Critical point theory for nondifferentiable functionals and applications

Critical Point Theory for Nonsmooth Energy Functionals and Applications

In this paper we prove an abstract result about the minimization of nonsmooth functionals. Then we obtain some existence results for Neumann problems with discontinuities.

Impossibility Results for Nondifferentiable Functionals∗

We examine challenges to estimation and inference when the objects of interest are nondifferentiable functionals of the underlying data distribution. This situation arises in a number of applications of bounds analysis and moment inequality models, and in recent work on estimating optimal dynamic treatment regimes. Drawing on earlier work relating differentiability to the existence of unbiased ...

Lecture 38: Applications of Critical Point Theory

1. Generalized Sphere Theorem As a first application of the critical point theory of distance funciton, we shall prove Theorem 1.1 (Grove-Shiohama). Let (M, g) be a complete simply connected Riemann-ian manifold with K > 1 4 and diam(M, g) ≥ π, then M is homeomorphic to S m. So Grove-Shiohama's theorem implies the sphere theorem. Proof. Let p, q ∈ M so that dist(p, q) = diam(M, g). In what foll...

Critical Point Theorems concerning Strongly Indefinite Functionals and Applications to Hamiltonian Systems

Let X be a Finsler manifold. We prove some abstract results on the existence of critical points for strongly indefinite functionals f ∈ C1(X ,R) via a new deformation theorem. Different from the works in the literature, the new deformation theorem is constructed under a new version of Cerami-type condition instead of Palais-Smale condition. As applications, we prove the existence of multiple pe...

Nondifferentiable Optimization: Introduction, Applications and Algorithms