On the validity of variational inequalities for obstacle problems with non-standard growth

TL;DR

This paper demonstrates that solutions to variational problems with non-standard growth satisfy the associated variational inequalities without smallness restrictions, using convex analysis and duality techniques.

Contribution

It establishes the validity of variational inequalities for obstacle problems with non-standard growth without smallness assumptions, expanding the theoretical understanding.

Findings

Solutions satisfy variational inequalities without smallness conditions

Duality formulas are established for approximating problems

Results apply to functions with finite energy as competitors

Abstract

The aim of the paper is to show that the solutions to variational problems with non-standard growth conditions satisfy a corresponding variational inequality without any smallness assumptions on the gap between growth and coercitivity exponents. Our results rely on techniques based on Convex Analysis that consist in establishing duality formulas and pointwise relations between minimizers and corresponding dual maximizers, for suitable approximating problems, that are preserved passing to the limit. In this respect we are able to show that the right class of competitors are the functions with finite energy, in agreement with the unconstrained results.