A Sobolev-like inequality for the Dirac operator

Abstract

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We finally specify a case where this equation can be solved.Comment: some typos corrected, the introduction has been rewritten and several references has been added, to appear in Journal of Functional Analysi

Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds

“…The first characterization is used in [3,4], while the second one is used, for example, in [5,6,28]. Writting ψ = D g S m ϕ for ϕ ∈ C ∞ (S m , S(S m )) and noting that ψ = D g S m ϕ = 0 if and only if ϕ = 0 (this is because (S m , g S m ) has positive scalar curvature and ker D g S m = {0} via the Lichnerowicz formula), and the density of C ∞ (S m , S(S m )) in W 1, 2m m+1 (S m , S(S m )), we have the equality in (4.6).…”

“…Ammann's result is recovered from our result: Ammann's solution corresponds to the mountain pass critical point of the dual action and such a critical point exists if the condition of Ammann is satisfied. See also [28], where Raulot proved the existence of a solution to the equation Dψ = H (x)|ψ| 2 m−1 ψ when D is invertible and a certain condition is satisfied for H . Both of the proofs of Ammann [4] and Raulot [28] rely on a subcritical approximation argument which is similar to Yamabe [33] and Aubin [8].…”

Rabinowitz–Floer homology for superquadratic Dirac equations on compact spin manifolds

“…In fact it coincides exactly with the Yamabe constant in that case. Now we can easily prove the following result (as in [24])…”

The boundary value problem for Dirac-harmonic maps

“…Some pertinent references are [3,4,5,9,23]. In our setting, for the spinor ψ we shall need a chirality boundary condition (first introduced by Gibbons-Hawking-Horowitz-Perry [10]).…”