains $Ω$. The main result is that the singularity of the solution along the border of the buried contact behaves exactly as the singularity for the solution of a mixed boundary value problem along the border between the Dirichlet and the Neumann boundary part.

Optimal Sobolev Regularity for Second Order Divergence Elliptic Operators on Domains with Buried Boundary Parts

TL;DR

This work establishes sharp Sobolev regularity for solutions to second-order divergence-form elliptic problems with discontinuous coefficients on domains with buried boundary parts. By localizing the problem, applying bi-Lipschitz transformations to map the geometry to model constellations, and exploiting symmetry to reduce to mixed Dirichlet–Neumann problems, the authors show that near the buried boundary border the solution inherits the same singularity structure as in the corresponding mixed boundary case. Consequently, for data in , the solution enjoys regularity in a neighborhood of the buried boundary, with the precise -range governed by local geometric and coefficient assumptions. These results provide a rigorous foundation for elliptic-operator analyses in semiconductor models with buried contacts and have potential implications for studying rigid inclusions in mechanics, while highlighting the intrinsic limits set by mixed-boundary singularities.

Abstract

We study the regularity of solutions of elliptic second order boundary value problems on a bounded domain in . The coefficients are not necessarily continuous and the boundary conditions may be mixed, i.e. Dirichlet on one part of the boundary and Neumann on the complementing part. The peculiarity is that is partly `buried' in in the sense that the topological interior of properly contains . The main result is that the singularity of the solution along the border of the buried contact behaves exactly as the singularity for the solution of a mixed boundary value problem along the border between the Dirichlet and the Neumann boundary part.
Paper Structure (13 sections, 22 theorems, 52 equations, 5 figures)

This paper contains 13 sections, 22 theorems, 52 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and general assumptions
  3. Notation, Preliminary Results, Model Constellations
  4. Notation
  5. Localization
  6. The geometric setting
  7. The Main Result
  8. Localization principles
  9. Auxiliary results
  10. The proof
  11. Regularity near points in $D^\parallel_d$ and $R_d$
  12. Regularity near points in $D^\parallel_c$ and $R_c$
  13. Concluding remarks

Key Result

Theorem 3.1

Let $\Lambda \subset \mathbb{R}^d$ be a domain and $D \subset \partial \Lambda$ a closed subset of the boundary. Suppose that $\rho$ is an elliptic coefficient function. Let $U_1,\ldots, U_n$ be an open covering of $\overline \Lambda$ and define $\Lambda_j = U_j \cap \Lambda$, $N_j = U_j \cap (\part

Figures (5)

  • Figure 1: The model set for the first case in \ref{['e-Neummanrand']}
  • Figure 2: The model set for the second case in \ref{['e-Neummanrand']}
  • Figure 5: $\mathfrak Q_-$ with the grey Neumann surface $M_-$; the remaining surfaces carry Dirichlet b.c..
  • Figure 6: $\mathfrak Q_-$ with the Neumann surface $M_-$ of \ref{['e-M8']}, consisting of the grey and the dotted part. The remaining surfaces carry Dirichlet b.c..
  • Figure 7: The case $\Lambda = \mathfrak C\setminus \Sigma_1$, $M=\emptyset$. The figure shows $\mathfrak C_-$ with the surface $\Sigma_1$ from $D^\parallel$ in black. The whole boundary is Dirichlet, on the crosshatched part due to antisymmetric reflection. The case $\Lambda=\mathfrak Q\setminus \Sigma_1$, $M=\emptyset$ gives an analogous picture for $\mathfrak Q_-$.

Theorems & Definitions (47)

  • Definition 2.1
  • Remark 2.2
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Definition 3.4
  • Lemma 3.5
  • proof
  • Definition 3.6
  • Lemma 3.7
  • ...and 37 more