We discuss some of the recent developments in variational methods while emphasizing new applications to nonlinear problems. We touch on several issues: (i) the formulation of variational set-ups which provide more information on the location of critical points and therefore on the qualitative properties of the solutions of corresponding Euler-Lagrange equations; (ii) the relationships between t...

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts reveals partly unexpected phenomena related to the specific nature of the norms induced by th...

Convergence of an approximate method for determining vibrational eigenpairs of an elastic solid containing an incompressible fluid is examined. The field variables are solid displacement and fluid pressure. We show that in suitable Sobolev spaces a variational formulation exists whose solution eigenvalues and eigenfunctions are identified with those of a compact operator. A nonconforming finite...

A connection between the maximum a posteriori (MAP) estimation and the variational formulation based on the minimization of a given variational integral subject to some noise constraints is established in this paper. A MAP estimator which uses a Markov or a maximum entropy random field model for the prior distribution can be viewed as a minimizer of a variational problem. Inspired by the maximu...

In this article we study the numerical approximation of a quasi-Newtonian Stokes flow problem where only the flow rates are specified at the inflow and outflow boundaries. A variational formulation of the problem, using Lagrange multipliers to enforce the stated flow rates, is given. Existence and uniqueness of the solution to the continuous, and discrete, variational formulations is shown. An ...

A variational formulation of the d-wave superconducting ground state and low-lying excitations in doped Mott insulators was developed building on early resonating valence bond ideas of P. W, Anderson. Analytical approximation schemes were developed to gain further insights into our recent variational Monte Carlo results. The effect of disorder on a two-dimensional Mott insulator was studied and...

A new system of extended general nonlinear regularized nonconvex set-valued variational inequalities is introduced, and the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities and the fixed point problems is verified. Then, by this equivalent formulation, a new perturbed projection iterative algorithm with mixed errors for finding a solut...

The smoothing properties of variational methods for the design of fair surfaces are considered using a PDE (partial differential equation) model. The PDE is the Euler-Lagrange equation that is obtained from the variational formulation of the surface design problem. The smoothing properties of the design scheme are considered by looking at the scheme's efficiency in reducing the amplitude of the...

In this note we consider a class of environmental games recently proposed in the literature and investigate them by using the powerful tools of variational inequalities. We also consider the case where some data of the problem can depend on a parameter and analyze the regularity of the solution with respect to the parameter. In view of applications to time-dependent or random models, we also in...

It is well known that the variational inequalities are equivalent to the fixed point problem. We use this alternative equivalent formulation to suggest and analyze some new proximal point methods for solving the variational inequalities. These new methods include the explicit, the implicit, and the extragradient methods as special cases. The convergence analysis of the new methods is considered...