A variational theory of the Hessian equation

Abstract

By studying a negative gradient flow of certain Hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous Hessian equations. To achieve this we derive uniform, first-and second-order a priori estimates for the elliptic and parabolic Hessian equations. Our results generalize well-known results for semilinear elliptic equations and the Monge-Ampère equation.

The k-Yamabe problem

“…A Sobolev type inequality for k-admissible functions was proved by Wang [75], and a Moser-Trudinger type inequality was proved by Tian and Wang [61]. The corresponding variational solutions, based on gradient flow approach, were obtained by Chou and Wang [16] …”

“…But the pure interior second derivative estimate (3.2) does not hold for equation (1.10). When the solution vanishes on the boundary, a Pogorelov type estimate was proved in [16]. (iii) In addition to the convex cones Γ k and −Γ k , equations (1.10) and (3.1) may have other elliptic branches, depending on k and n. Namely there are (nonconvex) cones Γ ⊂ R n such that equation (1.10) (and also (3.1)) is elliptic if the corresponding eigenvalues λ(D 2 u) ∈ Γ.…”

“…More general gradient flows were later used in [75] to prove the Sobolev type inequality (4.20) below and in [16] to prove the existence of min-max critical points of the functional corresponding to the k-Hessian equation (1.10 Proof. We reduce the proof of the lemma for k ≥ 2 to the case k = 1, namely (1.8).…”

The complex Hessian equation with infinite Dirichlet boundary value

“…See, for example, [2,5,6,14,15,16]. The existence of the classical solution for the Dirichlet problem of the eigenvalues of the real Hessian was proved by Caffarelli, Nirenberg and Spruck in [2].…”

“…In fact, the second inequality in (4.5) follows directly from Remark 2.8. To show the first one, we use the strict convexity of Ω to find the smallest positive number a, such that for anyz ∈ ∂Ω, there is a ball Moreover, by Theorem 3.1 of [15] and the methods in [5], the convergence is uniform in every compact set K ⊂ Ω and u ∈ C 0,1 loc (Ω). By the stability theorem of viscosity solutions under uniform convergence, we see that u is a viscosity Γ k -subharmonic solution of (1.7) and (1.8) which satisfies (1.15) by (4.6).…”