Elliptic operators with unbounded diffusion, drift and potential terms

Abstract

We prove that the realization $A_p$ in $L^p({\mathbb{R}}^N),1<p<\infty$, of the elliptic operator $A=(1+|x{|}^{\alpha })\Delta +b|x{|}^{\alpha -1}\frac{x}{|x|}\cdot \nabla -c|x{|}^{\beta }$ with domain $D(A_p)=\{u\in W^{2,p}({\mathbb{R}}^N)|Au\in L^p({\mathbb{R}}^N)\}$ generates a strongly continuous analytic semigroup $T(\cdot )$ provided that $\alpha >2,\beta >\alpha -2$ and any constants $b\in \mathbb{R}$ and $c>0$. This generalizes the recent results in [A.Canale, A. Rhandi, C. Tacelli, Ann. Sc. Norm. Super. Pisa CI. Sci. (5), 2016] and in [G.Metafune, C.Spina, C.Tacelli, Adv. Diff. Equat., 2014]. Moreover we show that $T(\cdot )$ is consistent, immediately compact and ultracontractive.

References in corpus (1)

• Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials