We consider the problem of finding a real number and a function u satisfying the PDE maxf u f; jDuj 1g D 0; x 2 R n :Here f is a convex, superlinear function. We prove that there is a unique such that the above PDE has a viscosity solution u satisfying lim jxj!1 u.x/=jxj D 1. Moreover, we show that associated to is a convex solution u with D 2 u 2 L 1 .R n / and give two min-max formulae for .has a probabilistic interpretation as being the least, long-time averaged (ergodic) cost for a singular control problem involving f .
For a bounded domain Ω ⊂ R n and p > n, Morrey's inequality implies that there is c > 0 such thatfor each u belonging to the Sobolev space W 1,p 0 (Ω). We show that the ratio of any two extremal functions is constant provided that Ω is convex. We also show with concrete examples why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this feature. As a by product, we obtain the uniqueness of an optimization problem involving the Green's function for the p-Laplacian.
In this paper, we construct a dumbbell domain for which the associated principal ∞-eigenvalue is not simple. This gives a negative answer to the outstanding problem posed in Juutinen et al.
We study the partial differential equationwhere u is the unknown function, L is a second-order elliptic operator, f is a given smooth function and H is a convex function. This is a model equation for HamiltonJacobi-Bellman equations arising in stochastic singular control. We establish the existence of a unique viscosity solution of the Dirichlet problem that has a Hölder continuous gradient. We also show that if H is uniformly convex, the gradient of this solution is Lipschitz continuous.
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