Gradient Hölder regularity for nonlocal double phase equations

Abstract: This paper is devoted to investigating the interior $C^{1, α}$ regularity of viscosity solutions to the nonlocal double phase equations $$\int_{\mathbb{R}^d} \left(\frac{|u(x)-u(y)|^{{p-2}(u(x)-u(y))}{|x-y|^{{d+sp}}+a(x,y)\frac{|u(x)-u(y)|^{{q-2}(u(x)-u(y))}{|x-y|^{{d+tq}}\right)\,dy=0,}}}}$$ where $2\le p\le q$, $s, t\in (0, 1)$ with $s\le t$, and $a(x, y)\ge0$. In the degenerate case, we solve the higher regularity issue raised by De Filippis-Palatucci [J. Differential Equations \textbf{267} (2019) 547--586]. By assuming the Lipschitz continuity of the modulating coefficient $a$, we are able to prove that the gradient of solution is Hölder continuous, provided the distance of $tq$ and $sp$ is suitably small. The core challenges consist in precisely characterizing the subtle interaction among the pointwise behaviour of the coefficient $a$, the growth exponents and the differentiability orders.

• The paper establishes sharp interior C¹,α regularity for viscosity solutions under precise structural constraints on the nonlocal operator.
• The paper employs advanced techniques, including modified De Giorgi iteration and Ishii–Lions methods, to manage mixed growth terms effectively.
• The paper provides explicit a priori estimates that control the impact of the nonlocal tail and variable modulating coefficients.

Gradient Hölder Regularity for Nonlocal Double Phase Equations

Introduction and Context

The paper "Gradient Hölder regularity for nonlocal double phase equations" (2604.22206) addresses a core open problem in the regularity theory of nonlocal equations with nonstandard (p, q)-growth. Specifically, the authors establish sharp interior $C^{1,\alpha}$ regularity for viscosity solutions to a broad class of nonlocal double phase equations characterized by mixed orders of differentiability and variable modulating coefficients. The considered nonlocal operator is

$$\mathrm{P.V.}\int_{\mathbb{R}^d}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}} + a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}} \,dy = 0$$

where $p\leq q$, $s\leq t$ in $(0,1)$, and $a(x,y)\geq0$ with suitable regularity and structure assumptions. This operator models nonlocal media with phase transitions between two different types of fractional elliptic behavior, regulated by the coefficient $a(x,y)$. The study is motivated by applications in homogenization theory and the need to understand the fine regularity properties of minimizers or viscosity solutions under degeneracies and non-uniform ellipticity.

Main Results

The primary achievement is a positive resolution to a regularity question raised by De Filippis-Palatucci [J. Differential Equations 267 (2019) 547–586]. The authors rigorously prove the following:

• Under symmetry, boundedness, translation invariance, and Lipschitz continuity on $a(x, y)$, and structural constraints between parameters ($tq \leq sp+\min\{1, q-p\}$), every $C^1$ viscosity solution to the nonlocal double phase equation in a domain $$\mathrm{P.V.}\int_{\mathbb{R}^d}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}} + a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}} \,dy = 0$$0 is locally $$\mathrm{P.V.}\int_{\mathbb{R}^d}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}} + a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}} \,dy = 0$$1 in $$\mathrm{P.V.}\int_{\mathbb{R}^d}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}} + a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}} \,dy = 0$$2, for some $$\mathrm{P.V.}\int_{\mathbb{R}^d}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}} + a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}} \,dy = 0$$3.
• The explicit a priori estimate is

$$\mathrm{P.V.}\int_{\mathbb{R}^d}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}} + a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}} \,dy = 0$$4

where the tail term quantifies the influence of the solution outside the local domain, and $$\mathrm{P.V.}\int_{\mathbb{R}^d}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}} + a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}} \,dy = 0$$5 depends only on the structure parameters and the Lipschitz seminorm of $$\mathrm{P.V.}\int_{\mathbb{R}^d}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}} + a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}} \,dy = 0$$6.

This establishes, for the first time, pointwise gradient Hölder continuity for a class of nonlocal energies with mixed growth, bridging a significant gap in analogy with the local case (cf. Colombo-Mingione [Arch. Ration. Mech. Anal. 215 (2015)]).

