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Interior Schauder estimates for fractional elliptic equations in nondivergence form

P. R. Stinga, M. Vaughan

TL;DR

This work proves sharp interior Schauder estimates for solutions of fractional nonlocal Poisson problems $(-a^{ij}(x)\partial_{ij})^s u=f$ in bounded domains with $0<s<1$ under minimal regularity on the coefficients. The authors develop an extension problem in one extra dimension with weight $z^{2-\frac{1}{s}}$ and introduce a novel $C_s$-viscosity framework tied to Monge--Ampère geometry, enabling a nonlocal Schauder theory, Harnack inequality, and Hopf lemma. Central to the method are MA-geometry Hölder spaces, intrinsic scaling, harmonic approximation, and a perturbative scheme that yields three levels of local approximation: constants, linear MA-polynomials, and MA-polynomials in the degenerate regime. The results advance the regularity theory for fractional nondivergence operators with rough coefficients and provide tools for applications in stochastic processes, membranes, and fractional Monge–Ampère problems. Finite-parameter Schauder and Harnack principles obtained here are expected to underpin further study of nonlocal equations in MA-type geometries.

Abstract

We obtain sharp interior Schauder estimates for solutions to nonlocal Poisson problems driven by fractional powers of nondivergence form elliptic operators $(-a^{ij}(x) \partial_{ij})^s$, for $0<s<1$, in bounded domains under minimal regularity assumptions on the coefficients $a^{ij}(x)$. Solutions to the fractional problem are characterized by a local degenerate/singular extension problem. We introduce a novel notion of viscosity solutions for the extension problem and implement Caffarelli's perturbation methodology in the corresponding degenerate/singular Monge--Ampère geometry to prove Schauder estimates in the extension. This in turn implies interior Schauder estimates for solutions to the fractional nonlocal equation. Furthermore, we prove a new Hopf lemma, the interior Harnack inequality and Hölder regularity in the Monge--Ampère geometry for viscosity solutions to the extension problem.

Interior Schauder estimates for fractional elliptic equations in nondivergence form

TL;DR

This work proves sharp interior Schauder estimates for solutions of fractional nonlocal Poisson problems in bounded domains with under minimal regularity on the coefficients. The authors develop an extension problem in one extra dimension with weight and introduce a novel -viscosity framework tied to Monge--Ampère geometry, enabling a nonlocal Schauder theory, Harnack inequality, and Hopf lemma. Central to the method are MA-geometry Hölder spaces, intrinsic scaling, harmonic approximation, and a perturbative scheme that yields three levels of local approximation: constants, linear MA-polynomials, and MA-polynomials in the degenerate regime. The results advance the regularity theory for fractional nondivergence operators with rough coefficients and provide tools for applications in stochastic processes, membranes, and fractional Monge–Ampère problems. Finite-parameter Schauder and Harnack principles obtained here are expected to underpin further study of nonlocal equations in MA-type geometries.

Abstract

We obtain sharp interior Schauder estimates for solutions to nonlocal Poisson problems driven by fractional powers of nondivergence form elliptic operators , for , in bounded domains under minimal regularity assumptions on the coefficients . Solutions to the fractional problem are characterized by a local degenerate/singular extension problem. We introduce a novel notion of viscosity solutions for the extension problem and implement Caffarelli's perturbation methodology in the corresponding degenerate/singular Monge--Ampère geometry to prove Schauder estimates in the extension. This in turn implies interior Schauder estimates for solutions to the fractional nonlocal equation. Furthermore, we prove a new Hopf lemma, the interior Harnack inequality and Hölder regularity in the Monge--Ampère geometry for viscosity solutions to the extension problem.
Paper Structure (23 sections, 34 theorems, 311 equations)

This paper contains 23 sections, 34 theorems, 311 equations.

Table of Contents

  1. Introduction
  2. Fractional power operators and extension problem
  3. Monge--Ampère setting
  4. Monge--Ampère geometry
  5. Monge--Ampère Hölder spaces
  6. Scaling in the Monge--Ampère geometry
  7. Viscosity solutions to the extension problem
  8. Definitions and preliminary results
  9. A stability result
  10. Analysis of harmonic functions
  11. Explicit barriers
  12. A Hopf lemma
  13. Regularity estimates for classical solutions
  14. Proof of Proposition \ref{['lem:H-classical']}
  15. Harnack inequality and Hölder regularity for viscosity solutions to the extension problem
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume that $\Omega \subset \mathbb{R}^n$ is a bounded domain satisfying the uniform exterior cone condition, $a^{ij}(x) \in C(\Omega) \cap L^\infty(\Omega)$ are symmetric and satisfy eq:ellipticity, and $f \in C_0(\Omega) \cap C^{0,\alpha}(\Omega)$ for some $0 < \alpha < 1$. Let $u \in \operatornam The constants $C$ above depend only on $n$, $s$, $\lambda$, $\Lambda$, $\alpha$, the modulus of con

Theorems & Definitions (72)

  • Theorem 1.1: Schauder estimates
  • Theorem 1.2
  • Remark 1.3: Harnack inequality and Hölder regularity for the fractional problem
  • Theorem 2.1: Particular case of Biswas-Stinga
  • Lemma 3.2
  • Remark 3.3
  • proof : Proof of Lemma \ref{['lem:balls-to-sections-away-from-0']}
  • Remark 3.4
  • Lemma 3.6
  • Lemma 3.7
  • ...and 62 more