Regularity of weak solutions to the model Venttsel problem for solutions of linear parabolic systems with nonsmooth in time principal matrix. $A(t)$-caloric method

TL;DR

This paper proves the Hölder continuity of weak solutions to a Venttsel boundary problem for linear parabolic systems with nonsmooth time-dependent coefficients, using the A(t)-caloric method under minimal assumptions.

Contribution

It establishes Hölder regularity for solutions with only boundedness assumptions on the principal matrices, advancing understanding of parabolic systems with nonsmooth coefficients.

Findings

Hölder continuity of weak solutions is proven.

Results hold under minimal boundedness assumptions.

The A(t)-caloric method is effectively applied.

Abstract

We consider a model Venttsel type problem for linear parabolic systems of equations. The Venttsel type boundary condition is fixed on the flat part of the lateral surface of a given cylinder. It is defined by parabolic operator (with respect to the tangential derivatives) and the conormal derivative. The H\"{o}lder continuity of a weak solution of the problem is proved under optimal assumptions on the data. In particular, only boundedness in the time variable of the principal matrices of the system and the boundary operator is assumed. All results are obtained by so-called A(t)-caloric method.