Boundary regularity of mixed local-nonlocal operators and its application

Let Omega be a bounded C-2 domain in R-n and u is an element of C(R-n) solves Delta u + aIu + C-0 vertical bar Du vertical bar >= -K in Omega, Delta u + aIu - C-0 vertical bar Du vertical bar <= K in Omega, u = 0 in Omega(c), in the viscosity sense, where 0 <= a < A(0), C-0 , K >= 0, and I is a suitable nonlocal operator. We show that u/delta is in C-kappa((Omega) over bar) for some kappa is an element of (0, 1), where delta(x) = dist(x, Omega(c)). Using this result, we also establish that u is an element of C-1,C-gamma ((Omega) over bar). Finally, we apply these results to study an overdetermined problem for mixed local-nonlocal operators.