These notes describe some aspects of partial diierential equations from the view point of the variational calculus. We begin by showing how a vari-ational problem (the study of critical points of an action functional) can be associated to a diierential equation. Here the idea is to deene the action in such a way that the associated Euler-Lagrange equation is the diierential equation of interest...

The scalar nonconvex variational problems of the minimum-energy type on Sobolev spaces are studied. As the Euler-Lagrange equation dramatically looses selectivity when extended in terms of the Young measures, the correct optimality conditions are sought by means of the convex compactification theory. It turns out that these conditions basically combine one part from the Euler-Lagrange equation ...

In 1940 (and 1964) S. M. Ulam proposed the well-known Ulam stability problem. In 1941 D. H. Hyers solved the Hyers-Ulam problem for linear mappings. In 1992 and 2008, J. M. Rassias introduced the Euler-Lagrange quadratic mappings and the JMRassias “product-sum” stability, respectively. In this paper we introduce an Euler-Lagrange type quadratic functional equation and investigate the JMRassias ...

Physical and geometric based variational techniques for surface construction have been shown to be advanced methods for designing high quality surfaces in the fields of CAD and CAGD. In this paper, we derive an Euler-Lagrange equation from a geometric invariant curvature integral functional–the integral about the mean curvature gradient. Using this Euler-Lagrange equation, we construct a sixtho...

-Using the He’s variational iteration method, it is possible to find the exact solutions or better approximate solutions of the partial differential equations. In this method, a correction functional is constructed by a general Lagrange multiplier, which can be identified via variational theory. In this paper, this method is used for solving a nonlinear partial differential equation, a three-di...

In this paper, smooth Chua’s equation is generalized to a higher order system from a special viewpoint of interconnected systems. Simple conditions for Lagrange stability are established. And a detailed Lagrange stable region analysis is given for the canonical Chua’s oscillator. In addition, a new nonlinearly coupled Chua’s circuit that appeared in the recent literature is also discussed and a...

The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is based on factorization of a non-homogeneous first order differential operator to products consisting of a scalar function, a homogeneous first order differential operator and the reciprocal of the scalar function. The factorization procedure is utiliz...