Hölder estimates for viscous Hamilton–Jacobi equations

Abstract

Presentation and statement of the results.We present here new results concerning regularity and a priori estimates for solutions (or subsolutions) of second order degenerate elliptic equations, whose simplest model is the followingHere Ω is an open bounded subset of R N , A : Ω → S + N is a continuous map into the space of symmetric non-negative matrices of order N , λ > 0 and p > 1 are given numbers and f : Ω → R is a continuous function. It is well-known that equations of the form (1) arise in connection to optimal control problems for degenerate diffusion processes governed by the Ito's equationwhere a(X t ) is interpreted as a feedback control and W t is a standard Brownian motion. Indeed, the classical Dynamic Programming argument (see [7]) indicates that given a cost functional of the formwhere τ x is the first exit time, then the value function u(x) =: inf a J(x, a) is a viscosity solution of equation (1), supplemented either by the Dirichlet boundary condition u = ϕ (exit-time problem) or by the state constraint condition τ x = +∞ (the process stays in Ω for all time, with probability one). Of course, both these conditions have to be formulated in a suitable generalized sense, which is possible in the framework of viscosity solutions (see [6]); indeed, one has to take into account the possibility that the boundary condition is not verified at every point (as for first order equations) but only in a relaxed sense. Similar problems have been investigated under various sets of assumptions (see e.g. [2], [3], [4], [8], [9], [12]). In particular, the state constraint problem for equation (1) was deeply studied in [10] in case of a purely Brownian motion (i.e. A(x) ≡ Id); it turns out that there is a significant difference between the cases p ≤ 2 and p > 2, since in this latter case the maximal solution of (1) is bounded and loss of boundary condition can occur in the study of the Dirichlet problem (see also [2]).Our main contribution concerns the case of superquadratic growth p > 2, where we prove that not only solutions, but even subsolutions of (1) are Hölder continuous. The precise result reads as follows:Theorem 1.