A logarithmic Hardy inequality |
| |
Authors: | Manuel del Pino Stathis Filippas Achilles Tertikas |
| |
Institution: | a Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile b CEREMADE, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France c Department of Applied Mathematics, University of Crete, Knossos Avenue, 714 09 Heraklion, Greece d Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion, Greece e Department of Mathematics, University of Crete, Knossos Avenue, 714 09 Heraklion, Greece |
| |
Abstract: | 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 by radial functions, in some range of the parameters. |
| |
Keywords: | Hardy inequality Sobolev inequality Interpolation Logarithmic Sobolev inequality Hardy-Sobolev inequalities Caffarelli-Kohn-Nirenberg inequalities Scale invariance Emden-Fowler transformation Radial symmetry Symmetry breaking |
本文献已被 ScienceDirect 等数据库收录! |
|