Local solvability of the k-Hessian equations

Abstract

Abstract. In this work, we study the existence of local solutions in R n to k-Hessian equation, for which the nonhomogeneous term f is permitted to change the sign or be non negative; if f is C ∞ , so is the local solution. We also give a classification for the second order polynomial solutions to the k−Hessian equation, it is the basis to construct the local solutions and obtain the uniform ellipticity of the linearized operators at such constructed local solutions.

Boundary pointwise C1, and C2, regularity for fully nonlinear elliptic equations

“…In this direction, Urbas [124,Theorem 3] proved global C 1 estimate. Tian, Wang and Wang [109] studied the local solvability of the k-Hessian equation in B r 0 for r 0 small. Maybe Theorem 4.1 can be applied to study the local solvability.…”

Existence and convexity of local solutions to degenerate Hessian equations

“…Remark that, in [20,21], we choose τ ∈ ∂Γ k (n) with σ k+1 (τ) < 0, so ϕ is not (k + 1)convex; also the solutions constructed in [3] must not be (k + 1)-convex, because in these cases every linearized operator (2.6) is uniformly elliptic. But in present work, we want to construct the local strictly convex solution, so we can't make that choice.…”

“…We also need to prove that the perturbation doesn't destruct the strictly convexity of ψ constructed by (2.7). Remark that the linearized equation is degenerate elliptic, so that there is a loss of the regularity for the à priori estimate of solution ρ m , but the coefficients of linearized operators depends on D 2 w m , so that we need to smoothing the solution ρ m to continue the iteration (2.8) for m ∈ N. This is quite different from the procedure of iteration used in [21] where the linearized equation is uniformly elliptic.…”

“…Which, in case v ∈ C 2 (E), is equivalent to t))z, this shows that the positive definiteness of Hessian matrix (D 2 v) is a sufficient condition for the strict convexity, but not necessary. However if u ∈ C 2 is a k-convex solution of S k [u] = f (y) ≥ 0 with 2 ≤ k < n and f (y 0 ) = 0, then S k+1 [u](y 0 ) > 0 will never occur, so that there are two possibilities: 1) S k+1 [u](y 0 ) < 0, in this case, u is not (k + 1)-convex; 2) S k+1 [u](y 0 ) = 0, in this case, it is shown in Theorem 1.1 of [21] that, if S k [u](y 0 ) = S k+1 [u](y 0 ) = 0, then S l [u](y 0 ) = 0 for k ≤ l ≤ n. In particular, if f ≡ 0, since the case S k+1 [u] > 0 will never occur, then either S k+1 [u] < 0 in some open subset of Ω or S k+1 [u] ≡ 0 in Ω itself, in the former case u is not (k + 1)−convex and let alone strictly convex; in the latter case, S l [u] ≡ 0 for k ≤ l ≤ n, then the graph (y; u(y)) for k = 2 has the vanishing sectional curvature and must be a cylinder or a plane, see [19] and [22], meanwhile the graph (y; u(y)) for k > 2, by Lemma 3.1 of [10], is a surface of constant nullity (at least) n − k + 1 and then is a (n − k + 1)-ruled surface. Therefore if f ≡ 0, the solution u to (1.1) is not strictly convex (at least) along the rulings.…”

On the local properties of the solutions to $\boldsymbol{k}$-Hessian equations

“…在文献 [5,6] 中, 我们选取 τ ∈ ∂Γ k (n) 满足 σ k+1 (τ ) < 0, 在此条件下 φ 不是 (k + 1)-凸. 同时, 文 献 [10] 中构造的解也不可能是 (k + 1)-凸的, 这是因为在某种意义下 "非 (k + 1)-凸" 与 "线性化算子 的一致椭圆型" 是等价的.…”