[Paper Review] Gradient estimates for singular $p$-Laplace type equations with measure data

Hongjie Dong, Hanye Zhu|arXiv (Cornell University)|Feb 17, 2021

Nonlinear Partial Differential Equations22 references4 citations

TL;DR

This paper establishes sharp interior and global gradient estimates for solutions to singular $p$-Laplace type equations with measure data for $p \in (1, \frac{3n-2}{2n-1}]$, a previously open range. It improves prior results by using the truncated Riesz potential $\mathbf{I}_1^R(|\mu|)$ instead of the Wolff potential, under weaker Dini-type continuity conditions on the coefficient and lower integrability assumptions on the modulus of continuity.

ABSTRACT

We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the $p$-Laplace equation $-Δ_p u=μ$ with $p\in (1,2)$. The cases when $p\in \big(2-\frac 1 n,2\big)$ and $p\in \big(\frac{3n-2}{2n-1},2-\frac{1}{n}\big]$ were studied in [9] and [22], respectively. In this paper, we improve the results in [22] and address the open case when $p\in \big(1,\frac{3n-2}{2n-1}\big]$. Interior and global modulus of continuity estimates of the gradients of solutions are also established.

Motivation & Objective

• Address the open problem of gradient estimates for singular $p$-Laplace equations with measure data when $p \in (1, \frac{3n-2}{2n-1}]$, a range not fully covered by prior work.
• Improve upon existing results in [22] by weakening the regularity assumptions on the vector field $A(x,\xi)$ and the modulus of continuity $\omega$
• Establish both interior and global modulus of continuity estimates for the gradient of solutions under minimal structural and regularity conditions.
• Provide pointwise gradient bounds using the truncated first-order Riesz potential $\mathbf{I}_1^R(|\mu|)$, which is more natural and stronger than the Wolff potential used previously.

Proposed method

• Derive interior pointwise gradient estimates using a modified iteration argument in the context of singular $p$-Laplace equations with $p \in (1,2)$.
• Employ the truncated Riesz potential $\mathbf{I}_1^R(|\mu|)(x) = \int_0^R \frac{|\mu|(B_t(x))}{t^{n-1}} \frac{dt}{t}$ as the main potential to control the gradient growth.
• Introduce a new maximal function $\mathbf{M}_1$ combining $L^{2-p}$-norms of $|\nabla u|+s$ and the $L^\infty$-norm of the Riesz potential to control oscillations.
• Use a dyadic decomposition and averaging technique over balls to control the oscillation of $\nabla u$, relying on the Dini condition $\int_0^1 \omega(r) \frac{dr}{r} < \infty$.
• Define a modified Wolff potential $\tilde{\mathbf{W}}_{1/p,p}^\rho(|\mu|)$ and Riesz potential $\tilde{\mathbf{I}}_1^\rho(|\mu|)$ to localize the estimates in small balls.
• Establish global modulus of continuity estimates by combining the decay of the oscillation of $\nabla u$ with the decay of the potential terms, using a parameter $\alpha_1 \in (0,\alpha)$ for Hölder-type control.

Experimental results

Research questions

• RQ1 Can sharp pointwise gradient estimates be established for singular $p$-Laplace equations with measure data when $p \in (1, \frac{3n-2}{2n-1}]$?
• RQ2 Can the dependence on the Wolff potential in prior results be replaced by the more natural and stronger Riesz potential $\mathbf{I}_1^R(|\mu|)$ under weaker regularity assumptions?
• RQ3 What is the optimal modulus of continuity for $\nabla u$ in terms of the measure data $\mu$ and the Dini modulus $\omega$?
• RQ4 Can the regularity assumptions on $A(x,\xi)$ and $\omega$ be weakened compared to previous works, particularly in the singular range $p \in (1,2)$?
• RQ5 How does the gradient's oscillation decay in small balls, and what role does the Dini condition on $\omega$ play in this decay?

Key findings

• The paper establishes a new interior pointwise gradient estimate: $|\nabla u(x)| \leq C[\mathbf{I}_1^R(|\mu|)(x)]^{1/(p-1)} + C\left(\fint_{B_R(x)}(|\nabla u|+s)^{2-p}\,dy\right)^{1/(2-p)}$ for $p \in (\frac{3n-2}{2n-1}, 2)$, valid at Lebesgue points.
• The Riesz potential $\mathbf{I}_1^R(|\mu|)$ provides a strictly stronger control than the Wolff potential $\mathbf{P}_\gamma^R(|\mu|)$ used in [22], as $\mathbf{I}_1^R(|\mu|) \leq C \mathbf{P}_\gamma^{2R}(|\mu|)^{1/\gamma}$ for $\gamma < 1$, and the new bound holds under weaker assumptions on $\omega$ and $A$.
• A global modulus of continuity estimate is derived: $|\nabla u(x) - \nabla u(y)| \leq C \left[ \left(\frac{\rho}{R}\right)^{\alpha_1} + \int_0^\rho \frac{\tilde{\omega}_1(t)}{t} dt \right] + C \|\tilde{\mathbf{W}}_{1/p,p}^\rho(|\mu|)\|_{L^\infty} + C \mathbf{M}_1^{2-p} \|\tilde{\mathbf{I}}_1^\rho(|\mu|)\|_{L^\infty}$, with $\rho = |x-y|$, under Dini continuity and $\alpha_1$-Hölder control.
• The constants in the estimates depend only on $n, p, \lambda, \omega$, and $R_0$, with no need for the stronger Dini-type condition $\int_0^1 \omega^\gamma(r) \frac{dr}{r} < \infty$ required in [22].
• The results extend to the full range $p \in (1, \frac{3n-2}{2n-1}]$, closing a gap in the literature for singular $p$-Laplace equations with measure data.
• The method relies on a novel use of dyadic decomposition and maximal function techniques to control oscillations, with the key innovation being the use of the Riesz potential instead of the Wolff potential in the singular regime.