# Elliptic operators with unbounded diffusion, drift and potential terms

## TL;DR

This paper proves that certain elliptic operators with unbounded coefficients generate analytic semigroups in L^p spaces, extending previous results and establishing properties like compactness and ultracontractivity.

## Contribution

It generalizes existing results by showing that elliptic operators with unbounded diffusion, drift, and potential terms generate strongly continuous analytic semigroups under specific conditions.

## Key findings

- Generates analytic semigroup in L^p for specified parameters
- Establishes the semigroup's consistency and immediate compactness
- Proves ultracontractivity of the semigroup

## 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.