Systems of fully nonlinear degenerate elliptic obstacle problems with Dirichlet boundary conditions

Abstract

In this paper, we prove existence and uniqueness of viscosity solutions to the following system: For i ∈ {1, 2,..., m} min F

y, x, ui(y, x), Dui(y, x), D2ui(y, x)

, ui(y, x) − max j=i

u j(y, x) − ci j(y, x)

= 0,(y, x) ∈ L ui(0, x) = gi(x), x ∈ , ¯ ui(y, x) = fi(y, x), (y, x) ∈ (0, L) × ∂ where ⊂ Rn is a bounded domain, L := (0, L)× and F : [0, L]×Rn×R×Rn×Sn → R is a general second-order partial differential operator which covers even the fully nonlinear case. (We will call a second-order partial differential operator F : [0, L] × Rn × R × Rn × Sn → R fully nonlinear if and only if, it has the following form F y, x, u, Dxu, D2 x xu

:= |α|=2 αα

y, x, u, Dxu, D2 x xu

Dαu(y, x) + α0 (y, x, u, Dxu) with the restriction that at least one of the functional coefficients αα, |α| = 2, contains a partial derivative term of second order.) Moreover, F belongs to an appropriate subclass of degenerate elliptic operators. Regarding uniqueness, we establish a comparison principle for viscosity sub and supersolutions of the Dirichlet problem. This system appears among others in the theory of the so-called optimal switching problems on bounded domains.