Fourth-order operators with unbounded coefficients

Abstract

We prove that operators of the form $A=-a(x{)}^2{\Delta }^2$, with $|Da(x)|\le ca(x{)}^{\frac{1}{2}}$, generate analytic semigroups in $L^p({\mathbb{R}}^N)$ for $1<p\le \infty$ and in $C_b({\mathbb{R}}^N)$. In particular, we deduce generation results for the operator $A:=-(1+|x{|}^2{)}^{\alpha }{\Delta }^2$, $0\le \alpha \le 2$. Moreover, we characterize the maximal domain of such operators in $L^p({\mathbb{R}}^N)$ for $1<p<\infty$.