Variational problems on product spaces Different obstacle constraints

TL;DR

This paper investigates two main variational minimization problems with different obstacle constraints in bounded open sets within one- and two-dimensional spaces, focusing on their mathematical properties and solutions.

Contribution

It introduces and analyzes two distinct variational problems with obstacle constraints on product spaces, expanding understanding of their solutions in low-dimensional settings.

Findings

Characterization of solutions in one-dimensional case

Existence results for two-dimensional problems

Comparison of obstacle constraints effects

Abstract

We study two principle minimizing problems, subject of different constraints. Our open sets are assumed bounded, except mentioning otherwise;precisely $\Omega=]0,1[^n \in {\mathbb{R}}^n , n=1 $ or $n=2$.