Positivity and symmetry of nonnegative solutions of semilinear elliptic equations on planar domains

We consider the Dirichlet problem for the semilinear equation u + f (u) = 0 on a bounded domain Ω ⊂ RN. We assume that Ω is convex in a direction e and symmetric about the hyperplane H = {x ∈ RN: x · e = 0}. It is known that if N 2 and Ω is of class C2, then any nonzero nonnegative solution is necessarily strictly positive and, consequently, it is reflectionally symmetric about H and decreasing in the direction e on the set {x ∈ Ω: x · e > 0}. In this paper, we prove the same result for a large class of nonsmooth planar domains. In particular, the result is valid if any of the following additional conditions on Ω holds: (i) Ω is convex (not necessarily symmetric) in the direction perpendicular to e, (ii) Ω is strictly convex in the direction e, (iii) Ω is piecewise-C1,1.