Projective Dirichlet boundary condition with applications to a geometric problem

Abstract

Given a domain Ω ⊂ Rⁿ, let λ > 0 be an eigenvalue of the elliptic operator L:= $\sum _{i,j = 1}^n\frac{\partial }{{\partial {x_i}}}({a^{ij}}\frac{\partial }{{\partial {x_j}}})$ on Ω for Dirichlet condition. For a function f ∈ L ²(Ω), it is known that the linear resonance equation Lu + λu = f in Ω with Dirichlet boundary condition is not always solvable. We give a new boundary condition P _λ(u|∂Ω) = 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 λ which satisfies ‖u‖_2,2 ≤ C(‖f‖₂ + ‖g‖_2,2) under suitable regularity assumptions on ∂Ω and L, where C is a constant depends only on n, Ω, and L. More a priori estimates, such as W ^2,p-estimates and the C ^2,α-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.