Elliptic and parabolic second-order PDEs with growing coefficients

TL;DR

This paper establishes that global Schauder estimates hold for second-order parabolic PDEs with unbounded, measurable-in-time, locally H"older continuous-in-space coefficients, demonstrating independence from lower order coefficient norms.

Contribution

It introduces a new localization procedure proving Schauder estimates are valid even with unbounded coefficients and shows the estimates' constants are independent of coefficient norms.

Findings

Global Schauder estimates hold under minimal regularity assumptions.

Constants in estimates are independent of lower order coefficient norms.

Provides a new proof of uniqueness applicable to unbounded coefficients.

Abstract

We consider a second-order parabolic equation in $\bR^{d+1}$ with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally H\"older continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the $L_{\infty}$-norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients.