\(L ^{p }\)-theory for second-order elliptic operators with unbounded coefficients
The paper deals with second-order elliptic operators with unbounded coefficients of the form \[ Au := -{\text{div}}(a\nabla u) + F \cdot \nabla u + Vu\quad \text{in}\;L^{p}(\mathbb{R}^{N}),\;1 < p < \infty. \] A new criterion for \(m\)-accretivity and \(m\)-sectoriality of \(A\) in \({L^{p}(\mathbb{R}^{N})}\) is given in terms of certain identities which behave like a sesquilinear form over \(L ^{p } \times L ^{p'}\). The result naturally extends the \textit{T. Kato} criterion from [Spectral theory of differential operators, Proc. Conf., Birmingham/ USA 1981, North-Holland Math. Stud. 55, 253--266 (1981; Zbl 0484.35027)] for the nonnegative selfadjointness to the case of \(p \neq 2.\) The simplicity of the approach adopted is illustrated by the typical example \({Au = -u'' + x^{3}u' + c |x|^{\gamma}u}\) in \({L^p(\mathbb{R})}.\)
scientific article; zbMATH DE number 6136964
scientific article