Additionally, the authors provide:

• Structural analysis of the delicate interaction between the nonlocality (parameters $$\mathrm{P.V.}\int_{\mathbb{R}^d}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}} + a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}} \,dy = 0$$7), the growth exponents $$\mathrm{P.V.}\int_{\mathbb{R}^d}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}} + a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}} \,dy = 0$$8, and the modulating coefficient $$\mathrm{P.V.}\int_{\mathbb{R}^d}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}} + a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}} \,dy = 0$$9.
• A full set of conditions ensuring the applicability of the De Giorgi method and perturbative techniques, closely controlling the lack of homogeneity and scaling invariance.

Technical Approach

The methodology blends several sophisticated ingredients:

• Linearization and Localization: Through localization of the directional derivatives and linearization around the solution, the authors derive a variable-coefficient nonlocal operator whose kernel inherits the complex structure of the original equation.
• Quantitative De Giorgi Iteration: Inspired by the recent advances for the fractional p-Laplacian (notably Giovagnoli–Jesus–Silvestre (Giovagnoli et al., 30 Sep 2025)), the authors devise a modified De Giorgi oscillation lemma for the nonlocal double phase context. The oscillation reduction employs precise estimates of the "nonlocal tail" and cone-type nondegeneracy arguments to manage the mixed growth terms.
• Perturbative and Approximation Arguments: The proofs rely on delicate approximation strategies to bridge the lack of natural scaling invariance, particularly crucial due to the variable coefficient $p\leq q$0 and the disparity between $p\leq q$1.
• Ishii–Lions' Method for Nonlocal Operators: The proof of the sharp gradient Hölder continuity, especially around the modulation zones, leverages the Ishii–Lions technique, carefully adapting it to handle nonlocality and the degenerate-elliptic setting.
• Patchwork of Local and Global Estimates: The arguments feature a dichotomy: either the gradient decays rapidly or approximates a constant vector closely in measure, allowing covering and rescaling arguments to establish uniform modulus of continuity.

Implications and Further Directions

Theoretical Advances

This work provides a comprehensive theory paralleling the celebrated local results for double phase problems, but extends them to the complex nonlocal field. The findings:

• Fill a crucial gap in nonlocal regularity, enabling a more complete Calderón–Zygmund and Schauder theory for hybrid nonlocal operators with variable exponents and coefficients.
• Clarify the role of the modulating coefficient: The necessity of translation invariance and Lipschitz continuity of $p\leq q$2 is sharply elucidated, demonstrating that weaker assumptions (e.g., mere Hölder regularity) are insufficient for gradient estimates.
• Sharpen the known thresholds: The explicit conditions on the relation between $p\leq q$3, and the regularity of $p\leq q$4 ensure optimality and are in line with counterexamples in the local case.

Practical Relevance

The results have direct implications for:

• Mathematical models of strongly anisotropic or multi-phase composite materials where nonlocal interactions and phase transitions are prominent.
• Analysis of variational problems relevant to material science, image processing, and nonlocal flows, providing assurance of solution regularity, which is critical for numerical simulations and theoretical developments.
• Progress on numerics and stochastic game approaches involving nonlocal, mixed-differentiability equations, where gradient bounds are key.

Future Developments

This work raises several avenues for further exploration:

• Boundary regularity: While the current results are interior, extensions to boundary Hölder gradient regularity under optimal geometric conditions would be of high importance.
• Relaxation of $p\leq q$5 regularity: Investigating whether partial results hold under weaker assumptions or alternative variational frameworks.
• Sharpness of thresholds and counterexamples: The necessity and sufficiency of parameter constraints, especially in borderline regimes ($p\leq q$6, $p\leq q$7), merit further analysis.
• Extension to parabolic and higher-order nonlocal systems: The techniques developed provide a roadmap for tackling related problems in time-dependent or system contexts.

Numerical Results and Notable Claims

While the paper emphasizes non-quantitative analytic estimates, the strong claims include:

• Hölder continuity of the gradient holds universally under the prescribed structural and regularity assumptions, even in the fully degenerate regime.
• The equivalence of weak and viscosity solution notions in this context is established or referenced, a crucial aspect for well-posedness and robustness of the results.

Conclusion

The paper delivers a definitive and technically robust analysis of the interior gradient Hölder continuity for viscosity solutions to nonlocal double phase equations, placing the theory of such operators on par with the most advanced developments in the nonlocal regularity landscape. The innovative adaptation of De Giorgi and Ishii–Lions techniques to accommodate mixed growth and nonlocality sets a strong precedent for further work in nonlocal, degenerate, and variable-coefficient PDEs (2604.22206).