[Paper Review] Gradient regularity in mixed local and nonlocal problems

Cristiana De Filippis, Giuseppe Mingione|arXiv (Cornell University)|Apr 13, 2022

Differential Equations and Boundary Problems4 citations

TL;DR

This paper establishes the first optimal Hölder regularity of the gradient for minimizers of mixed local-nonlocal variational problems involving $p$-growth and fractional $\gamma$-growth, under the condition $p > s\gamma$. The authors develop a novel comparison method combining local and nonlocal Caccioppoli-type estimates, proving that $Du$ is locally $C^{1,\alpha}$-regular in $\Omega$ when $f \in L^d(\Omega)$ for $d > n$, extending classical regularity theory to nonlinear mixed operators.

ABSTRACT

Minimizers of functionals of mixed local and nonlocal type are locally $C^{1,β}$-regular.

Motivation & Objective

• To establish optimal $C^{1,\alpha}$ regularity of the gradient for minimizers of mixed local and nonlocal variational problems involving $p$-growth and fractional $\gamma$-growth.
• To extend the classical De Giorgi-Nash-Moser theory to nonlinear, degenerate mixed operators where $p \neq \gamma$, which had not been previously addressed.
• To develop a robust analytical framework that handles the interplay between local $W^{1,p}$ and nonlocal $W^{s,\gamma}$ capacities under the critical condition $p > s\gamma$.
• To generalize results beyond the $p=\gamma$ case, which had been the focus in prior literature, and to prove maximal regularity in the nonlinear regime.

Proposed method

• A new comparison argument is introduced, comparing the minimizer $u$ to a local $p$-harmonic function $h$ in small balls, using a weighted $V_\mu$-difference estimate to control the gradient oscillation.
• The method relies on a refined Caccioppoli-type inequality for the nonlocal term, derived from the Euler-Lagrange equation of the functional, and combined with energy estimates in the local $W^{1,p}$-setting.
• A key technical step involves the use of the $V_\mu$-transformation, which linearizes the $p$-growth structure and allows for optimal decay estimates in the gradient difference $|Du - Dh|^{p}$.
• The proof establishes a decay estimate for the oscillation of $Du$ via a Campanato-type characterization, using the decay of $\int_{B_t} |Du - (Du)_{B_t}|^p \,dx$ and a careful choice of scaling parameters.
• The analysis incorporates a double-scale decay: one from the local $p$-harmonic approximation $h$, and another from the nonlocal term, both controlled via the condition $p > s\gamma$, ensuring the local term dominates.
• The method is flexible and applies verbatim to general mixed equations, not just variational problems, by relying on the Euler-Lagrange equation and energy-type estimates.

Experimental results

Research questions

• RQ1 Can the gradient of minimizers of mixed local-nonlocal functionals with $p \neq \gamma$ achieve $C^{1,\alpha}$ regularity, even when the nonlocal term has lower integrability than the local one?
• RQ2 What is the optimal regularity threshold for the gradient when the local $p$-energy and nonlocal $\gamma$-energy are coupled, and how does the condition $p > s\gamma$ affect this?
• RQ3 Can the classical De Giorgi-Nash-Moser theory be extended to nonlinear mixed operators where the local and nonlocal terms have different growths ($p \neq \gamma$)?
• RQ4 Is it possible to achieve maximal regularity (i.e., $C^{1,\alpha}$) for the gradient in mixed problems when the nonlocal term is degenerate or singular, provided the local term dominates in capacity?
• RQ5 How can one control the oscillation of the gradient in mixed problems using a comparison method that simultaneously handles local and nonlocal contributions?

Key findings

• The gradient $Du$ of any minimizer $u$ of the functional $\mathcal{F}(w) = \int_\Omega (|Dw|^p - fw)\,dx + \iint_{\mathbb{R}^n \times \mathbb{R}^n} \frac{|w(x)-w(y)|^\gamma}{|x-y|^{n+s\gamma}}\,dx\,dy$ is locally Hölder continuous in $\Omega$ under the condition $p > s\gamma$, $p,\gamma > 1$, $0 < s < 1$, and $f \in L^d(\Omega)$ for $d > n$.
• The result holds even when $p \neq \gamma$, which is a novel contribution, as prior results were limited to $p = \gamma$ or only $L^\infty$ and $C^{0,\alpha}$ regularity.
• The local $C^{1,\alpha}$ regularity of $Du$ is established via a Campanato-type integral characterization, with the Hölder exponent $\alpha$ depending on $n$, $p$, $\gamma$, $s$, and the integrability of $f$, specifically $\alpha = \sigma_2 \alpha_0 / (2\sigma_2 p + 4n)$.
• The proof achieves optimal decay rates for the oscillation of $Du$, with $\int_{B_t} |Du - (Du)_{B_t}|^p \,dx \leq c t^{\alpha p}$, confirming the $C^{1,\alpha}$-regularity of $Du$ in $\Omega_0 \Subset \Omega$.
• The method is robust and extends to general non-variational mixed equations, as it relies only on the Euler-Lagrange equation and energy estimates, not on full variational structure.
• The condition $p > s\gamma$ is essential: it ensures that the $W^{1,p}$-capacity dominates the $W^{s,\gamma}$-capacity, so that the local term controls the regularity, which is the key to the gradient regularity result.