Estimates in $L_{p}$ for solutions of SPDEs with coefficients in Morrey classes

TL;DR

This paper derives $L_{p}$-norm estimates for solutions of divergence form SPDEs with coefficients in Morrey classes, extending classical results to more general coefficient conditions, even without stochastic terms.

Contribution

It introduces $L_{p}$ estimates for SPDE solutions with Morrey class coefficients, a novel extension beyond traditional $L_{p}$-space assumptions.

Findings

Established $L_{p}$-norm bounds for solutions and derivatives.

Results are new even in deterministic cases.

Extended estimates to coefficients in Morrey classes.

Abstract

For solutions of a certain class of SPDEs in divergence form we present some estimates of their $L_{p}$-norms and the $L_{p}$-norms of their first-order derivatives. The main novelty is that the low-order coefficients are supposed to belong to certain Morrey classes instead of $L_{p}$-spaces. Our results are new even if there are no stochastic terms in the equation.