We consider a non-local isoperimetric problem arising as the sharp interface limit of the Ohta–Kawasaki free energy introduced to model microphase separation of diblock copolymers. We perform a second order variational analysis that allows us to provide a quantitative second order minimality condition. We show that critical configurations with positive second variation are indeed strict local minimizers of the problem. Moreover, we provide, via a suitable quantitative inequality of isoperimetric type, an estimate of the deviation from minimality for configurations close to the minimum in the

We study the evolution of a system of n particles {(. That system is a conservative system with a Hamiltonian of the formwhere W 2 is the Wasserstein distance and µ is a discrete measure concentrated on the set {(. Typically, µ(0) is a discrete measure approximating an initial L ∞ density and can be chosen randomly. When d = 1, our results prove convergence of the discrete system to a variant of the semigeostrophic equations. We obtain that the limiting densities are absolutely continuous with respect to Lebesgue measure. When {ν n } ∞ n=1converges to a measure concentrated on a special d-dimensional sets, we obtain the Vlasov-Monge-Ampère (VMA) system. When, d = 1 the VMA system coincides with the standard Vlasov-Poisson system.

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Abstract. We consider a functional which models an elastic body with a cavity. We show that if a critical point has positive second variation then it is a strict local minimizer. We also provide a quantitative estimate.

For H ∈ C 2 (R N ×n ) and u : Ω ⊆ R n → R N , consider the systemWe construct D-solutions to the Dirichlet problem for (1), an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our Dsolutions are W 1,∞ -submersions and are obtained without any convexity hypotheses for H, through a result of independent interest involving existence of strong solutions to the singular value problem for general dimensions n = N . α = 1, ..., N . Our general notation is either self-explanatory or a convex combination of standard symbolisations as e.g. in [E, D, EG, DM2]. The system (1.1) is the