We derive a new variational principle in optics. We first formulate the principle for paraxial waves and then generalize it to arbitrary waves. The new principle, unlike the Fermat principle, concerns both the phase and the intensity of the wave. In particular, the principle provides a method for finding the ray mapping between two surfaces in space from information on the wave's intensity ther...

This paper is concerned with the existence of two non-trivial weak solutions for a p(x)-Kirchho type problem by using the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle and the theory of the variable exponent Sobolev spaces.

X iv :p hy si cs /9 90 30 14 v1 [ ph ys ic s. cl as sph ] 8 M ar 1 99 9 Unified Descriptions Of All Differential Variational Principles Y. C. Huang Z. X. Liu X. G. Li Department of Applied Physics, Beijing Polytechnic University, Beijing 100022, P.R. China Department of Physics, Henan Normal University, Xingxiang 453002, P. R. China INFN, Sezione di Catania, Corso Italia 57, I-95129 Catania, It...

We prove a variational principle in reflexive Banach spaces X with KadecKlee norm, which asserts that any Lipschitz (or any proper lower semicontinuous bounded from below extended real-valued) function in X can be perturbed with a parabola in such a way that the perturbed function attains its infimum (even more can be said — the infimum is well-posed). In addition, we have genericity of the poi...

We consider the following nonlinear singular elliptic equation −div (|x| −2a ∇u) = K(x)|x| −bp |u| p−2 u + λg(x) in R N , where g belongs to an appropriate weighted Sobolev space, and p denotes the Caffarelli–Kohn– Nirenberg critical exponent associated to a, b, and N. Under some natural assumptions on the positive potential K(x) we establish the existence of some λ 0 > 0 such that the above pr...

This paper is concerned with the existence of nontrivial solutions for a class of degenerate quasilinear elliptic systems involving critical Hardy-Sobolev type exponents. The lack of compactness is overcame by using the Brezis-Nirenberg approach, and the multiplicity result is obtained by combining a version of the Ekeland’s variational principle due to Mizoguchi with the Ambrosetti-Rabinowitz ...

A general variational principle of classical fields with a Lagrangian containing field quantity and its derivatives of up to the N-order is presented. Noether’s theorem is derived. The generalized Hamilton-Jacobi’s equation for the Hamilton’s principal functional is obtained. These results are surprisingly in great harmony with each other. They will be applied to general relativity in the subse...

I find conditions under which the ”Weak Energy Principle” of Katz, Inagaki and Yahalom (1993) gives necessary and sufficient conditions. My conclusion is that, necessary and sufficient conditions of stability are obtained when we have only two mode coupling in the gyroscopic terms of the perturbed Lagrangian. To illustrate the power of this new energy principle, I have calculated the stability ...

In this paper, we establish a variant of Ekeland’s variational principle. This result suggest to introduce a generalization of the famous PalaisSmale condition. An example is provided showing how it is used to give the existence of minimizer for functions for which the Palais-Smale condition and the one introduced by Cerami are not satisfied.