Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials

Anna Canale; Francesco Pappalardo; Ciro Tarantino

doi:10.3934/cpaa.2020274

Article Contents

2021, Volume 20, Issue 1: 405-425. Doi: 10.3934/cpaa.2020274

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Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials

1.
Dipartimento di Ingegneria dell'Informazione ed Elettrica e Matematica Applicata, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, SA, Italy
2.
Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli Federico Ⅱ, Complesso Universitario Monte S. Angelo, Via Cintia, 80126 Naples, Italy
3.
Dipartimento di Scienze Economiche e Statistiche, Università degli Studi di Napoli Federico Ⅱ, Complesso Universitario Monte S. Angelo, Via Cintia, 80126 Naples, Italy

^* Corresponding author
^* Corresponding author

Received: February 29, 2020

Revised: August 31, 2020

Early access: November 2020

Published: January 2021

The first two authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

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Abstract

The main results in the paper are the weighted multipolar Hardy inequalities

$ \begin{equation*} c\int_{\mathbb{R}^N}\sum\limits_{i = 1}^n\frac{\varphi^2}{|x-a_i|^2}\,\mu(x)dx \leq\int_{\mathbb{R}^N}|\nabla \varphi |^2\mu(x)dx+ K\int_{\mathbb{R}^N} \varphi^2\mu(x)dx, \end{equation*} $

in $ \mathbb{R}^N $ for any $ \varphi $ in a suitable weighted Sobolev space, with $ 0<c\le c_{o,\mu} $, $ a_1,\dots,a_n\in \mathbb{R}^N $, $ K $ constant. The weight functions $ \mu $ are of a quite general type.

The paper fits in the framework of Kolmogorov operators defined on smooth functions

$ \begin{equation*} Lu = \Delta u+\frac{\nabla \mu}{\mu}\cdot\nabla u, \end{equation*} $

perturbed by multipolar inverse square potentials, and related evolution problems. Necessary and sufficient conditions for the existence of exponentially bounded in time positive solutions to the associated initial value problem are based on weighted Hardy inequalities. For constants $ c $ beyond the optimal Hardy constant $ c_{o,\mu} $ we are able to show nonexistence of positive solutions.

Keywords:

Mathematics Subject Classification: Primary: 35K15, 35K65, 35B25; Secondary: 34G10, 47D03.

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