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Regularity for Doubly Nonlinear Equations in the Mixed Regime

Simone Ciani, Eurica Henriques, Mariia Savchenko, Igor I. Skrypnik, Yevgeniia Yevgenieva

Abstract

We study the local Hölder continuity of nonnegative solutions to doubly nonlinear equations by introducing a new technique that allows us to treat the cases where the equation is both singular and degenerate, up to specific Barenblatt numbers. Our argument relies on a new integral $L^1$-$L^1$ Harnack estimate, of independent interest.

Regularity for Doubly Nonlinear Equations in the Mixed Regime

Abstract

We study the local Hölder continuity of nonnegative solutions to doubly nonlinear equations by introducing a new technique that allows us to treat the cases where the equation is both singular and degenerate, up to specific Barenblatt numbers. Our argument relies on a new integral - Harnack estimate, of independent interest.
Paper Structure (32 sections, 17 theorems, 153 equations, 1 figure)

This paper contains 32 sections, 17 theorems, 153 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Origins and framing of the topic
  3. Main Result
  4. Novelty and Significance
  5. Structure of the paper
  6. Notations and Preliminary Results
  7. Notations
  8. Auxiliary Lemma
  9. Local Energy Estimates
  10. De Giorgi Type Lemmas
  11. Expansion of Positivity
  12. Other integral estimates
  13. $L^{1}_{\mathrm{loc}}-L^{1}_{\mathrm{loc}}$ Harnack Inequalities
  14. Reduction of the Oscillation
  15. Case \ref{['eq1.7']}: $\lambda<1$ and $p+N(\lambda-1)>0$
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $u$ be a nonnegative, local weak solution to eq1.1-eq1.2 in $\Omega_T$ and assume also that one of the following conditions holds or or then $u$ is locally Hölder continuous in $\Omega_T.$

Figures (1)

  • Figure 1: Illustration of the novelty of the present results. The hatched region represents the range of exponents $(p,m)$ for which the local Hölder continuity of solutions to \ref{['eq1.0']} was established in the existing literature. The solid gray region represents the range covered by the present work (for the case $N=3$).

Theorems & Definitions (27)

  • Definition 1.1
  • Remark 1.1
  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 17 more