Dual variational methods for a nonlinear Helmholtz system

Variational Methods for Nonlinear Elliptic Eigenvalue Problems

In the present note, we give a simple general proof for the existence of solutions of the following two types of variational problems: PROBLEM A. To minimize fa F(x> u, • • • , Du)dx over a subspace VofW>*(tt). PROBLEM B. TO minimize ƒ« F(x, w, • • • , Du)dx for u in V with / a G(x, u, • • • , D^u)dx^c. The solution of the first problem yields a weak solution of a corresponding elliptic boundar...

Adaptive frame methods for nonlinear variational problems

In this paper we develop adaptive numerical solvers for certain nonlinear variational problems. The discretization of the variational problems is done by a suitable frame decomposition of the solution, i.e., a complete, stable, and redundant expansion. The discretization yields an equivalent nonlinear problem on the space of frame coefficients. The discrete problem is then adaptively solved usi...

A Survey of Direct Methods for Solving Variational Problems

This study presents a comparative survey of direct methods for solving Variational Problems. Thisproblems can be used to solve various differential equations in physics and chemistry like RateEquation for a chemical reaction. There are procedures that any type of a differential equation isconvertible to a variational problem. Therefore finding the solution of a differential equation isequivalen...

Dd Methods for a System of Coupled Helmholtz Equations

Primal–dual Methods for Nonlinear Constrained Optimization

. . . If a function of several variables should be a maximum or minimum and there are between these variables one or several equations, then it will be suffice to add to the proposed function the functions that should be zero, each multiplied by an undetermined quantity, and then to look for the maximum and the minimum as if the variables were independent; the equation that one will find combin...