In this paper we establish new restrictions on symplectic embeddings of certain convex domains into symplectic vector spaces. These restrictions are stronger than those implied by the Ekeland-Hofer capacities. By refining an embedding technique due to Guth, we also show that they are sharp.

In this article, we study the multiplicity of solutions for a class fourth-order elliptic equations with concave and convex nonlinearities in $\mathbb{R}^N$. Under appropriate assumption, prove that there are at least two equation by Nehari manifold Ekeland variational principle, one which is ground state solution.

We extend the classical Neyman-Pearson theory for testing composite hypotheses versus composite alternatives, using a convex duality approach as in Witting (1985). Results of Aubin & Ekeland (1984) from non-smooth convex analysis are employed, along with a theorem of Komll os (1967), in order to establish the existence of a max-min optimal test and to investigate its properties. The theory is i...

· M. Y. Brenier Directeur de Recherches CNRS, Université de Nice-Sophia-Antipolis · M. G. Buttazzo Professeur Università di Pisa · M. G. Carlier Professeur Université Paris-Dauphine · M. J.-A. Carrillo Profesor Investigador ICREA, Universitat Autònoma de Barcelona · M. I. Ekeland Professeur U. Paris-Dauphine et U. of British Columbia · M. P.-L. Lions Professeur Collège de France · M. J.-M. More...

We consider the classical inverse mapping theorem of Nash and Moser from angle some recent development by Ekeland authors. Geometrisation tame estimates coupled with certain ideas coming variational analysis when applied to a directionally differentiable produces very general surjectivity result and, if injectivity can be ensured, expected Lipschitz-like continuity inverse. also present brief a...

We consider quasilinear strongly resonant problems with discontinuous right-hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points. We prove the existence of at least three nontrivial solutions. Our approach uses the nonsmooth critical point theory for locally Lipschitz functionals due to Chang (1981) and ...

In this paper we will investigate the existence of multiple solutions for the problem (P ) −∆pu+ g(x, u) = λ1h(x) |u|p−2 u, in Ω, u ∈ H 0 (Ω) where ∆pu = div ( |∇u|p−2 ∇u ) is the p-Laplacian operator, Ω ⊆ IR is a bounded domain with smooth boundary, h and g are bounded functions, N ≥ 1 and 1 < p < ∞. Using the Mountain Pass Theorem and the Ekeland Variational Principle, we will show the existe...

We propose a deep learning method for the numerical solution of partial differential equations that arise as gradient flows. The relies on Brezis–Ekeland principle, which naturally defines an objective function to be minimized, and so is ideally suited machine approach using neural networks. describe our in general framework illustrate with help example implementation heat equation space dimens...

A general abstract duality result is proposed for equations which are governed by the sum of two operators (possibly multivalued). It allows to unify a large number of variational duality principles, including the Clarke-Ekeland least dual action principle and the Singer-Toland duality. Moreover, it offers a new duality approach to some central questions in the theory of variational inequalitie...