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$\mathrm{L}^p$-based Sobolev theory on closed manifolds of minimal regularity: Scalar Elliptic Equations

Gonzalo A. Benavides, Ricardo H. Nochetto, Mansur Shakipov

Abstract

This paper and its follow-up arXiv:2508.11109 are concerned with the well-posedness and $\mathrm{L}^p$-based Sobolev regularity for appropriate weak formulations of a family of prototypical PDEs posed on manifolds of minimal regularity. In particular, the domains are assumed to be compact, connected $d$-dimensional manifolds without boundary of class $C^k$ and $C^{k-1,1}$ ($k \geq 1$) embedded in $\mathrm{R}^{d+1}$. The focus of this program is on the $\mathrm{L}^p$-based theory that is sharp with respect to the regularity of the source terms and the manifold. In the present paper, we focus our attention on the case of general scalar elliptic problems. We first establish $\mathrm{L}^p$-based well-posedness and higher regularity for the purely diffusive problems with variable coefficients by localizing and rewriting these equations in flat domains to employ the Calderón--Zygmund theory, combined with duality arguments. We then invoke the Fredholm alternative to derive analogous results for general scalar elliptic problems, underscoring the subtle differences that the geometric setting entails compared to the theory in flat domains.

$\mathrm{L}^p$-based Sobolev theory on closed manifolds of minimal regularity: Scalar Elliptic Equations

Abstract

This paper and its follow-up arXiv:2508.11109 are concerned with the well-posedness and -based Sobolev regularity for appropriate weak formulations of a family of prototypical PDEs posed on manifolds of minimal regularity. In particular, the domains are assumed to be compact, connected -dimensional manifolds without boundary of class and () embedded in . The focus of this program is on the -based theory that is sharp with respect to the regularity of the source terms and the manifold. In the present paper, we focus our attention on the case of general scalar elliptic problems. We first establish -based well-posedness and higher regularity for the purely diffusive problems with variable coefficients by localizing and rewriting these equations in flat domains to employ the Calderón--Zygmund theory, combined with duality arguments. We then invoke the Fredholm alternative to derive analogous results for general scalar elliptic problems, underscoring the subtle differences that the geometric setting entails compared to the theory in flat domains.
Paper Structure (16 sections, 25 theorems, 122 equations)

This paper contains 16 sections, 25 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminary notation
  3. Background
  4. Statement of the main results
  5. Preliminaries on differential geometry
  6. Sobolev spaces on Γ
  7. Definition of Sobolev spaces on Γ
  8. Comparison between different definitions
  9. Sobolev embeddings on Γ
  10. The scalar elliptic theory
  11. Lᵖ-based regularity for problems in divergence form
  12. Lᵖ-based regularity for general scalar elliptic problems
  13. Extensions of \ref{['thm:gsep-wp']} and \ref{['thm:gsep-reg']}
  14. The biharmonic problem on Γ
  15. Variational problems in reflexive Banach spaces
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $p \in (1,\infty)$. Assume $\Gamma$ is of class $C^1$ and let $\boldsymbol{\mathsf{A}} \in \mathbb{C}(\Gamma)$ and that $\boldsymbol{\mathsf{A}}$ is strictly elliptic in $\Gamma$; that is, there is $\Lambda > 0$ such that for a.e. $\mathbf{x} \in \Gamma$ and for each $\boldsymbol{\xi} \in \mathrm{R}^{d+1}$ that is tangential to $\Gamma$ at the point $\mathbf{x}$. Then for each $f \in (\mathrm

Theorems & Definitions (48)

  • Theorem 1.1: well-posedness of the operator $-\operatorname{div}_\Gamma(\boldsymbol{\mathsf{A}} \nabla_\Gamma \cdot)$ in $\mathrm{W}^{1,p}_\#(\Gamma)$
  • Theorem 1.2: $\mathrm{W}^{m+2,p}$-regularity for the operator $-\operatorname{div}_\Gamma(\boldsymbol{\mathsf{A}} \nabla_\Gamma \cdot)$ on $\Gamma$
  • Theorem 1.3: well-posedness of general scalar elliptic equations
  • Theorem 1.4: higher regularity for general scalar elliptic equations
  • Remark 2.1: independence of parametrizations
  • Definition 3.1: Sobolev spaces on $\Gamma$
  • Proposition 3.2: independence of parametrizations
  • proof
  • Definition 3.3: differential operators on Sobolev spaces on $\Gamma$
  • Lemma 3.4: product rule on $\mathrm{W}^{m,p}(\Gamma)$
  • ...and 38 more