Hessian Equations with Elementary Symmetric Functions

Abstract

Abstract. We consider the Dirichlet problem for two types of degenerate elliptic Hessian equations . New results about solvability of the equations in the C 1,1 space are provided.

A class of curvature type equations

“…There is an interesting PDE problem in finding the optimal power p ℓ in the requirement of α 1/p ℓ ℓ ∈ C 1,1 to obtain the above C 1,1 estimate for the degenerate fully nonlinear equations. For Krylov's equation (1.4), C 1,1 estimates were proved with the assumption that p l ≥ k − l + 1 for 0 ≤ l ≤ k − 1, and this was weakened to be p l ≥ k − l by Dong [13] and hence partially confirmed a question proposed by Krylov in [32]. Here, in the above theorem, no assumption is required on the coefficient of…”

Representation and regularity for the Dirichlet problem for real and complex degenerate Hessian equations

“…The following two lemmas has been mostly established in [Don06] when the hyperbolic polynomial H m is the m-th elementary symmetric polynomial of the usual eigenvalues of the matrix. See Lemmas 4.3 and 4.5 in [Don06]. We extend them to any homogeneous hyperbolic polynomial.…”

“…Wang provided a new and shorter proof of the boundary second derivative estimate obtained in [Kry95b], which resulted in the global C 1,1regularity for degenerate real Hessian equations under general settings. In [Don06], under weaker regularity assumptions on f , H. Dong obtained the global C 1,1 -regularity result for degenerate real Hessian equations described by elementary symmetric functions of eigenvalues of u xx .…”

Unilateral Global Interval Bifurcation for the Hessian Equation and Its Applications

“…Here, e k-Hessian equations are a type of important fully nonlinear equations. Historically, there are a large amount of papers in the literature on the existence, regularity, and the qualitative properties of solutions for the k-Hessian equation; see [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references therein. Wang [14] has proved the existence of the first eigenvalue λ 1 of problem (1) with λa(x) � λ k and g � 0.…”