# Anisotropic Orlicz-Sobolev spaces of vector valued functions and Lagrange equations

## TL;DR

This paper investigates properties of anisotropic Orlicz-Sobolev spaces for vector functions, establishing a variational framework for Lagrangian systems with conditions ensuring well-defined and differentiable functionals.

## Contribution

It introduces a variational setting for Lagrangian systems within anisotropic Orlicz-Sobolev spaces and provides conditions for the functional's proper mathematical behavior.

## Key findings

- Properties of anisotropic Orlicz and Orlicz-Sobolev spaces are characterized.
- A variational framework for certain Lagrangian systems is developed.
- Conditions for the functional to be well-defined and differentiable are established.

## Abstract

In this paper we study some properties of anisotropic Orlicz and anisotropic Orlicz-Sobolev spaces of vector valued functions for a special class of G-functions. We introduce a variational setting for a class of Lagrangian Systems. We give conditions which ensure that the principal part of variational functional is finitely defined and continuously differentiable on Orlicz-Sobolev space.