A logarithmic Hardy inequality

Manuel del Pino (1) , Jean Dolbeault (2) , Stathis Filippas (3) , Achiles Tertikas (3)

1 DIM - Departamento de Ingeniería Matemática [Santiago]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Department of Applied Mathematics [Heraklion]

We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with super-quadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy. This inequality differs from standard logarithmic Sobolev inequalities in the sense that the measure is neither Lebesgue's measure nor a probability measure. All terms are scale invariant. After an Emden-Fowler transformation, the inequality can be rewritten as an optimal inequality of logarithmic Sobolev type on the cylinder. Explicit expressions of the sharp constant, as well as minimizers, are established in the radial case. However, when no symmetry is imposed, the sharp constants are not achieved among radial functions, in some range of the parameters.

Mots clés: Hardy inequality; Sobolev inequality; interpolation; logarithmic Sobolev inequality; Hardy-Sobolev inequalities; Caffarelli-Kohn-Nirenberg inequalities; scale invariance; Emden-Fowler transformation; radial symmetry; symmetry breaking

Domaines: Equations aux dérivées partielles [math.AP]

Manuel del Pino, Jean Dolbeault, Stathis Filippas, Achiles Tertikas. A logarithmic Hardy inequality. Journal of Functional Analysis, 2010, 259 (8), pp.2045-2072. ⟨10.1016/j.jfa.2010.06.005⟩. ⟨hal-00438199⟩