Characterization of the Solvability of Generalized Constrained Variational Equations

Manuel Ruiz Galán

doi:10.1155/2012/247425

Abstract

In a general context, that of the locally convex spaces, we characterize the existence of a solution for certain variational equations with constraints. For the normed case and in the presence of some kind of compactness of the closed unit ball, more specifically, when we deal with reflexive spaces or, in a more general way, with dual spaces, we deduce results implying the existence of a unique weak solution for a wide class of linear elliptic boundary value problems that do not admit a classical treatment. Finally, we apply our statements to the study of linear impulsive differential equations, extending previously stated results.

Highlights

It is common knowledge that in studying differential problems, variational methods have come to be essential
In this paper we replace the Lax-Milgram theorem with a characterization of the unique solvability of a certain type of variational equation with constraints. Such a constrained variational equation arises naturally; for instance, when in the variational formulation of an elliptic partial differential equation, its essential boundary constraints are treated as constraints
When the function data do not belong to Hilbert spaces, these results do not apply. For this reason we study a more general type of variational equation with constraints, whose most important particular case relies on this construction: given a reflexive Banach space X, normed spaces Y, Z, and W, continuous bilinear forms a : X × Y → R, b : Y × Z → R, and c : X × W → R, and continuous linear functionals y0∗ : Y → R and w0∗ : W → R, denoting

Summary

Introduction

It is common knowledge that in studying differential problems, variational methods have come to be essential. Such a constrained variational equation arises naturally; for instance, when in the variational formulation of an elliptic partial differential equation, its essential boundary constraints are treated as constraints This result allows us to consider problems without data functions in a Hilbert space, which is beyond the control of the classical theory 3, section II.[1] Proposition 1.1 , 4, Lemma 4.67. When the function data do not belong to Hilbert spaces, these results do not apply For this reason we study a more general type of variational equation with constraints, whose most important particular case relies on this construction: given a reflexive Banach space X, normed spaces Y , Z, and W, continuous bilinear forms a : X × Y → R, b : Y × Z → R, and c : X × W → R, and continuous linear functionals y0∗ : Y → R and w0∗ : W → R, denoting. We assume that all the spaces are real, our results are valid and adapted to the complex case

Variational Inequalities with Constraints in Lcs

Constrained Variational Equations in Reflexive Banach Spaces

A Word on Dual Banach Spaces

Application to Linear Impulsive Differential Equations