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Higher order interpolative geometries and gradient regularity in evolutionary obstacle problems

Sunghan Kim, Kaj Nyström

Abstract

We prove new optimal $C^{1,α}$ regularity results for obstacle problems involving evolutionary $p$-Laplace type operators in the degenerate regime $p > 2$. Our main results include the optimal regularity improvement at free boundary points in intrinsic backward $p$-paraboloids, up to the critical exponent, $α\leq 2/(p-2)$, and the optimal regularity across the free boundaries in the full cylinders up to a universal threshold. Moreover, we provide an intrinsic criterion by which the optimal regularity improvement at free boundaries can be extended to the entire cylinders. An important feature of our analysis is that we do not impose any assumption on the time derivative of the obstacle. Our results are formulated in function spaces associated to what we refer to as higher order or $C^{1,α}$ intrinsic interpolative geometries.

Higher order interpolative geometries and gradient regularity in evolutionary obstacle problems

Abstract

We prove new optimal regularity results for obstacle problems involving evolutionary -Laplace type operators in the degenerate regime . Our main results include the optimal regularity improvement at free boundary points in intrinsic backward -paraboloids, up to the critical exponent, , and the optimal regularity across the free boundaries in the full cylinders up to a universal threshold. Moreover, we provide an intrinsic criterion by which the optimal regularity improvement at free boundaries can be extended to the entire cylinders. An important feature of our analysis is that we do not impose any assumption on the time derivative of the obstacle. Our results are formulated in function spaces associated to what we refer to as higher order or intrinsic interpolative geometries.
Paper Structure (33 sections, 31 theorems, 278 equations)

This paper contains 33 sections, 31 theorems, 278 equations.

Table of Contents

  1. Introduction
  2. Modulus of continuity
  3. Intrinsic space-time cylinders and $p$-paraboloids
  4. Higher order intrinsic interpolative geometries
  5. Function spaces
  6. Main results
  7. Proofs: arguing from the smallest degenerate scale
  8. Organization of the paper
  9. Review of the literature, our contribution and a discussion of sharpness
  10. Optimal regularity and the intrinsic geometry: a discussion of sharpness
  11. Preliminaries
  12. Concept of solutions
  13. Technical tools
  14. Gradient estimates for weak solutions
  15. Interior gradient estimates revisited
  16. ...and 18 more sections

Key Result

Theorem 1.4

Let $H$ be as in generalH, with $(a,s)$ as in asp and eq:a-C2, let $\psi$ be an obstacle in $\mathcal{O}$, and let $u$ be a weak solution to $\max\{H u, \psi-u\} = 0$ in $\mathcal{O}$. Then there exists $\alpha_h \equiv \alpha_h(n,p,\nu,L)$, $\alpha_h \in(0,1)$, such that if $\omega$ verifies eq:om for some ${\bar{r}}>0$ and $\bar{\mu}>1$, then $u\in \tilde{C}_s^{1,\omega}(x_0,t_0;{\bar{r}},\bar{

Theorems & Definitions (62)

  • Definition 1.1: Pointwise regularity
  • Definition 1.2: Pointwise regularity in backward $p$-paraboloids
  • Definition 1.3: Full regularity
  • Theorem 1.4: Regularity in the interior
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9: Regularity improvement at free boundaries
  • Corollary 1.10
  • ...and 52 more