Nonlinear Stochastic Partial Differential Equations with Singular Diffusivity and Gradient Stratonovich Noise

We study existence and uniqueness of a variational solution in terms of stochastic variational inequalities (SVI) to stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative Stratonovich gradient-type noise. We derive a commutator relation for the unbounded noise coefficients in terms of a geometric Killing vector condition. The drift term is given by the total variation flow, respectively, by a singular p-Laplace-type operator. We impose nonlinear zero Neumann boundary conditions and precisely investigate their connection with the coefficient fields of the noise. This solves an open problem posed in [Barbu, Brzeźniak, Hausenblas, Tubaro; Stoch. Proc. Appl., 123 (2013)] and [Barbu, Röckner; J. Eur. Math. Soc., 17 (2015)].