Fourth-order operators with unbounded coefficients

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