We consider a variational principle suitable for tracking boundaries of gas bubbles in liquids and its consequences. The principle is derived from a generalization of the Principle of Stationary Action applied to a Riemannian manifold of volume preserving flow maps. The dual variational principle for the markers tracking the free surfaces is induced by the Wasserstein-Kantorovich metric. This l...

The functional type extension of Ekeland’s variational principle [7] due to Zhong [18] is logical equivalent with it.

This paper is concerned with necessary and sufficient conditions for near-optimal singular stochastic controls for systems driven by a nonlinear stochastic differential equations (SDEs in short). The proof of our result is based on Ekeland’s variational principle and some dilecate estimates of the state and adjoint processes. This result is a generalization of Zhou’s stochastic maximum principl...

The field equations associated with the Born-Infeld-Einstein action including matter are derived using a Palatini variational principle. Scalar, electromagnetic, and Dirac fields are considered. It is shown that an action can be chosen for the scalar field that produces field equations identical to the usual Einstein field equations minimally coupled to a scalar field. In the electromagnetic an...

We introduce several new kinds of inferior and superior limits and corresponding kinds of semicontinuity of a set-valued map. Together with the known concepts of semicontinuity, they can be used to have a clearer insight of local behaviors of maps. Then we investigate all major semicontinuity properties of solution maps to a general quasivariational inclusion. Consequences are derived for sever...

We show that some measures suuering from the so-called Renormalization Group pathologies satisfy a variational principle and that the corresponding limit of the pressure, with boundary conditions in a set of measure 1, is proportional to the pressure of the Ising model.

The variational principle states that if a differentiable functional F attains its minimum at some point zi, then F’(C) = 0; it has proved a valuable tool for studying partial differential equations. This paper shows that if a differentiable function F has a finite lower bound (although it need not attain it), then, for every E > 0, there exists some point u( where 11 F’(uJj* < l , i.e., its de...