We establish for $2 \le k \le n-1$ the strict concavity of the function $f_k(λ)=\log(σ_k(λ))$ on a subset of the positive cone $Γ_n=\{λ=(λ_{1}, λ_{2}, \cdots,λ_{n})\in \mathbb{R}^n; λ_j>0,j=1,\cdots, n\}$ where $σ_{k}(λ)$ is the basic symmetric polynomial of degree $k,$ $2 \leq k \leq n.$ Then we apply the result to study the so-called $d-$concavity of the $k-$Hessian type function $F_{k}(R)=\log \left(S_{k}(R)\right),$ where $S_{k}(R)=σ_{k}(λ(R)), λ(R)= \left(λ_{1}, λ_{2}, \cdots, λ_{n}\right) \in \mathbb{C}^{n}$ is eigenvalue-vector of $R \in \mathbb{R}^{n \times n},$ $R=ω+β, ω^{T}=ω, ω>0, \quad β^{T}=-β.$ The $d-$concavity will be used in our next paper to study the existence of admissible solutions to the Dirichlet problem for the augmented nonsymmetric $k-$Hessian type equations.
We introduce the so-called $d$-concavity, $d \geq 0,$ and prove that the nonsymmetric Monge-Ampère type function of matrix variable is concave in an appropriate unbounded and convex set. We prove also the comparison principle for nonsymmetric Monge-Ampère type equations in the case when they are so-called $δ$-elliptic with respect to compared functions with $0 \leq δ<1$.
Using the established $d$-concavity of the $k$-Hessian type functions $F_k(R)=\log(S_k(R)),$ whose variables are nonsymmetric matrices, we prove $ C^{2, α}(\overlineΩ) $ estimates for strictly $(δ, \widetildeγ_k) $-admissible solutions to the Dirichlet problem without the well-known regularity condition. A necessary condition for the existence of strictly $δ$-admissible solutions to the equations is given. By the method of continuity, we provide some sufficient conditions for the unique solvability in the class of strictly $(δ,\widetildeγ_k)$-admissible solutions to the Dirichlet problem, prov…
The main result of this paper gives a plenary proof on the curvature estimates for $k$ curvature equations with general right hand sides with $n<2k$ based on a concavity inequality. We further give a explicit lower bound of the inequality.
In this paper, we establish a curvature estimate for semi-convex solutions of Hessian equations in hyperbolic space. We also obtain a curvature estimate for admissible solutions to prescribed curvature measure type problem in hyperbolic space. A crucial ingredient in both estimates is a concavity inequality for Hessian operator.
For a sufficiently regular open bounded set $D \subset R^2$ let us consider the equation $(-Δ)^{1/2} φ(x) = 1$, $x \in D$ with the Dirichlet exterior condition $φ(x) = 0$, $x \in D^c$. $φ$ is the expected value of the first exit time from $D$ of the Cauchy process in $R^2$. We prove that if $D \subset R^2$ is a convex bounded domain then $φ$ is concave on $D$. To show it we study the Hessian matrix of the harmonic extension of $φ$. The key idea of the proof is based on a deep result of Hans Lewy concerning determinants of Hessian matrices of harmonic functions.
In this paper, we continue our investigations into the global theory of oblique boundary value problems for augmented Hessian equations. We construct a global barrier function in terms of an admissible function in a uniform way when the matrix function in the augmented Hessian is only assumed regular. This enables us to derive global second derivative estimates in terms of boundary estimates which are then obtained by strengthening the concavity or monotonicity conditions in our previous work on the strictly regular case. Finally we give some applications to existence theorems which embrace st…
We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form $$f(λ(D^2u))=g(x),$$ with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli--Nirenberg--Spruck \cite{Caffarelli1985}, Trudinger \cite{Trudinger1995} and many others, and there had been significant discussions on the solvability of the classical Dirichlet problem via the continuity method, under the assumption that $f$ is a concave function. In this paper, based on the Perron's method, we establish an exterior existence and uniquen…
A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains
University of Science and Technology of China（中国科学技术大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Università degli Studi di Firenze（佛罗伦萨大学）
In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $γ$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of \textbf{symmetric} convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex $K$ and $L,$ $$ γ(λK+(1-λ)L)^{\frac{1}{2n}}\geq λγ(K)^{\frac{1}{2n}}+(1-λ)γ(L)^{\frac{1}{2n}}. $$ Further, we show that under suitable dimension-free uniform bounds on the Hessian of the potential, the log-concavity of even measures can be strengthened to $p$-concavity, with $p>…
