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

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    * Corresponding author 

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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  • 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.

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


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