Essential self-adjointness of second-order elliptic operator with measurable coefficients
Let us consider the matrix \((a_{kj}(x))\), \(k,j=1,...,\ell\), the elements of which are real measurable functions in \({\mathbb{R}}^{\ell}\) satisfying \[ a_{kj}(x)=a_{jk}(x),\quad \sum_{j}\xi^ 2_ j=\sum_{k,j}a_{kj}(x)\xi_ k\xi_ j, \] for all \(\xi \in {\mathbb{R}}^{\ell}\) and almost all \(x\in {\mathbb{R}}^{\ell}\). There is proved the essential self-adjointness of \[ L=-\sum_{k,j}(\partial /\partial x_ k)a_{kj}(x)\partial /\partial x_ j+V,\quad V\in L^ 2,\quad V\geq 0. \] This is the first result in this direction if \(\ell \geq 4\) and the coefficients \(a_{kj}\) are locally unbounded functions.
scientific article; zbMATH DE number 3962265