Doubly nonlinear stochastic evolution equations

Abstract

We present an existence theory for martingale and strong solutions to doubly nonlinear evolution equations in a separable Hilbert space in the form $$d(Au) + Bu\,dt \ni F(u)\,dt + G(u)\,dW$$ where both $A$ and $B$ are maximal monotone operators, possibly multivalued, $F$ and $G$ are Lipschitz-continuous, and $W$ is a cylindrical Wiener process. Via regularization and passage-to-the-limit we show the existence of martingale solutions. The identification of the limit is obtained by a lower-semicontinuity argument based on a suitably generalized It\^o's formula. If either $A$ or $B$ is linear and symmetric, existence and uniqueness of strong solutions follows. Eventually, several applications are discussed, including doubly nonlinear stochastic Stefan-type problems.