On Well-Posedness of Some Constrained Variational Problems

Savin Treanţă

doi:10.3390/math9192478

Abstract

By considering the new forms of the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity of the considered scalar multiple integral functional, in this paper we study the well-posedness of a new class of variational problems with variational inequality constraints. More specifically, by defining the set of approximating solutions for the class of variational problems under study, we establish several results on well-posedness.

Highlights

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In this paper, motivated and inspired by the above research papers, we study the well-posedness property for new constrained variational problems, implying second-order multiple integral functionals and partial derivatives
The framework is based on function spaces of infinite-dimension and multiple integral-type functionals. This element is completely new for the well-posed optimization problems

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. In this paper, motivated and inspired by the above research papers, we study the well-posedness property for new constrained variational problems, implying second-order multiple integral functionals and partial derivatives. In this regard, we formulate new forms of monotonicity, lower semicontinuity, hemicontinuity, and pseudomonotonicity for the considered multiple integral-type functional. The framework is based on function spaces of infinite-dimension and multiple integral-type functionals This element is completely new for the well-posed optimization problems. The present paper is structured as follows: In Section 2, we formulate the problem under study and introduce the new forms of monotonicity, lower semicontinuity, hemicontinuity, and pseudomonotonicity for the considered multiple integral-type functional.

Preliminaries and Problem Formulation

Conclusions