In this article, we prove uniqueness of symmetric positive solutions of the variational ODE system −w′′ + aw − wv = 0 −v′′ + bv − w2 2 = 0, where a and b are positive constants.

We present a general theorem on the existence of nite element minimizers for the approximation of variational problems of multiple inte-grals. Our theorem applies to variational problems for which a minimum does not exist in innnite-dimensional spaces of functions. Such problems occur in models for microstructure in martensitic and ferromagnetic crystals .

We consider the Cahn-Hilliard equation in one space dimension with scaling parameter ε, i.e. ut = (W ′(u) − εuxx)xx, where W is a nonconvex potential. In the limit ε ↓ 0, under the assumption that the initial data are energetically well-prepared, we show the convergence to a Stefan problem. The proof is based on variational methods and exploits the gradient flow structure of the Cahn-Hilliard e...

We show that the variational representations for f -divergences currently used in the literature can be tightened. This has implications to a number of methods recently proposed based on this representation. As an example application we use our tighter representation to derive a general f -divergence estimator based on two i.i.d. samples and derive the dual program for this estimator that perfo...

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In this paper we study Riemannanian manifolds (M, g) equipped with a smooth measure m. In particular, we show that Riemannian invariants of (M, g) give rise to conformal densities of the Riemannian measure space (M, g,m). This leads to a natural definition of the Ricci and scalar curvatures of RM -spaces, both of which are conformally invariant. We also study some natural variational integrals.

After presenting an overview about variational problems on probability measures for functionals involving transport costs and extra terms encouraging or discouraging concentration, we look for optimality conditions, regularity properties and explicit computations in the case where Wasserstein distances and interaction energies are considered.

We consider several differential-topological invariants of compact 4-manifolds which directly arise from Riemannian variational problems. Using recent results of Bauer and Furuta [5, 4], we compute these invariants in many cases that were previously intractable. In particular, we are now able to calculate the Yamabe invariant for many connected sums of complex surfaces.