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Elliptic operators with unbounded diffusion, drift and potential terms (1705.08007v1)

Published 22 May 2017 in math.AP

Abstract: We prove that the realization $A_p$ in $Lp(\mathbb{R}N),\,1<p<\infty$, of the elliptic operator $A=(1+|x|{\alpha})\Delta+b|x|{\alpha-1}\frac{x}{|x|}\cdot \nabla-c|x|{\beta}$ with domain $D(A_p) ={ u \in W{2,p}(\mathbb{R}N)\, |\, Au \in Lp(\mathbb{R}N)}$ 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.

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