Some results on second-order elliptic operators with polynomially growing coefficients in $L^p$-spaces

TL;DR

This paper investigates second-order elliptic operators with polynomially growing coefficients in $L^p$-spaces, establishing conditions for their minimal realizations to generate analytic semigroups using quadratic form methods.

Contribution

It provides new results on the generation of analytic semigroups by elliptic operators with polynomial growth coefficients in $L^p$-spaces, including conditions for minimal realizations.

Findings

Operators generate analytic $C_0$-semigroups for all $p eq 1, ext{and} eq ext{infinity}$

Conditions identified for the minimal realization to be the operator's closure

Quadratic form methods effectively used for analysis

Abstract

In this paper we study minimal realizations in $L^p(\mathbb{R}^N)$ of the second order elliptic operator \begin{equation*} { A_{b,c}} := (1+|x|^\alpha)\Delta + b|x|^{\alpha-2}x\cdot\nabla - c |x|^{\alpha-2} - |x|^{\beta} , \quad x \in \mathbb{R}^N, \end{equation*} where $N\geq3$, $\alpha\in[0,2)$, $\beta >0$, and $b, c$ are real numbers. We use quadratic form methods to prove that $\left(A_{b,c},C_c^\infty\left(\mathbb{R}^N\setminus \{0\}\right)\right)$ admits an extension that generates an analytic $C_0-$semigroup for all $p\in(1,\infty)$. Moreover, we give conditions on the coefficients under which this extension is precisely the closure of $\left(A_{b,c},C_c^\infty\left(\mathbb{R}^N\setminus \{0\}\right)\right)$.