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Solvability of the nonlinear Dirichlet problem with integro-differential operators. (English) Zbl 1386.60234

The authors consider the following problem \[ \begin{aligned} F(u,x) + u(x)-l(x) = 0, \quad x \in O \\ u(x) = g(x), \quad x \in O^c \end{aligned} \] where \[ \begin{aligned} (u,x) = -\inf_{a \in [\underline{a},\overline{a}]} H(u,x,a) - {\mathcal I}(u,x) \end{aligned} \] Here, \(\underline{a} \leq \overline{a}\) are given numbers and \[ \begin{aligned} {\mathcal I}(u,x) = \int_{\mathbb{R}^d} (u(x+y)-u(x) -Du(x)y I_{B_1}(y)) \nu(dy) \end{aligned} \] with \[ \begin{aligned} H(u,x,a) = \frac{1}{2} \mathrm{tr}(A(a) D^2u(x)) + b(a) Du(x) \end{aligned} \] \(A(a) = \sigma'(a) \sigma(a)\) and \(\nu(\cdot)\) is a Lévy measure on \(\mathbb{R}^d\). The aim of this paper is to find a sufficient condition of the existence and uniqueness of the viscosity solution for the above Dirichlet problem.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
47G20 Integro-differential operators
93E20 Optimal stochastic control
60J75 Jump processes (MSC2010)
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35J60 Nonlinear elliptic equations
35J66 Nonlinear boundary value problems for nonlinear elliptic equations
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