In this article we consider a class of non-degenerate elliptic operators obtained by superpositioning the Laplacian and a general nonlocal operator. We study the existence-uniqueness results for Dirichlet boundary value problems, maximum principles and generalized eigenvalue problems. As applications to these results, we obtain Faber-Krahn inequality and a one-dimensional symmetry result related to the Gibbons' conjecture. The latter results substantially extend the recent results of Biagi et. al. [7,9] who consider the operators of the form −∆ + (−∆) s with s ∈ (0, 1).

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Let Ω be a bounded C 2 domain in R n and uin viscosity sense, where 0 ≤ a ≤ A0, C0, K > 0, and I is a suitable nonlocal operator. We show that u/δ is in C κ ( Ω) for some κ ∈ (0, 1), where δ(x) = dist(x, Ω c ). Using this result we also establish that u ∈ C 1,γ ( Ω). Finally, we apply these results to study an overdetermined problem for mixed local-nonlocal operators.

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