Smooth approximation of selfadjoint differential operators of divergence form with bounded measurable coefficients

Yu. B. Orochko

Sbornik. Mathematics, Tome 37 (1980) no. 3, pp. 389-401

Résumé

Suppose that $S=-\sum_{j,k=1}^m\frac\partial{\partial x_j}a_{jk}(x)\frac\partial{\partial x_k}+1$ is a uniformly elliptic expression in $\mathbf R^m$, $m\geqslant1$, with real measurable coefficients, and $A$ is the selfadjoint operator associated with the sesquilinear form $a[f,g]$ in $L_2(\mathbf R^m)$ constructed from $S$; $a[f,g]$ is the limit of a sequence $a_n[f,g]$ ($n=1,2,\dots$) of analogous forms constructed from expressions of the type $S$, but with smooth coefficients. For forms in abstract Hilbert space we prove theorems that imply the strong convergence of $\Phi(A_n)$ ($A_n$ is the operator associated with $a_n$, and $\Phi(\lambda)$ is a bounded continuous function on the half-line $\lambda\geqslant0$) to $\Phi(A)$ as $n\to\infty$. Applications to the spectral theory of the operator $A$ are given. Bibliography: 14 titles.