Stochastic variational inequalities associated with elasto-plastic torsion
The aim of the article is the existence and uniqueness theorem of the initial value problem for the stochastic evolution variational inequality \[ \frac{\partial u}{\partial t}-\triangle u+\partial I_{\mathfrak{R}}(u)\ni \frac{\partial M}{\partial t} \; \;\; u(0)=u_0 \in \mathfrak{R}=\{v\in H_0^1(G):|\nabla v(x)|\leq 1\} \eqno{(1)} \] for almost all \(x\) where \(\partial I_{\mathfrak{R}}(\cdot)\) denotes the subdifferential of the indicator function \(I_{\mathfrak{R}}(\cdot)\) and \(\mathfrak{R}\) is a closed convex subset of a certain functional space. In (1) \(\mathfrak{R}\) is associated with the elasto-plastic torsion problem. The right-hand side of (1) represents a random noise, where \(M=M(t)\) is a certain Hilbert space valued continuous martingale. It is seeking for a stochastic process \(u=u(t)\) defined on the time interval \([0,T]\) such that \(u(t)\in \mathfrak{R}\) for almost all \(t\) and (1) is satisfied in an appropriate sense with probability one. Following to the article [\textit{D. Nualart} and \textit{E. Pardoux}, Probab. Theory Relat. Fields 93, No. 1, 77--89 (1992; Zbl 0767.60055)], the author reduces this problem to the deterministic problem \[ \frac{\partial }{\partial t}(u-M)-\triangle u \in -\partial I_{\mathfrak{R}}(u) \] with essential modification, since the right-hand-side of (1) is not an ordinary function with respect to time variable and \(M(t)\) is only Hölder continuous by \(t\) and is not of bounded variation. The assumption \(M\in C([0,T];C_0^1(\overline{G}))\) allows to establish the uniqueness of solution. The existence of the solution is proved then at the usage of stochastic integrals. It is shown also that this stochastic process is a Markov one and has an invariant measure on \(\mathfrak{R}\).
scientific article; zbMATH DE number 6351687