Local solvability for a class of evolution equations (Q971819)

The authors consider a second-order partial differential operator \[ Q=- \sum^n_{j,k=1} \partial_{x_j}(\phi(x)^d a_{j, k}(x)\partial_{x_k})- \sum^n_{j=1} b_j(x)\partial_{x_j}+ c(x) \] in an open set \(Q\) of the origin in \(\mathbb{R}^n\), where \(d\in \mathbb{N}\) and the coefficients satisfy \(\phi(0)= 0\) and \(d\phi\neq 0\) on \(\phi^{-1}(0)\). \[ A(x)(\xi)= \sum a_{jk}(x)\xi_j \xi_k\geq 0 \] for \(x\in \Omega\), \(A(0)\) has rank \(n-1\), \(A(d\phi)= 0\) for all \(x\in \Omega\) and \(\theta= \sum b_j\partial_{x_j}\phi(0)\neq 0\) Then they prove that if \(d\) is even then \(Q\) is locally solvable near the origin and if \(d\) is odd then is locally solvable near the origin if and only if \(\theta< 0\). Furthermore, they prove that either \(d\) is even or if \(d\) is odd and \(\theta< 0\) then the solvability of \(Q\) occurs in the \(L^2\) sense.