Elliptic operators with unbounded diffusion and drift coefficients in $L^p$ spaces

Abstract

We prove that, for $\alpha > 2$, $\frac{N}{N-2+c} < p <\infty$, the operator $Lu=(1+|x|^\alpha)\Delta u+c|x|^{\alpha-1}\tfrac{x}{|x|}\cdot\nabla$ generates an analytic semigroup in $L^p$, which is contractive if and only if $p \ge \frac{N+\alpha-2}{N-2+c}$. Moreover, for $\alpha <\frac{N}{p'}+c$, we provide an explicit description of the domain. Spectral properties of the operator $L$ and kernel estimates are also obtained.