On the Sobolev trace Theorem for variable exponent spaces in the critical range

TL;DR

This paper investigates the Sobolev Trace Theorem in variable exponent spaces at critical exponents, establishing conditions for extremal existence based on domain and exponent properties.

Contribution

It provides new criteria linking domain geometry and variable exponents to the existence of extremals in the Sobolev Trace Theorem at critical ranges.

Findings

Conditions on the best constant for extremal existence

Local criteria on exponents and domain geometry

Existence results for extremals in variable exponent spaces

Abstract

In this paper we study the Sobolev Trace Theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. Then we give local conditions on the exponents and on the domain (in the spirit of Adimurthy and Yadava) in order to satisfy such conditions, and therefore to ensure the existence of extremals.