Carleman condition for a degenerate second-order differential operator
In \(R^ N\), \(N\geq 2\) a degenerate elliptic operator \[ Lu(x)=- \sum^{N}_{i=1}(\partial /\partial x_ i)(\alpha^ 2(x)\partial u(x)/\partial x_ i)+\beta (x)u(x) \] is investigated where \(\alpha (x)\in C^{\infty}(R^ N),\quad \beta (x)=\alpha^ 2(x)-\alpha (x)\Delta \alpha (x)+q(x),\) \(| \alpha (x)| \geq 0\), \(q(x)\in L_ 2(R^ N).\) It is proved that if degeneration is not very strong, namely, \(\alpha^{-1}(x)\in L_ 2(R^ N)\), then the minimal operator generated by the given elliptic expression is of Carleman type. The estimates are obtained for the kernel of the inverse which is an integral of the minimal operator.
scientific article; zbMATH DE number 4091987
scientific article