Scale invariant elliptic operators with singular coefficients

TL;DR

This paper characterizes when a class of scale-invariant elliptic operators with singular coefficients generate semigroups in L^p spaces, providing explicit conditions on parameters and describing the domain of the generator.

Contribution

It offers a complete characterization of the generation of semigroups by these operators with singular coefficients, including domain description and parameter conditions.

Findings

Semigroup generation conditions depend on parameters D_c and roots s_i.

Explicit domain characterization of the generator in L^p.

Conditions involve inequalities relating N/p, roots s_i, and parameters.

Abstract

We show that a realization of the operator $L=|x|^\alpha\Delta +c|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -b|x|^{\alpha-2}$ generates a semigroup in $L^p(\mathbb {R}^N)$ if and only if $D_c=b+(N-2+c)^2/4 > 0$ and $s_1+\min\{0,2-\alpha\}<N/p<s_2+\max\{0,2-\alpha\}$, where $s_i$ are the roots of the equation $b+s(N-2+c-s)=0$, or $D_c=0$ and $s_0+\min\{0,2-\alpha\} \le N/p \le s_0+\max\{0,2-\alpha\}$, where $s_0$ is the unique root of the above equation. The domain of the generator is also characterized.