Boundary regularity of mixed local-nonlocal operators and its application

Abstract

Let ΩΩ be a bounded C2C2 domain in RnRn and u∈C(Rn)u∈C(Rn) solves Δu+aIu+C0|Du|≥−KinΩ,Δu+aIu−C0|Du|≤KinΩ,u=0inΩc,Δu+aIu+C0|Du|≥−KinΩ,Δu+aIu−C0|Du|≤KinΩ,u=0inΩc, in the viscosity sense, where 0≤a≤A00≤a≤A0, C0,K≥0C0,K≥0, and I is a suitable nonlocal operator. We show that u/δu/δ is in Cκ(Ω¯)Cκ(Ω¯) for some κ∈(0,1)κ∈(0,1), where δ(x)=dist(x,Ωc)δ(x)=dist⁡(x,Ωc). Using this result, we also establish that u∈C1,γ(Ω¯)u∈C1,γ(Ω¯).Finally, we apply these results to study an overdetermined problem for mixed local-nonlocal operators