Solution of Dirichlet's problem for the equation Δu =?1 in a convex region

Let u be a solution of the following boundary-value problem: u¦ = 0, where is a closed convex curve and u = –1 in the region D bounded by . Then u has only one local maximum, and all its level curves are convex.Translated from Matematicheskie Zametki, Vol. 9, No. 1, pp. 89–92, January, 1971.