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Existence and regularity results for a Neumann problem with mixed local and nonlocal diffusion

Craig Cowan, Mohammad El Smaily, Pierre Aime Feulefack

TL;DR

This work addresses the well-posedness of a mixed local-nonlocal elliptic problem with drift and a nonlocal Neumann boundary condition. By leveraging an extension $\widetilde{u}$ and a carefully constructed functional-analytic framework, it derives regularity $u\in W^{2,p}(\Omega)$ for $f\in L^p(\Omega)$ under precise ranges of the fractional order $s$ and integrability exponent $p$, and proves existence and uniqueness via a continuation (deformation) argument in $\gamma\in[0,1]$. The analysis hinges on detailed regularity estimates for the nonlocal extension and on integration-by-parts formulas in the fractional setting, plus a maximum principle to guarantee uniqueness. The results extend well-posedness theory for purely local or purely nonlocal problems to the mixed diffusion setting with a nonlocal boundary, contributing to the understanding of regularity and spectral properties of mixed-order operators with drift.

Abstract

In this paper, we consider an elliptic problem driven by a mixed local-nonlocal operator with drift and subject to nonlocal Neumann condition. We prove the existence and uniqueness of a solution $u\in W^{2,p}(Ω)$ of the considered problem with $L^p$-source function when $p$ and $s$ are in a certain range.

Existence and regularity results for a Neumann problem with mixed local and nonlocal diffusion

TL;DR

This work addresses the well-posedness of a mixed local-nonlocal elliptic problem with drift and a nonlocal Neumann boundary condition. By leveraging an extension and a carefully constructed functional-analytic framework, it derives regularity for under precise ranges of the fractional order and integrability exponent , and proves existence and uniqueness via a continuation (deformation) argument in . The analysis hinges on detailed regularity estimates for the nonlocal extension and on integration-by-parts formulas in the fractional setting, plus a maximum principle to guarantee uniqueness. The results extend well-posedness theory for purely local or purely nonlocal problems to the mixed diffusion setting with a nonlocal boundary, contributing to the understanding of regularity and spectral properties of mixed-order operators with drift.

Abstract

In this paper, we consider an elliptic problem driven by a mixed local-nonlocal operator with drift and subject to nonlocal Neumann condition. We prove the existence and uniqueness of a solution of the considered problem with -source function when and are in a certain range.
Paper Structure (18 sections, 5 theorems, 113 equations)

This paper contains 18 sections, 5 theorems, 113 equations.

Table of Contents

  1. Introduction and main results
  2. The domain and the coefficients.
  3. The normal derivative of $u$ on $\partial \Omega$.
  4. Some functional spaces
  5. Estimates on $\widetilde{u}$
  6. Case (i): $0<s<1/2$.
  7. Case (ii): $s=1/2.$
  8. Case (iii): ${1}/{2}<s<1$.
  9. The $A_x^1$ term.
  10. The $A_x^i$ term for $i=2,3,4$.
  11. The case $s=\frac{1}{2}$.
  12. The case $s\in (1/2,1)$.
  13. Case (i).
  14. Case (ii).
  15. Case (iii).
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\Omega$ be an open bounded set of $\mathbb{R}^N$ with smooth boundary and $f\in L^p(\Omega).$ Then,

Theorems & Definitions (10)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: AFR22
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof