On the local solvability of elliptic equations on compact manifolds
The interesting paper under review deals with local solvability of nonlinear elliptic equations over compact manifolds. Precisely, the local image of a nonlinear elliptic operator over a compact manifold is a submanifold described by a full set of independent equations if and only if the corank of the linearized operator is constant. In case this is not verified, the author exhibits a higher order infinitesimal invariant, the epidimension, which forces the number of independent equations to decrease. It is shown that the epidimension of a natural operator with enough symmetry must either vanish or be maximal in which case the local image admits no equation. Moreover, it is shown that, in general, a local nonlinear version of the Fredholm scheme, which always exists, encodes the maximal number of independent equations. Finally, the underdetermined elliptic case is discussed and a conjecture for it is stated.
scientific article; zbMATH DE number 5368313
scientific article