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Higher order H{ö}lder approximation by solutions of second order elliptic equations

Grigori Rozenblum, Nikolay Shirokov

TL;DR

This work develops a higher-order Hölder theory for approximation by solutions of a second-order elliptic equation on ${\mathbf N}-2$-AD regular sets. By introducing the space ${\mathcal H}_{\mathcal L}^{\mathbf r+\omega}({\mathbf K})$ via local ${\mathcal L}$-harmonic approximants and a compatibility condition, the authors prove a global approximation result: for each $\delta$, there exists a global ${\mathcal L}$-harmonic function $v_{\delta}$ on ${\mathbf K}_{\delta}$ with $|v_{\delta}-f|\le c\delta^{\mathbf r}\omega(\delta)$ on ${\mathbf K}$, and $|v_{\delta}-v_{\delta/2}|$ obeys the same rate. The core method combines a careful extension $f_0$ using a specially constructed kernel $K$ and Green-function estimates to express $f_0$ as an ${\mathcal L}$-potential, with a decomposition that yields $v_{\delta}$ and its error control. Under higher smoothness of the coefficients, derivatives of $v_{\delta}$ converge to surrogate derivatives $f_{(\alpha)}$, enabling generalized Taylor-type expansions of $f$. A counterexample demonstrates the necessity of the local-${\mathcal L}$-harmonic approximation in the general (wild-K) setting, clarifying the limits of local-to-global transfer in irregular geometries.

Abstract

For a given second order elliptic operation $\mathcal{L}$ in a domain $Ω\subset{\mathbb{R}}^\mathbf{N}$, $\mathbf{N}\ $, and a compact set $\mathbf{K}\subsetΩ$, order $\mathbf{N}$-$2$-Ahlfors-David regular, we define the space $\mathcal{H}^{\mathbf{r}+ω}_{\mathcal{L}}(\mathbf{K})$ of continuous functions $f(x),\, x\in\mathbf{K}$, admitting, for any $δ>0$, a local approximation in the $δ$-neighborhood of any point $x\in\mathbf{K}$, with $δ^{\mathbf{r}}ω(δ)$-error estimate, by solutions of the equation $\mathcal{L} u=0$. For such functions, we prove the existence of a global approximation $v_δ$ on $\mathbf{K}$ with the same order of error estimate, by a solution of the same equation in a $δ$-neighborhood of $\mathbf{K}$. A number of properties of these functions $v_δ$ and their derivatives are established.

Higher order H{ö}lder approximation by solutions of second order elliptic equations

TL;DR

This work develops a higher-order Hölder theory for approximation by solutions of a second-order elliptic equation on -AD regular sets. By introducing the space via local -harmonic approximants and a compatibility condition, the authors prove a global approximation result: for each , there exists a global -harmonic function on with on , and obeys the same rate. The core method combines a careful extension using a specially constructed kernel and Green-function estimates to express as an -potential, with a decomposition that yields and its error control. Under higher smoothness of the coefficients, derivatives of converge to surrogate derivatives , enabling generalized Taylor-type expansions of . A counterexample demonstrates the necessity of the local--harmonic approximation in the general (wild-K) setting, clarifying the limits of local-to-global transfer in irregular geometries.

Abstract

For a given second order elliptic operation in a domain , , and a compact set , order --Ahlfors-David regular, we define the space of continuous functions , admitting, for any , a local approximation in the -neighborhood of any point , with -error estimate, by solutions of the equation . For such functions, we prove the existence of a global approximation on with the same order of error estimate, by a solution of the same equation in a -neighborhood of . A number of properties of these functions and their derivatives are established.
Paper Structure (32 sections, 19 theorems, 185 equations)

This paper contains 32 sections, 19 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. The approximation problem
  3. The main results.1
  4. The ideas of the proof
  5. The main result. 2
  6. Structure of the paper
  7. Conventions
  8. Some preparatory facts
  9. Geometry considerations
  10. Estimates of some integrals
  11. Estimates of derivatives of the Green function
  12. The generating kernel and the extension
  13. The extension operator
  14. Construction of the kernel $K$
  15. Estimates of derivatives of $K(x,y)$. First order derivatives
  16. ...and 17 more sections

Key Result

Theorem 1.2

Let ${\mathbf K}$ be ${\mathbf N}$-$2$-AD regular. Suppose that the coefficients of the operator ${\mathcal{L}}$ belong to $C^3$. Then function $f$ defined on ${\mathbf K}$ belongs to the class ${\mathcal{H}}_{{\mathcal{L}}}^{{\mathbf r}+\omega}({\mathbf K})$ if and only if for any $\delta< \frac{1}

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.2
  • Corollary 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 17 more