Dimensional Universality of Schauder Estimates Constants for Fourth Order Heat-Type Equations

TL;DR

This paper introduces a novel method to derive Schauder estimates for multidimensional fourth order heat-type equations, leveraging one-dimensional estimates and probabilistic constructions, revealing a universality of constants across dimensions.

Contribution

It presents a new approach combining existing results and probabilistic methods to establish Schauder estimates with universal constants for complex heat equations.

Findings

Multidimensional Schauder estimates can be derived from one-dimensional cases.

Constants in Schauder estimates are universal across dimensions for these equations.

The method bridges deterministic and probabilistic techniques for PDE analysis.

Abstract

A new method to compute Schauder Estimates for multidimensional fourth order heat-type equations is proposed. In particular, we show how knowing Schauder or Sobolev estimates for the one-dimensional fourth order heat equation allows to derive their multidimensional analogs for equations with time inhomogeneous coefficients with the same constants as in the case of the one-dimensional heat equation. Our method relies on a merger between (Krylov and Priola, 2017), where they actually showed the same result for the classical second order heat equation and (Funaki, 1979), where a probabilistic construction of solutions for the fourth order heat equation is presented.