Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces

Simona Fornaro; Federica  Gregorio; Abdelaziz Rhandi

doi:10.3934/cpaa.2016040

Article Contents

2016, Volume 15, Issue 6: 2357-2372. Doi: 10.3934/cpaa.2016040

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Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces

1.
Università degli Studi di Pavia, Dipartimento di Matematica “F. Casorati”, via Ferrata 1, 27100 Pavia
2.
Dipartimento di Fisica, Universitá degli Studi di Salerno, via Giovanni Paolo II, 132, 84084, Fisciano (Sa)
3.
Dipartimento di Ingegneria dell'Informazione e Matematica Applicata, Università degli Studi di Salerno, Via Ponte Don Melillo, 84084 Fisciano (Sa)

Received: February 29, 2016

Revised: May 31, 2016

Published: October 2016

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Abstract

In this paper we give sufficient conditions on $\alpha \ge 0$ and $c\in R$ ensuring that the space of test functions $C_c^\infty(R^N)$ is a core for the operator \begin{eqnarray} L_0u=(1+|x|^\alpha )\Delta u+\frac{c}{|x|^2}u=:Lu+\frac{c}{|x|^2}u, \end{eqnarray} and $L_0$ with a suitable domain generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p(R^N), 1 < p < \infty$. The proofs are based on some $L^p$-weighted Hardy's inequality and perturbation techniques.

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Mathematics Subject Classification: Primary: 47D06, 35P05, 35J70; Secondary: 35K65, 34G10.

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