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Higher regularity estimates for solutions to $\infty$-Laplacian-type models

João Vitor da Silva, Makson S. Santos, Mayra Soares

TL;DR

The paper addresses higher regularity for inhomogeneous $\Delta_{\infty}$-Laplacian models at interior critical points. It develops a blow-up/renormalization framework in a one-phase tangential regime together with smallness assumptions on the forcing to derive a pointwise decay $|u(x)| \le C\,|x-x_0|^{\alpha}$ with $\alpha = \dfrac{4-\kappa}{3-(m+\kappa)}>1$ at $x_0$ in the $1$-nullity set $C_{\mathcal{Z}}(u)$, plus $C^{[\alpha],\alpha-[\alpha]}$ regularity under additional tangential structure. The work also develops reflecting estimates between the positive and negative parts and a non-degeneracy result under a natural forcing lower bound, with extensions to $(3-\gamma)$-homogeneous operators and Hénon-type variants. Overall, these results advance interior regularity theory for inhomogeneous infinity-Laplacian equations, with implications for obstacle-type, free-boundary, and dead-core problems.

Abstract

In this work, we tackle the higher regularity estimates of solutions to inhomogeneous $\infty-$Laplacian equations at interior critical points. Our estimates provide smoothness properties better than the corresponding available regularity for the model with bounded forcing terms. We explore several scenarios, thereby obtaining improved regularity estimates, which depend on the universal parameters of the model. Our findings connect with nowadays well-known estimates developed for obstacle and dead-core type problems.

Higher regularity estimates for solutions to $\infty$-Laplacian-type models

TL;DR

The paper addresses higher regularity for inhomogeneous -Laplacian models at interior critical points. It develops a blow-up/renormalization framework in a one-phase tangential regime together with smallness assumptions on the forcing to derive a pointwise decay with at in the -nullity set , plus regularity under additional tangential structure. The work also develops reflecting estimates between the positive and negative parts and a non-degeneracy result under a natural forcing lower bound, with extensions to -homogeneous operators and Hénon-type variants. Overall, these results advance interior regularity theory for inhomogeneous infinity-Laplacian equations, with implications for obstacle-type, free-boundary, and dead-core problems.

Abstract

In this work, we tackle the higher regularity estimates of solutions to inhomogeneous Laplacian equations at interior critical points. Our estimates provide smoothness properties better than the corresponding available regularity for the model with bounded forcing terms. We explore several scenarios, thereby obtaining improved regularity estimates, which depend on the universal parameters of the model. Our findings connect with nowadays well-known estimates developed for obstacle and dead-core type problems.
Paper Structure (9 sections, 23 theorems, 124 equations, 1 table)

This paper contains 9 sections, 23 theorems, 124 equations, 1 table.

Table of Contents

  1. Introduction
  2. Literature review
  3. Theory of existence/uniqueness and regularity to infinity-Laplacian
  4. Elliptic diffusion models with free boundaries
  5. Tools and auxiliary results
  6. Regularity estimates under "smallness regimes"
  7. Regularity estimates in specific dimensions
  8. Reflecting estimates: Proof of Theorem \ref{['ThmPositive-Part']}
  9. Non-degeneracy of solutions: Proof of Theorem \ref{['ThmNon-Degen']}

Key Result

Theorem 1.1

Let $u \in C^0(B_1)$ be a viscosity solution to P with $x_0 \in\mathrm{C}_{\mathcal{Z}}(u)$. There exist $\varepsilon > 0$ such that if one of the following conditions holds true, then given $r \in \left(0,\frac{1}{2}\right)$, we can find a universal constant $\mathrm{C}>0$ such that for $\alpha>1$ as in eq_ouralpha.

Theorems & Definitions (36)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: Non-degeneracy
  • Remark 2
  • Theorem 2.1: Aronsson Aronsson68
  • Theorem 2.2: RTU15
  • Theorem 2.3: RTU15
  • ...and 26 more