Nonlocal boundary value problems of a stochastic variational inequality modeling an elasto-plastic oscillator excited by a filtered noise

In the literature, failure risk analysis on most elasto-perfectly-plastic oscillators is essentially focused on those excited by white noise, which is rather restrictive from the modeling perspective. Our present article is motivated by the study of the probability distribution of the solution of a stochastic variational inequality modeling an elasto-plastic oscillator excited by a filtered noise. We introduce a class of partial differential equations (PDEs) with nonlocal Dirichlet conditions and we establish the unique existence of solutions of these PDEs by extending the method developed in [A. Bensoussan and J. Turi, Applied and Numerical Partial Differential Equations, Comput. Methods Appl. Sci. 15, Springer, New York, 2009, pp. 9-23]. A major mathematical challenge here is to carry out the analysis of boundary value problems for elliptic equations in dimension two rather than that in dimension one.