Fourth-order operators with unbounded coefficients in $L^1$ spaces

Abstract

We prove that operators of the form $A=-a(x{)}^2\hat{I}{”}^2$, with suitable growth conditions on the coefficient $a(x)$, generate analytic semigroups in $L^1({\mathbb{R}}^N)$. In particular, we deduce generation results for the operator $A:=-(1+|x{|}^2{)}^{\hat{I}}\pm \hat{I}{”}^2$, $0\le \hat{I}\pm \le 2$. Moreover, we characterise the maximal domain of $A$ in $L^1({\mathbb{R}}^N)$.

arXiv admin note: text overlap with arXiv:2401.14187