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Well-posedness of renormalized solutions for a stochastic $ p $-Laplace equation with $ L^1 $-initial data

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  • We consider a $ p $-Laplace evolution problem with stochastic forcing on a bounded domain $ D\subset\mathbb{R}^d $ with homogeneous Dirichlet boundary conditions for $ 1<p<\infty $. The additive noise term is given by a stochastic integral in the sense of Itô. The technical difficulties arise from the merely integrable random initial data $ u_0 $ under consideration. Due to the poor regularity of the initial data, estimates in $ W^{1,p}_0(D) $ are available with respect to truncations of the solution only and therefore well-posedness results have to be formulated in the sense of generalized solutions. We extend the notion of renormalized solution for this type of SPDEs, show well-posedness in this setting and study the Markov properties of solutions.

    Mathematics Subject Classification: 35K92, 60H15.


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