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Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations

Prashanta Garain

TL;DR

This work addresses the qualitative theory of weak solutions to mixed anisotropic and nonlocal quasilinear equations, uniting local and nonlocal operators via the anisotropic p-Laplacian $-H_p$ and the fractional p-Laplacian $(-\Delta_p)^s$. The authors develop a comprehensive toolkit built around a Picone identity tailored to the mixed operator, yielding Brezis-Oswald-type results, Sturmian comparisons, eigenvalue structure, Hardy inequalities, and interrelation results for singular systems, alongside robust regularity results. In Part II, they extend energy methods to both elliptic and doubly nonlinear parabolic problems, proving Caccioppoli-type estimates, weak Harnack inequalities, and semicontinuity together with detailed pointwise behavior and De Giorgi-type expansions, all in the mixed local-nonlocal anisotropic setting. Together these results deepen the understanding of existence, regularity, and asymptotic properties of solutions to mixed anisotropic/nonlocal equations, with potential applications to physical models exhibiting both local and nonlocal diffusion under anisotropic effects.

Abstract

This article is divided into two parts. In the first part, we examine the Brezis-Oswald problem involving a mixed anisotropic and nonlocal $p$-Laplace operator. We establish results on existence, uniqueness, boundedness, and the strong maximum principle. Additionally, for certain mixed anisotropic and nonlocal $p$-Laplace equations, we prove a Sturmian comparison theorem, establish comparison and nonexistence results, derive a weighted Hardy-type inequality, and analyze a system of singular mixed anisotropic and nonlocal $p$-Laplace equations. A key component of our approach is the use of the Picone identity, which we adapt from the local and nonlocal cases. In the second part of the article, we focus on regularity estimates. In the elliptic setting, we establish a weak Harnack inequality and semicontinuity results. We also consider a class of doubly nonlinear mixed anisotropic and nonlocal parabolic equations, proving semicontinuity results and analyzing the pointwise behavior of solutions. These results rely on appropriate energy estimates, De Giorgi-type lemmas, and positivity expansions. Finally, we derive various energy estimates, which may be of independent interest.

Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations

TL;DR

This work addresses the qualitative theory of weak solutions to mixed anisotropic and nonlocal quasilinear equations, uniting local and nonlocal operators via the anisotropic p-Laplacian and the fractional p-Laplacian . The authors develop a comprehensive toolkit built around a Picone identity tailored to the mixed operator, yielding Brezis-Oswald-type results, Sturmian comparisons, eigenvalue structure, Hardy inequalities, and interrelation results for singular systems, alongside robust regularity results. In Part II, they extend energy methods to both elliptic and doubly nonlinear parabolic problems, proving Caccioppoli-type estimates, weak Harnack inequalities, and semicontinuity together with detailed pointwise behavior and De Giorgi-type expansions, all in the mixed local-nonlocal anisotropic setting. Together these results deepen the understanding of existence, regularity, and asymptotic properties of solutions to mixed anisotropic/nonlocal equations, with potential applications to physical models exhibiting both local and nonlocal diffusion under anisotropic effects.

Abstract

This article is divided into two parts. In the first part, we examine the Brezis-Oswald problem involving a mixed anisotropic and nonlocal -Laplace operator. We establish results on existence, uniqueness, boundedness, and the strong maximum principle. Additionally, for certain mixed anisotropic and nonlocal -Laplace equations, we prove a Sturmian comparison theorem, establish comparison and nonexistence results, derive a weighted Hardy-type inequality, and analyze a system of singular mixed anisotropic and nonlocal -Laplace equations. A key component of our approach is the use of the Picone identity, which we adapt from the local and nonlocal cases. In the second part of the article, we focus on regularity estimates. In the elliptic setting, we establish a weak Harnack inequality and semicontinuity results. We also consider a class of doubly nonlinear mixed anisotropic and nonlocal parabolic equations, proving semicontinuity results and analyzing the pointwise behavior of solutions. These results rely on appropriate energy estimates, De Giorgi-type lemmas, and positivity expansions. Finally, we derive various energy estimates, which may be of independent interest.
Paper Structure (17 sections, 56 theorems, 231 equations)

This paper contains 17 sections, 56 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Part-I
  3. Brezis-Oswald problem
  4. Comparison results
  5. Eigenvalue problem
  6. Weighted Hardy type inequality
  7. System of singular nonlinear equations
  8. Non existence of positive super-solutions
  9. Part-II
  10. Elliptic case
  11. Caccioppoli type energy estimates
  12. Weak Harnack inequality
  13. Semicontinuity
  14. Doubly nonlinear parablic case
  15. Caccioppoli type energy estimates
  16. ...and 2 more sections

Key Result

Lemma 1.3

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $1<p<\infty,\,0<s<1$. Then there exists a positive constant $C=C(N,p,s)$ such that for every $u\in W^{1,p}(\Omega)$.

Theorems & Definitions (90)

  • Remark 1.1
  • Remark 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Definition 1.5
  • Remark 1.6
  • Lemma 1.7
  • Lemma 1.8
  • Lemma 1.9
  • Lemma 1.10
  • ...and 80 more