We consider a function which is a viscosity solution of a uniformly elliptic equation only at those points where the gradient is large. We prove that the Hölder estimates and the Harnack inequality, as in the theory of Krylov and Safonov, apply to these functions.
In this paper we continue the study of the Dirichlet problem for an elliptic equation on a domain in R3 which was begun in [5]. For R > 0 let ΩR be the ball of radius R centered at the origin with boundary ∂Ω R. The Dirichlet problem we are concerned with is the following:(-Δ - b(x).∇) u(x) = f(x), x ∈ Ω R,with zero boundary conditionsu(x) = 0, x ∈ ∂Ω R.
In this paper we are concerned with studying the Dirichlet problem for an elliptic equation on a domain in R3. For simplicity we shall assume that the domain is a ball ΩR of radius R. Thus:ΩR = {x ∈ R3 : |x| < R}.The equation we are concerned with is given by(-Δ - b(x).∇) u(x) = f(x), x ∈ ΩR,with zero Dirichlet boundary conditions.
On montre que les solutions faibles de l’équation , où est une mesure positive négligeant les polaires, vérifient une inégalité de Harnack. On s’occupe également des sursolutions dont on fait la représentation intégrale a l’aide d’une fonction de Green. Comme les solutions sont discontinues, on est amené à utiliser les formules probabilistes.