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

TL;DR

This paper establishes conditions under which certain elliptic operators with unbounded diffusion and inverse square potentials generate well-behaved semigroups in $L^p$ spaces, using Hardy inequalities and perturbation methods.

Contribution

It provides new criteria for the generation of $C_0$-semigroups by elliptic operators with unbounded coefficients and inverse square potentials in $L^p$ spaces.

Findings

Identifies conditions on parameters for the operator to generate a positive $C_0$-semigroup.

Uses $L^p$-weighted Hardy inequalities to establish core properties.

Demonstrates the applicability of perturbation techniques in this setting.

Abstract

In this paper we give sufficient conditions on $\alpha \geq 0$ and $c\in \mathbb{R}$ ensuring that the space of test functions $C_c^\infty(\mathbb{R}^N)$ is a core for the operator $$L_0u=(1+|x|^\alpha )\Delta u+\frac{c}{|x|^2}u=:Lu+\frac{c}{|x|^2}u,$$ and $L_0$ with suitable domain generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p(\mathbb{R}^N),\,1<p<\infty$. The proofs are based on some $L^p$-weighted Hardy's inequality and perturbation techniques.