Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Schauder estimates for elliptic equations degenerating on lower dimensional manifolds

Gabriele Cora, Gabriele Fioravanti, Stefano Vita

Abstract

In this paper we begin exploring a local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold $$ -\mathrm{div}(|y|^aA(x,y)\nabla u)=|y|^af+\mathrm{div}(|y|^aF)\qquad\mathrm{in \ } B_1\subset\mathbb R^d, $$ where $z=(x,y)\in\mathbb R^{d-n}\times\mathbb R^n$, $2\leq n\leq d$ are two integers and $a\in\mathbb R$. Such equations are a prototypical example of elliptic equations spoiling their uniform ellipticity on the (possibly very) thin characteristic manifold $Σ_0=\{|y|=0\}$ of dimension $0\leq d-n\leq d-2$, having $$λ|y|^a|ξ|^2\leq |y|^aA(x,y)ξ\cdotξ\leqΛ|y|^a|ξ|^2.$$ Whenever $a+n>0$, the weak solutions with a homogeneous conormal boundary condition at $Σ_0$ are provided to be $C^{0,α}$ or even $C^{1,α}$ regular up to $Σ_0$. Our approach relies on a regularization-approximation scheme which employs domain perforation, very fine blow-up procedures, and a new Liouville theorem in the perforated space. Our theory extends to the case of equations degenerating on suitably smooth curved manifolds.

Schauder estimates for elliptic equations degenerating on lower dimensional manifolds

Abstract

In this paper we begin exploring a local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold where , are two integers and . Such equations are a prototypical example of elliptic equations spoiling their uniform ellipticity on the (possibly very) thin characteristic manifold of dimension , having Whenever , the weak solutions with a homogeneous conormal boundary condition at are provided to be or even regular up to . Our approach relies on a regularization-approximation scheme which employs domain perforation, very fine blow-up procedures, and a new Liouville theorem in the perforated space. Our theory extends to the case of equations degenerating on suitably smooth curved manifolds.

Paper Structure

This paper contains 29 sections, 35 theorems, 480 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Functional setting
  3. Weighted Sobolev spaces
  4. Hardy-Poincaré and Poincaré inequalities
  5. Weighted capacity
  6. H=W property
  7. Trace inequalities and compact embeddings
  8. Poincaré-Wirtinger inequality and 2-admissible weights
  9. Functional setting with stability in perforated domains
  10. Functional spaces on perforated domains
  11. Smooth functions
  12. Weighted $L^p$ spaces
  13. Weighted Sobolev spaces
  14. Stable Hardy-Poincaré and Poincaré inequalities
  15. Stable trace inequalities on the boundary of the hole
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $a+n>0$, $p>(d+a_+)/2$, $q>d+a_+$. Let $A$ be a uniformly elliptic matrix satisfying eq:unif:ell and eq:unif:ell:loc with constants $0<\lambda\leq\lambda_*\leq\Lambda_*\leq\Lambda$. Let $\alpha_*=\alpha_*(n,a,\lambda_*/\Lambda_*)$ be defined as in alphastar2. Let Let $A$ be continuous with $f\in L^{p,a}(B_1)$, $F\in L^{q,a}(B_1)^d$ and $u$ be a weak solution to generalPDE in $B_1$. Then, $u\

Figures (2)

  • Figure 1: This image describes Case 1 when $n=d=2$. In this case we have $d_k/r_k\to\infty$ and $\tilde{z}_k=z_k$.
  • Figure 2: The images on the left and on the right describe respectively Case 2 and Case 3 when $n=d=2$. In Case 2 we have $d_k/r_k\leq c$, $|y_k|/r_k\to\infty$ and $\tilde{z}_k=z^0_k$. In Case 3 we have $|y_k|/r_k\leq c$ and $\tilde{z}_k=0$.

Theorems & Definitions (78)

  • Theorem 1.1: Hölder $C^{0,\alpha}$ estimate
  • Theorem 1.2: Schauder $C^{1,\alpha}$ estimate
  • Theorem 1.3: Stable regularity estimates in perforated domains
  • Theorem 1.4: Liouville
  • Remark 1.5
  • Remark 2.1
  • Proposition 2.2: Hardy-Poincaré inequality
  • Proposition 2.3: Poincaré inequality
  • Lemma 2.4: Capacitary ranges
  • proof
  • ...and 68 more