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Optimality in nonlocal time-dependent obstacle problems

Ioannis Athanasopoulos, Luis Caffarelli, Emmanouil Milakis

Abstract

This paper showcases the effectiveness of the quasiconvexity property in addressing the optimal regularity of the temporal derivative and establishes conditions for its continuity in nonlocal time-dependent obstacle problems.

Optimality in nonlocal time-dependent obstacle problems

Abstract

This paper showcases the effectiveness of the quasiconvexity property in addressing the optimal regularity of the temporal derivative and establishes conditions for its continuity in nonlocal time-dependent obstacle problems.
Paper Structure (4 sections, 7 theorems, 190 equations)

This paper contains 4 sections, 7 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Preliminaries. Statement of the problem.
  3. Optimality of time derivative
  4. Hölder continuity of time derivative

Key Result

Lemma 3.1

Let $Q_1:=B_1\times (-1,0]$ with $B_1:=\{x\in \mathbb R^n:|x|\leq 1\}$. Suppose that $0<\underset{Q_1}\max~v^\varepsilon\leq 1$ where $v^\varepsilon$ is a solution to derprob, then there exists a constant $\sigma>0$, independent of $\varepsilon$, such that implies that $v^\varepsilon\geq 1/2$ in $Q_{1/2}$.

Theorems & Definitions (15)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 5 more