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Local Hölder regularity for bounded, signed solutions to nonlocal Trudinger equations

Karthik Adimurthi

Abstract

We prove local Hölder regularity for bounded and sign-changing weak solutions to nonlocal Trudinger equations of the form \[ (|u|^{p-2}u)_t + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}} = 0, \] in the range $1< p<\infty$ and $s \in (0,1)$. One of the main difficulties in extending the local theory to the nonlocal Trudinger equation is that when $0 \ll u \ll \infty$ locally, a crucial change of variable is unavailable in the nonlocal case due to the presence of the Tail term. We adapt several new ideas developed in the past few years to prove the required Hölder regularity.

Local Hölder regularity for bounded, signed solutions to nonlocal Trudinger equations

Abstract

We prove local Hölder regularity for bounded and sign-changing weak solutions to nonlocal Trudinger equations of the form in the range and . One of the main difficulties in extending the local theory to the nonlocal Trudinger equation is that when locally, a crucial change of variable is unavailable in the nonlocal case due to the presence of the Tail term. We adapt several new ideas developed in the past few years to prove the required Hölder regularity.

Paper Structure

This paper contains 43 sections, 30 theorems, 210 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A brief history of the problem
  3. On historical development of intrinsic scaling
  4. Our approach to proving Hölder regularity to .
  5. Notations
  6. Function spaces
  7. Definitions
  8. Auxiliary Results
  9. Preliminary Estimates
  10. Shrinking Lemma
  11. Tail Estimates
  12. Expansion of Positivity
  13. De Giorgi Lemma
  14. Expansion of positivity in time
  15. A Shrinking Lemma
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $p\in(1,\infty)$, $s\in(0,1)$ and let $u\in L^p(I;W^{s,p}_{\text{loc}}(\Omega))\cap L^\infty(I;L^2_{\text{loc}}(\Omega))\cap L^\infty(I;L^{p-1}_{sp}(\mathbb{R}^n))$ be any bounded, sign-changing weak solution to maineq. Then $u$ is locally Hölder continuous in $\Omega_T$, i.e., there exist const for any $(x,t) \in B_{\frac{1}{2} R}(0)\times (-(\tfrac{1}{2} R)^{sp},0)$ and Here $R$ is any fixe

Figures (4)

  • Figure 1: Measure to pointwise bound in \ref{['Prop:1:1']}
  • Figure 2: De Giorgi lemma
  • Figure 3: Expansion of positivity in time
  • Figure 6: \ref{['assump1']}

Theorems & Definitions (72)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 1.3
  • Definition 1.4
  • Theorem 1.5
  • Lemma 1.6
  • proof
  • Lemma 1.7
  • Lemma 1.8
  • Lemma 1.9
  • ...and 62 more