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

Abstract

In this paper we give sufficient conditions on α ≥ 0 and c ∈ R ensuring that the space of test functions C ∞ c (R N ) is a core for the operatorand L 0 with a suitable domain generates a quasi-contractive and positivity preserving C 0 -semigroup in L p (R N ), 1 < p < ∞. The proofs are based on some L p -weighted Hardy's inequality and perturbation techniques.2000 Mathematics Subject Classification. 35P05, 35J70, 35K65. Key words and phrases. Inverse square potential, positivity preserving C 0 -semigroup, core, dissipative and dispersive operator, Hardy's inequality, unbounded diffusion.The 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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