Lipschitz potential estimates for diffusion with jumps
Nirjan Biswas, Harsh Prasad · Nonlinear Differential Equations and Applications NoDEA · 2025
For $$p \in (1, \infty )$$ p ∈ ( 1 , ∞ ) and $$s \in (0,1)$$ s ∈ ( 0 , 1 ) , we consider the following mixed local-nonlocal equation P $$\begin{aligned} - \Delta _p u + (-\Delta _p)^s u = f \; \text {in} \; \Omega , \end{aligned}$$ - Δ p u + ( - Δ p ) s u = f in Ω , where $$\Omega \subset \mathbb {R}^d$$ Ω ⊂ R d is a bounded domain and the function $$f \in L_{loc}^1(\Omega )$$ f ∈ L loc 1 ( Ω ) . Depending on the dimension d, we prove gradient potential estimates of weak solutions to (P) for the entire ranges of p and s. As a byproduct, we recover the corresponding estimates in the purely diffusive setup, providing connections between the local and nonlocal aspects of the equation. Our results are new, even for the linear case $$p=2$$ p = 2 .