Obstacle problem for evolution equations involving measure data and operator corresponding to semi-Dirichlet form

Abstract

Abstract. In the paper, we consider the obstacle problem, with one and two irregular barriers, for semilinear evolution equation involving measure data and operator corresponding to a semi-Dirichlet form. We prove the existence and uniqueness of solutions under the assumption that the right-hand side of the equation is monotone and satisfies mild integrability conditions. To treat the case of irregular barriers, we extend the theory of precise versions of functions introduced by M. Pierre. We also give some applications to the so-called switching problem.

Asymptotics for logistic-type equations with Dirichlet fractional Laplace operator

“…The proof is completed by showing that (i) is equivalent to (iii). To see this, we first note that by [31, Theorem 6.2, Chapter 3], there exists a unique solution to (1.2), and by [26,Theorem 5.3], there exists a unique solution to the problem (iii) (i.e. a unique function v (T ) satisfying (a), (b) for some uniformly integrable martingale M ).…”

The obstacle problem for stochastic porous media equations

“…Qiu [43] expanded to backward stochastic partial differential equations. [35,34,33,27,20,17] applied the method of probabilistic interpretation of the solution using backward doubly stochastic differential equation. It is worth noting that the probabilistic interpretation method is still feasible for nonlinear stochastic partial differential equations.…”

Smooth measures and capacities associated with nonlocal parabolic operators

“…There has recently been increasing interest in semilinear evolution problems of the form − ∂u ∂t − L t u = f (·, u) + µ, u(T, ·) = ϕ, (1.3) involving operators L t associated with a (possibly nonlocal) Dirichlet form and bounded measure that do not charge the sets of zero parabolic capacity associated with ∂ t + L t (see [12,11,13] and the references therein). Motivated by possible applications to problems of the form (1.3), in the present paper we investigate the structure of such measures.…”