Removability of a Level Set for Solutions of Quasilinear Equations

In this paper, we study the removability of a level set for the solutions of quasilinear elliptic and parabolic equations of the second order. We show, under rather general assumptions on the coefficients of the equation, that if a function u ∈ C 1(Ω) is a viscosity solution to the equation in the set then u is, in fact, a solution in the whole domain Ω. In addition to the linear equations in nondivergence form with Lipschitz coefficients, our results cover, for example, the p-Laplace equation, the minimal surface equation, the Burgers equation, and the heat equation.