Cordes conditions and some alternatives for parabolic equations and discontinuous diffusion

TL;DR

This paper explores conditions ensuring existence and regularity of solutions for parabolic equations with discontinuous coefficients, extending classical results and applying them to diffusion processes.

Contribution

It introduces Cordes-type conditions for parabolic equations with discontinuous coefficients, providing new existence and regularity results.

Findings

Existence of solutions under Cordes conditions.

Regularity results for solutions with discontinuous coefficients.

Application to diffusion process modeling.

Abstract

The paper considers parabolic equations in non-divergent form with discontinuous coefficients at higher derivatives. Their investigation is most complicated because, in general, in the case of discontinuous coefficients, the uniqueness of a solution for nonlinear parabolic or elliptic equations can fail, and there is no a priory estimate for partial derivatives of a solution. In this paper, existence and regularity results are obtained under some Cordes type restrictions on the coefficients. The results are applied to diffusion processes.