A stochastic approach to a multivalued Dirichlet-Neumann problem. (English) Zbl 1195.35192
Summary: We prove the existence and uniqueness of a viscosity solution of the parabolic variational inequality (PVI) with a mixed nonlinear multivalued Neumann-Dirichlet boundary condition:
\[ \begin{cases} \frac{\partial u(t,x)}{\partial t}-{\mathcal L}_tu(t,x)+ \partial\varphi(u(t,x))\ni f(t,x,u(t,x),(\nabla u\sigma)(x,t)), &t>0,\;x\in{\mathcal D},\\ \frac{\partial u(t,x)}{\partial n}+ \partial\psi(u(t,x))\ni g(t,x,u(t,x)), &t>0,\;x\in Bd({\mathcal D}),\\ u(0,x)= h(x), &x\in \overline{\mathcal D}, \end{cases} \]
where \(\partial\varphi\) and \(\partial\psi\) are subdifferential operators and \({\mathcal L}_t\) is a second-differential operator given by
\[ {\mathcal L}_tv(x)= \frac12 \sum_{i,j=1}^d (\sigma\sigma^*)_{ij}(t,x) \frac{\partial^2v(x)}{\partial x_i\partial x_j}+ \sum_{i=1}^d b_i(t,x) \frac{\partial v(x)}{\partial x_i}. \]
The result is obtained by a stochastic approach. First we study the following backward stochastic generalized variational inequality:
\[ \begin{cases} dY_t+F(t,Y_t,Z_t)dt+ G(t,Y_t)dA_t\in \partial\varphi(Y_t)dt+ \partial\psi(Y_t)dA_t+ Z_tdW_t, &0\leq t\leq T,\\ Y_T=\xi, \end{cases} \]
where \((A_t)_{t\geq0}\) is a continuous one-dimensional increasing measurable process, and then we obtain a Feynman-Kaç representation formula for the viscosity solution of the PVI problem.
\[ \begin{cases} \frac{\partial u(t,x)}{\partial t}-{\mathcal L}_tu(t,x)+ \partial\varphi(u(t,x))\ni f(t,x,u(t,x),(\nabla u\sigma)(x,t)), &t>0,\;x\in{\mathcal D},\\ \frac{\partial u(t,x)}{\partial n}+ \partial\psi(u(t,x))\ni g(t,x,u(t,x)), &t>0,\;x\in Bd({\mathcal D}),\\ u(0,x)= h(x), &x\in \overline{\mathcal D}, \end{cases} \]
where \(\partial\varphi\) and \(\partial\psi\) are subdifferential operators and \({\mathcal L}_t\) is a second-differential operator given by
\[ {\mathcal L}_tv(x)= \frac12 \sum_{i,j=1}^d (\sigma\sigma^*)_{ij}(t,x) \frac{\partial^2v(x)}{\partial x_i\partial x_j}+ \sum_{i=1}^d b_i(t,x) \frac{\partial v(x)}{\partial x_i}. \]
The result is obtained by a stochastic approach. First we study the following backward stochastic generalized variational inequality:
\[ \begin{cases} dY_t+F(t,Y_t,Z_t)dt+ G(t,Y_t)dA_t\in \partial\varphi(Y_t)dt+ \partial\psi(Y_t)dA_t+ Z_tdW_t, &0\leq t\leq T,\\ Y_T=\xi, \end{cases} \]
where \((A_t)_{t\geq0}\) is a continuous one-dimensional increasing measurable process, and then we obtain a Feynman-Kaç representation formula for the viscosity solution of the PVI problem.
MSC:
| 35K85 | Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 49J40 | Variational inequalities |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35D40 | Viscosity solutions to PDEs |
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