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 \( i\in \left\{ 1,2,\dots ,m\right\} \)

$$\begin{aligned}{} & {} \min \biggl \{ F\bigl ( y,x,u_{i}(y,x),D u_{i}(y,x),D^2 u_{i}(y,x)\bigl ), u_{i}(y,x)-\max _{j\ne i}\bigl ( u_{j}(y,x)-c_{ij}(y,x)\bigl )\biggl \}\\{} & {} =0, \left( y,x \right) \in \Omega _{L}\\{} & {} u_{i}(0,x)=g_{i}(x), x\in \bar{\Omega },\ u_i(y,x)=f_i(y,x), (y,x)\in (0,L)\times \partial {\Omega } \end{aligned}$$

where \( \Omega \subset \mathbb {R}^n \) is a bounded domain, \( \Omega _{L}:=(0,L)\times \Omega \) and \( F:\left[ 0,L\right] \times \mathbb {R}^n\times \mathbb {R}\times \mathbb {R}^n\times \mathcal {S}^n\rightarrow \mathbb {R}\) is a general second-order partial differential operator which covers even the fully nonlinear case. (We will call a second-order partial differential operator \(F:\left[ 0,L\right] \times \mathbb {R}^n\times \mathbb {R}\times \mathbb {R}^n\times \mathcal {S}^n\rightarrow \mathbb {R}\) fully nonlinear if and only if, it has the following form

$$\begin{aligned} F \left( y,x,u,D_x u,D_{xx}^2 u\right) :=\sum _{|\alpha |=2}\alpha _{\alpha }\left( y,x,u,D_x u,D_{xx}^2 u \right) D^{\alpha }u(y,x)+\alpha _{0}\left( y,x,u,D_x u \right) \end{aligned}$$

with the restriction that at least one of the functional coefficients \( \alpha _{\alpha },\ |\alpha |=2, \) 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.