Compactness methods for $\Gamma^{1,\alpha}$ boundary Schauder estimates in Carnot groups

TL;DR

This paper establishes Schauder estimates for boundary regularity of solutions to perturbed horizontal Laplacians in Carnot groups, crucial for understanding subelliptic free boundary problems.

Contribution

It introduces compactness methods to prove boundary Schauder estimates for $ abla^{1,eta}$ regularity in Carnot groups with minimally smooth domains.

Findings

Proves boundary Schauder estimates for $ abla^{1,eta}$ regularity.

Addresses minimally smooth boundary domains in Carnot groups.

Provides tools applicable to subelliptic free boundary problems.

Abstract

The aim of this paper is to prove $\Gamma^{1,\alpha}$ Schauder estimates near a $C^{1,\alpha}$ non-characteristic portion of the boundary for $\Gamma^{0, \alpha}$ perturbations of horizontal Laplaceans in Carnot groups. This situation of minimally smooth domains presents itself naturally in the study of subelliptic free boundary problems of obstacle type, see [15].