Day 4 Titles and Abstracts



Optimal transport, smoothness and cut locus

Cedric Villani
Ecole Normale Superiore, Lyon, France and Institute Henri Poincaré
villani -at- ihp.jussieu.fr

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The problem of smoothness of optimal transport mixes smooth and nonsmooth analysis together with geometry in an incredibly tight way, which I will review. As a consequence, an unexpected result of stability of the cut locus near the round sphere will be presented.

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Smoothing estimates for dispersive equations

Michael Ruzhansky
Imperial College, London, UK
m.ruzhansky -at- imperial.ac.uk

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We present two new methods: canonical transforms and comparison principles, for the analysis of smoothing estimates for a variety of dispersive equations with constant coefficients. These will allow us not only to establish new estimates, but to relate the known estimates to each other. The lecture will be based on the joint work with Mitsuru Sugimoto (Nagoya).

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Existence and non-existence of TV bounds for scalar conservation laws with discontinuous flux

G.D.Veerappa Gowda
Tata Institute of Fundamental Research
Centre for Applicable Mathematics,
gowda -at- math.tifrbng.res.in

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For the scalar conservation laws with discontinuous flux, an infinite family of $ (A,B)$-interface entropies are introduced and each one of them has been shown to form an $ L^1$-contraction semigroup. One of the main unsettled questions concerning conservation law with discontinuous flux is boundedness of total variation of the solution. In this talk, we discuss this particular issue in detail and produce a counter example to show that the solution, in general, have unbounded total variation near the interface. In fact this example illustrates that smallness of BV norm of the initial data is immaterial. We hereby settled the question of determining for which of the aforementioned $ (A,B)$ pairs, the solution will have bounded total variation in the case of strictly convex fluxes.

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A posteriori error estimates for Hamilton-Jacobi equations

B. Cockburn
University of Minnesota, USA
cockburn -at- math.umn.edu

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Being able to compare the viscosity solution of a Hamilton-Jacobi equation with an arbitrary continuous function is essential for many practical applications. We show how to estimate the uniform norm of the difference of the viscosity solution and any continuous function for steady-state and time-dependent model Hamilton-Jacobi equations. The estimate is independent of the space dimension and of the convexity properties of the Hamiltonian. Numerical experiments showing the quality of the estimate are presented.

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Some elliptic problem with exponential nonlinearies

Marcello Lucia
City University, New York, USA
mlucia -at- math.csi.cuny.edu

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I will present some uniqueness and existence results for some problems involving exponential nonlinearities that arise in geometry or physics.

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Multiple positive solutions for quasilinear degenerate elliptic equations with convex-concave type nonlinearities

K. Sreenadh
Indian Institute of Technology, Delhi
sreenadh -at- gmail.com

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We discuss the existence of multiple positive solutions for p-Laplacian type equations on bounded domains with combination of convex-concave nolinearities. A brief survey and recent developments for Dirichlet and co-normal boundary value problems with exponential growth nonlinearities will be discussed.

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Nonexistence of k-peaks solutions for some elliptic problem in strictly convex domains.

Massimo Grossi
University of Rome-1, Italy
grossi -at- mat.uniroma1.it

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let us consider the positive solution with zero Dirichlet boundary data of the equation

$\displaystyle -\Delta u=u^{\frac{n+2}{n-2}-\epsilon}$

in a strictly convex domain. We show that, for $ \epsilon$ small enough, there exists only the solution with one peak. The same happens, in the two dimensional case, for the equation

$\displaystyle -\Delta u=\lambda e^u$

when $ \lambda$ is small enough.

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