Given a domain Omega subset of R-n, let lambda > 0 be an eigenvalue of the elliptic operator L := Sigma(n)(i,j=1) partial derivative/partial derivative x(i) (a(ij) partial derivative/partial derivative x(j)) on Omega for Dirichlet condition. For a function f is an element of L-2 (Omega), it is known that the linear resonance equation Lu + lambda u - f in Omega with Dirichlet boundary condition is not always solvable. We give a new boundary condition P-lambda(u vertical bar partial derivative Omega) = g, called to be projective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to lambda which satisfies parallel to u parallel to(2,2) <= C(parallel to f parallel to(2) + parallel to g parallel to(2,2)) under suitable regularity assumptions on partial derivative Omega and L, where C is a constant depends only on n, Omega, and L. More a priori estimates, such as W-2,W-p-estimates and the C-2,C-alpha-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean (Cartan) torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.

## Edit Comment