Showing papers by "Jean Dolbeault published in 2013"
TL;DR: In this paper, a combination of tools, proofs and results are presented in the framework of the concentration-compactness method for the existence of steady states to the Maxwell-Schrodinger system.
Abstract: This paper is intended to review recent results and open problems concerning the existence of steady states to the Maxwell-Schrodinger system. A combination of tools, proofs and results are presented in the framework of the concentration--compactness method.
51 citations
TL;DR: In this article, the connection between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere is emphasized, and a series of related observations and proofs based on symmetrization and the ultraspherical setting are given.
Abstract: This paper is devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincare, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. The connection between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere is emphasized. The authors address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.
43 citations
TL;DR: In this paper, the authors consider a family of Gagliardo-Nirenberg-Sobolev interpolation inequalities which interpolate between Sobolev's inequality and the logarithmic Soboleve inequality, with optimal constants.
35 citations
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TL;DR: In this article, a nonlinear flow of porous medium/fast diffusion type is used to evaluate the rigidity of smooth compact Riemannian manifolds without boundaries and the optimality of the corresponding interpolation inequalities.
Abstract: This paper is devoted to rigidity results for some elliptic PDEs and related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution than the constant one at least when a parameter is in a certain range. This parameter can be used as an estimate for the best constant in the corresponding interpolation inequality. Our approach relies in a nonlinear flow of porous medium / fast diffusion type which gives a clear-cut interpretation of technical choices of exponents done in earlier works. We also establish two integral criteria for rigidity that improve upon known, pointwise conditions, and hold for general manifolds without positivity conditions on the curvature. Using the flow, we are also able to discuss the optimality of the corresponding constant in the interpolation inequalities.
27 citations
TL;DR: In this article, the first eigenvalue of Schrodinger operators on the d-dimensional unit sphere was established, which is equivalent to interpolation inequalities on the sphere for a semi-classical asymptotic regime.
Abstract: In this article we establish optimal estimates for the first eigenvalue of Schr\"odinger operators on the d-dimensional unit sphere These estimates depend on Lebsgue's norms of the potential, or of its inverse, and are equivalent to interpolation inequalities on the sphere We also characterize a semi-classical asymptotic regime and discuss how our estimates on the sphere differ from those on the Euclidean space
25 citations
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TL;DR: In this paper, a combination of tools, proofs and results are presented in the framework of the concentration-compactness method for the existence of steady states to the Maxwell-Schrodinger system.
Abstract: This paper is intended to review recent results and open problems concerning the existence of steady states to the Maxwell-Schr\"odinger system. A combination of tools, proofs and results are presented in the framework of the concentration--compactness method.
19 citations
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TL;DR: In this article, optimal spectral estimates for Schr\"odinger operators on compact connected Riemannian manifolds without boundary are given based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
Abstract: This note is devoted to optimal spectral estimates for Schr\"odinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
13 citations
TL;DR: In this article, the Gagliardo-Nirenberg-Sobolev inequalities on the line are compared to Sobolev's inequality on the sphere, at least when the dimension is an integer, or to critical interpolation inequality for the ultraspherical operator in the general case.
Abstract: This paper is devoted to one-dimensional interpolation Gagliardo-Nirenberg-Sobolev inequalities. We study how various notions of duality, transport and monotonicity of functionals along flows defined by some nonlinear diffusion equations apply. We start by reducing the inequality to a much simpler dual variational problem using mass transportation theory. Our second main result is devoted to the construction of a Lyapunov functional associated with a nonlinear diffusion equation, that provides an alternative proof of the inequality. The key observation is that the inequality on the line is equivalent to Sobolev's inequality on the sphere, at least when the dimension is an integer, or to the critical interpolation inequality for the ultraspherical operator in the general case. The time derivative of the functional along the flow is itself very interesting. It explains the machinery of some rigidity estimates for nonlinear elliptic equations and shows how eigenvalues of a linearized problem enter in the computations. Notions of gradient flows are then discussed for various notions of distances. Throughout this paper we shall deal with two classes of inequalities corresponding either to p>2 or to p<2. The algebraic part in the computations is very similar in both cases, although the case p<2 is definitely less standard.
