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Gradient Hölder regularity in mixed local and nonlocal linear parabolic problem

Stuti Das

Abstract

We prove the local Hölder regularity of weak solutions to the mixed local nonlocal parabolic equation of the form \begin{equation*} u_t-Δu+\text{P.V.}\int_{\mathbb{R}^{n}} {\frac{u(x,t)-u(y,t)}{{\left|x-y\right|}^{n+2s}}}dy=0, \end{equation*} where $0<s<1$; for every exponent $α_0\in(0,1)$. Here, $Δ$ is the usual Laplace operator. Next, we show that the gradients of weak solutions are also $α$-Hölder continuous for some $α\in (0,1)$. Our approach is purely analytic and it is based on perturbation techniques.

Gradient Hölder regularity in mixed local and nonlocal linear parabolic problem

Abstract

We prove the local Hölder regularity of weak solutions to the mixed local nonlocal parabolic equation of the form \begin{equation*} u_t-Δu+\text{P.V.}\int_{\mathbb{R}^{n}} {\frac{u(x,t)-u(y,t)}{{\left|x-y\right|}^{n+2s}}}dy=0, \end{equation*} where ; for every exponent . Here, is the usual Laplace operator. Next, we show that the gradients of weak solutions are also -Hölder continuous for some . Our approach is purely analytic and it is based on perturbation techniques.
Paper Structure (16 sections, 17 theorems, 104 equations)

This paper contains 16 sections, 17 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Function spaces
  5. Weak solutions
  6. Steklov averages
  7. Statements of the main results
  8. Auxiliary results
  9. Fundamental Estimates
  10. Caccioppoli estimate
  11. Improving the Caccioppoli estimate
  12. Difference estimate
  13. Merging Caccioppoli and difference estimates
  14. Decay estimate for the non-local integral
  15. Hölder continuity
  16. ...and 1 more sections

Key Result

Lemma 2.3

Let $v \in L^{ r}(0,T;L^q(\Omega))$. Then, as $h\to 0$, $v_h$ converges to $v$ in $L^{ r}(0,T-\varepsilon;L^q(\Omega))$ for every $\varepsilon \in(0, T)$. If $v \in C(0, T ; L^q(\Omega))$, then as $h\to 0$, $v_h(.,t)$ converges to $v(\cdot, t)$ in $L^q(\Omega)$ for every $t \in(0, T-\varepsilon), \f

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 15 more