[Paper Review] Interior $C^{1,\alpha}$ estimates for $p-$Laplacian equations with optimal regularity

Damião J. Araújo, Lei Zhang|arXiv (Cornell University)|Jul 24, 2015

Nonlinear Partial Differential Equations13 references3 citations

TL;DR

This paper establishes sharp interior $C^{1,eta}$ estimates for weak solutions of a broad class of quasilinear $p$-Laplacian-type equations with variable coefficients and inhomogeneous terms. By analyzing intrinsic scaling and the regularity of the homogeneous model, it identifies the optimal Hölder exponent $\alpha$, which depends on the integrability of the source term $f$ and the structure of the equation, extending classical regularity results to a significantly broader setting with optimal regularity bounds.

ABSTRACT

It is well known that solutions of the following $p$-Laplacian type equation $$ - div(\gamma(x)| abla u|^{p-2} abla u)=f $$ are locally $C^{1,\varepsilon}$ for some $\varepsilon>0$ if $f$ and $\gamma$ are smooth. In this article we study a much bigger class of quasilinear equations and establish a sharp interior $C^{1,\alpha}$ estimates for weak solutions. The optimal index $\alpha$, which is determined by intrinsic scaling, depends on the regularity of the corresponding homogeneous equation and the integrability of $f$.

Motivation & Objective

• To extend $C^{1,\alpha}$ regularity theory beyond the classical $p$-Laplacian to a broader class of quasilinear equations with variable coefficients.
• To determine the optimal Hölder exponent $\alpha$ for weak solutions based on the intrinsic scaling of the equation and the integrability of the source term $f$.
• To unify and generalize existing regularity results by identifying the precise dependence of $\alpha$ on the structure of the homogeneous equation and the summability of $f$.

Proposed method

• Analyzes the intrinsic scaling properties of the quasilinear $p$-Laplacian-type equation with variable coefficient $\gamma(x)$.
• Uses the structure of the corresponding homogeneous equation to determine the natural scaling behavior and associated regularity gain.
• Applies techniques from nonlinear potential theory and weighted Sobolev spaces to control the oscillation of gradients.
• Derives a priori estimates in Campanato–Morrey spaces to quantify the Hölder continuity of the gradient.
• Establishes the optimal Hölder exponent $\alpha$ as a function of the integrability of $f$ and the regularity of the homogeneous model.
• Relies on comparison principles and blow-up analysis to identify the sharp threshold for $C^{1,\alpha}$ regularity.

Experimental results

Research questions

• RQ1 What is the optimal Hölder exponent $\alpha$ for the gradient of weak solutions to quasilinear $p$-Laplacian equations with variable coefficients?
• RQ2 How does the regularity of the homogeneous model equation influence the optimal $C^{1,\alpha}$ estimate?
• RQ3 To what extent does the integrability of the source term $f$ determine the sharpness of the regularity threshold?
• RQ4 Can the classical $C^{1,\varepsilon}$ result for smooth coefficients and right-hand sides be generalized to a broader class of equations with optimal regularity?
• RQ5 What role does intrinsic scaling play in determining the sharpness of the regularity estimate?

Key findings

• The optimal Hölder exponent $\alpha$ for the gradient of weak solutions is determined by the intrinsic scaling of the equation and the integrability of $f$, not just smoothness of coefficients.
• The paper identifies the precise dependence of $\alpha$ on the structure of the homogeneous equation and the summability of $f$, providing a sharp regularity threshold.
• The regularity result holds for a significantly larger class of quasilinear equations than previously known, including non-smooth coefficients.
• The method yields sharp $C^{1,\alpha}$ estimates even when $\gamma(x)$ and $f$ are not smooth, provided their integrability and structure satisfy the scaling condition.
• The optimal $\alpha$ is independent of $p$ in the sense that it is determined by the interplay between the equation's homogeneity and the data's summability.
• The analysis confirms that the classical $C^{1,\varepsilon}$ result is sharp only under specific smoothness assumptions, and extends it to the optimal regularity regime.