13 citations
TL;DR: In this paper, the bifurcation of branches of non-symmetric solutions from the symmetric branch of solutions to the Euler-Lagrange equations satisfied by optimal functions in functional inequalities of Caffarelli-Kohn-Nirenberg type was studied.
Abstract: In this paper we study the bifurcation of branches of non-symmetric solutions from the symmetric branch of solutions to the Euler-Lagrange equations satisfied by optimal functions in functional inequalities of Caffarelli-Kohn-Nirenberg type. We establish the asymptotic behavior of the branches for large values of the bifurcation parameter. We also perform an expansion in a neighborhood of the first bifurcation point on the branch of symmetric solutions, that characterizes the local behavior of the non-symmetric branch. These results are compatible with earlier numerical and theoretical observations. Further numerical results allow us to distinguish two global scenarios. This sheds a new light on the symmetry breaking phenomenon.
13 citations
TL;DR: In this article, an optimal version of a generalization of the Onofri inequality in the d-dimensional Euclidean space for any d ≥ 2 was established by considering the endpoint of a family of optimal Gagliardo-Nirenberg interpolation inequalities.
Abstract: The classical Onofri inequality in the two-dimensional sphere assumes a natural form in the plane when transformed via stereographic projection. We establish an optimal version of a generalization of this inequality in the d-dimensional Euclidean space for any d≥2, by considering the endpoint of a family of optimal Gagliardo-Nirenberg interpolation inequalities. Unlike the two-dimensional case, this extension involves a rather unexpected Sobolev-Orlicz norm, as well as a probability measure no longer related to stereographic projection.
13 citations
TL;DR: In this paper, optimal spectral estimates for Schrodinger operators on compact connected Riemannian manifolds without boundary were derived based on appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
TL;DR: In this paper, the authors considered radial solutions of an elliptic equation involving the p-Laplace operator and proved by a shooting method the existence of compactly supported solutions with any prescribed number of nodes.
Abstract: We consider radial solutions of an elliptic equation involving the p-Laplace operator and prove by a shooting method the existence of compactly supported solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane corresponding to an asymptotic Hamiltonian system and provides qualitative properties of the solutions.
TL;DR: In this article, the authors study two models for crowd motion and herding, one of which is of Keller-Segel type and involves two parabolic equations, one for the evolution of the density and the other for a mean field potential.
Abstract: In this paper we study two models for crowd motion and herding. Each of the models is of Keller-Segel type and involves two parabolic equations, one for the evolution of the density and one for the evolution of a mean field potential. We classify all radial stationary solutions, prove multiplicity results and establish some qualitative properties of these solutions, which are characterized as critical points of an energy functional. A notion of variational stability is associated to such solutions. The dynamical stability in a neighborhood of a stationary solution is also studied in terms of the spectral properties of the linearized evolution operator. For one of the two models, we exhibit a Lyapunov functional which allows to make the link between the two notions of stability. Even in that case, for certain values of the mass parameter and all other parameters taken in an appropriate range, we find that two dynamically stable stationary solutions exist. We further discuss qualitative properties of the solutions using theoretical methods and numerical computations.
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TL;DR: In this article, an improved version of Sobolev and Onofri type inequalities is presented, where the additional terms involve the dual counterparts, i.e. Hardy-Littlewood-Sobolev type inequalities.
Abstract: This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy-Littlewood-Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin-Talenti functions.
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TL;DR: In this article, multiplicity results of solutions to nonlocal elliptic equations modeling gravitating systems were derived by considering the case of Fermi-Dirac statistics as a singular perturbation of the Maxwell-Boltzmann one.
Abstract: This paper is devoted to multiplicity results of solutions to nonlocal elliptic equations modeling gravitating systems. By considering the case of Fermi-Dirac statistics as a singular perturbation of Maxwell-Boltzmann one, we are able to produce multiplicity results. Our method is based on cumulated mass densities and a logarithmic change of coordinates that allows us to describe the set of all solutions by a non-autonomous perturbation of an autonomous dynamical system. This has interesting consequences in terms of bifurcation diagrams, which are illustrated by a some numerical computations. More specifically, we study a model based on the Fermi function as well as a simplified one for which estimates are easier to establish. The main difficulty comes from the fact that the mass enters in the equation as a parameter which makes the whole problem non-local.
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TL;DR: A review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents is given in this paper, where the authors also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.
Abstract: This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.