Maximal \(L^p\) regularity for elliptic equations with unbounded coefficients (Q1601668)

The authors deal with the following equation in \(\mathbb{R}^N\): \[ \lambda \varphi-N_0 \varphi=f, \quad \lambda> 0,\tag{1} \] where \(N_0\) is the elliptic operator defined by \[ N_0\varphi= \frac 12\Delta \varphi-\langle DU, D\varphi \rangle,\quad \varphi\in C_0^\infty (\mathbb{R}^N),\tag{2} \] and \(U:\mathbb{R}^N \to \mathbb{R}\) is of class \(C^3\) satisfying \[ \int_{\mathbb{R}^N} e^{-2U(y)}dy< +\infty. \tag{3} \] Moreover, they prove, under suitable assumptions, existence and uniqueness of a solution of (1). Let now \(p>1\) and denote by \(N_p\) the closure of \(N_0\) in \(L^p(\mathbb{R}^n, \nu)\). The main goal of the paper is to prove that \(N_p\) is \(m\)-dissipative and its domain of definition is given by \(D(N_p)=W^{2,p} (\mathbb{R}^N, \nu)\), where \(\nu\) is the Borel probability measure on \(\mathbb{R}^N\) defined as \(\nu(dx)= {e^{-2U(x)} dx\over \int_{\mathbb{R}^N} e^{-2U(y)}dy}\).