On the null-spaces of first-order elliptic partial differential operators in $R\textbf {^{n}}$

Abstract:

The objects to be studied are the null-spaces of linear first-order elliptic partial differential operators with domain ${H_1}({R^n};{C^k})$ in ${L_2}({R^n};{C^k})$, the first-order coefficients of which become constant and the zero-order coefficients of which vanish outside a compact set in ${R^n}$. An example is given of an operator of this type which has a nontrivial null-space. It is shown that the dimension of the null-space of such an operator is finite for any number n of independent variables, and that this dimension is an upper-semicontinuous function of the operator in a certain sense.