Existence and multiplicity of weak solutions for an elliptic system is studied. By using Ekeland’s variational principle and the mountain pass theorem, we prove existence of at least three weak solutions. AMS Subject Classifications: 35J40, 35J67.

where K, h ∈ L(R), a(x) is a positive bounded function, and f(s) is asymptotically linear with respect to s at infinity, that is f(s)/s goes to a constant as s → +∞. Under suitable assumptions on K, a and f , we prove that the problem (P) has at least two positive solutions for |K|2 and |h|2 small by using Ekeland’s variational principle and Mountain pass theorem.

We propose a very weak type of generalized distance called weak τ -function and use it to weaken the assumptions about lower semicontinuity in existing formulations of Ekeland’s variational principle for a kind of minimizers of a set-valued mapping, which is different from the Pareto minimizer, and in recent results which are equivalent to Ekeland’s variational principle.

In this article, we establish necessary conditions for optimality in stochastic control of systems governed by stochastic differential equations with nonsmooth coefficients. The approach used is based on the approximation of the nonsmooth coefficient by smooth one which generate a sequence of smooth control problems. Ekeland’s variational principle is then applied to obtain a sequence of nearly...

In this paper, we take a fresh look at using spectral analysis for assessing locking phenomena in finite element formulations. We propose to “measure” by comparing the difference between eigenvalue and mode error curves computed on coarse meshes with “asymptotic” “overkill” meshes, both plotted respect normalized number. To demonstrate intimate relation membrane accuracy, focus example of circu...

We describe a dissipation principle/variational principle which may be useful in modeling motion in small viscous systems and provide brief illustrations to brownian motor or molecular rachet situations which are found in intracellular transport. Monge-Kantorovich mass transport and Wasserstein metric play an interesting role in these developments. Some properties of the system that ensure the ...

We consider a class of alloys and ceramics with equilibria described by non-attainable infima of non-quasiconvex variational integrals. Such situations frequently arise when atomic lattice structure plays an important role at the mesoscopic continuum level. We prove that standard variational approaches associated with gradient based relaxation of non-quasiconvex integrals in Banach or Hilbert s...

Abstract. We use a new variational method—based on the theory of anti-selfdual Lagrangians developed in [2] and [3]—to establish the existence of solutions of convex Hamiltonian systems that connect two given Lagrangian submanifolds in R . We also consider the case where the Hamiltonian is only semi-convex. A variational principle is also used to establish existence for the corresponding Cauchy...

A recently developed variational principle of virtual dissipation along with a new approach to the thermodynamics of open systems is applied to coupled mass and heat transfer in a porous solid containing a fluid. General differential field equations are derived directly from the variational principle. A general energy flux theorem is formulated. Vapor-liquid phase transition and capillary conde...