Abstract
In this work, we study the existence of $C^{\infty}$ local solutions to $2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case that the right hand side function $f$ possibly vanishes, changes the sign, is positively or negatively defined. We also give the convexities of solutions which are related with the annulation or the sign of right-hand side function $f$. The associated linearized operator are uniformly elliptic.
Highlights
We are interested by the following k-Hessian equation
S k[u] = σk(λ(D2u)), k = 1, . . . , n, where λ(D2u) = (λ1, λ2, . . . , λn), λ j is the eigenvalue of the Hessian matrix (D2u), and σk(λ) =
We say that a function u is a local solution of (1.1) near y0 ∈ Ω, if there exists a neighborhood of y0, Vy0 ⊂ Ω such that u ∈ C2(Vy0) satisfies the equation (1.1) on Vy0
Summary
Elliptical k- Hessian equation, existence of local solutions, convexities. We say that a function u is a local solution of (1.1) near y0 ∈ Ω, if there exists a neighborhood of y0, Vy0 ⊂ Ω such that u ∈ C2(Vy0) satisfies the equation (1.1) on Vy0. We study the existence of C∞-local solution of the following 2-Hessian equation in R3, (1.3) The equation (1.3) is uniformly elliptic with respect to the above local solutions.
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