Onofri inequalities and rigidity results
Copyright policy )Onofri inequalities and rigidity results
This paper is devoted to the Moser-Trudinger-Onofri inequality on smooth compact connected Riemannian manifolds. We establish a rigidity result for the Euler-Lagrange equation and deduce an estimate of the optimal constant in the inequality on two-dimensional closed Riemannian manifolds. Compared to existing results, we provide a non-local criterion which is well adapted to variational methods, introduce a nonlinear flow along which the evolution of a functional related with the inequality is monotone and get an integral remainder term which allows us to discuss optimality issues. As an important application of our method, we also consider the non-compact case of the Moser-Trudinger-Onofri inequality on the two-dimensional Euclidean space, with weights. The standard weight is the one that is computed when projecting the two-dimensional sphere using the stereographic projection, but we also give more general results which are of interest, for instance, for the Keller-Segel model in chemotaxis.
Elliptic equations on manifolds, general theory, compact Riemannian manifold, Sobolev inequality, optimal constant, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, Nonlinear elliptic equations, Ricci tensor, 510, Mathematics - Analysis of PDEs, 515, Laplace-Beltrami operator, FOS: Mathematics, Nonlinear parabolic equations, nonlinear diffusion, semilinear elliptic equation, uniqueness, Heat and other parabolic equation methods for PDEs on manifolds, Analyse, carré du champ, rigidity, Poincaré inequality, Moser-Trudinger-Onofri inequality, Analysis of PDEs (math.AP)
Elliptic equations on manifolds, general theory, compact Riemannian manifold, Sobolev inequality, optimal constant, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, Nonlinear elliptic equations, Ricci tensor, 510, Mathematics - Analysis of PDEs, 515, Laplace-Beltrami operator, FOS: Mathematics, Nonlinear parabolic equations, nonlinear diffusion, semilinear elliptic equation, uniqueness, Heat and other parabolic equation methods for PDEs on manifolds, Analyse, carré du champ, rigidity, Poincaré inequality, Moser-Trudinger-Onofri inequality, Analysis of PDEs (math.AP)
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