Renormalized solutions for stochastic $ p $-Laplace equations with $ L^1 $-initial data: The case of multiplicative noise

Abstract

<p style='text-indent:20px;'>We consider a <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplace evolution problem with multiplicative noise on a bounded domain <inline-formula><tex-math id="M4">\begin{document}$ D\subset\mathbb{R}^d $\end{document}</tex-math></inline-formula> with homogeneous Dirichlet boundary conditions for <inline-formula><tex-math id="M5">\begin{document}$ 1&lt;p&lt;\infty $\end{document}</tex-math></inline-formula>. The random initial data is merely integrable. Consequently, the key estimates are available with respect to truncations of the solution. We introduce the notion of renormalized solutions for multiplicative stochastic <inline-formula><tex-math id="M6">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplace equations with <inline-formula><tex-math id="M7">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula>-initial data and study existence and uniqueness of solutions in this framework.</p>