[Paper Review] Convexity of solutions and Brunn-Minkowski inequalities for Hessian equations in $\\R^3$

Paolo Salani|arXiv (Cornell University)|Jan 3, 2011

Geometric Analysis and Curvature Flows6 citations

TL;DR

This paper establishes new convexity properties of solutions to Hessian equations in $\mathbb{R}^3$ using Minkowski addition of convex functions, proving that $v = -\log(-u)$ is convex for solutions to $S_2(D^2u) = \Lambda_2(\Omega)(-u)^2$ and $v = -\sqrt{-u}$ is convex for $S_2(D^2u) = 1$. It further proves a Brunn-Minkowski inequality for the $2$-torsional rigidity functional $\tau_2(\Omega)$, extending classical inequalities to fully nonlinear Hessian operators in three dimensions.

ABSTRACT

By using Minkowski addition of convex functions, we prove convexity and rearrangement properties of solutions to some Hessian equations in $\\R^3$ and Brunn-Minkowski and isoperimetric inequalities for related functionals.

Motivation & Objective

• To establish convexity properties of solutions to Hessian equations $S_k(D^2u) = f$ in $\mathbb{R}^3$ for $k=2$, particularly for Dirichlet problems with power-type nonlinearities.
• To extend Brunn-Minkowski and isoperimetric inequalities to the $k$-torsional rigidity functional $\tau_k(\Omega)$ in three dimensions.
• To provide new proofs of existing convexity results using Minkowski addition of convex functions, offering a macroscopic alternative to the constant rank technique.
• To explore the possibility of a Minkowski problem for the $2$-torsional rigidity functional $\tau_2(\Omega)$ via representation formulas and variational methods.

Proposed method

• Uses Minkowski addition of convex functions to analyze the convexity of transformed solutions $v = -\log(-u)$ and $v = -\sqrt{-u}$.
• Applies variational techniques to define and study the $k$-torsional rigidity functional $\tau_k(\Omega)$ via infimum over $k$-convex functions vanishing on $\partial\Omega$.
• Derives a representation formula for $\tau_2(\Omega)$ in terms of the support function and $(n-k)$-area measure on the unit sphere, linking it to the solution's gradient on the boundary.
• Establishes a Brunn-Minkowski inequality for $\tau_2(\Omega)$ by leveraging the convexity of transformed solutions and Minkowski's mixed volume theory.
• Generalizes results to a broader class of nonlinearities $S_2(D^2u) = \lambda (-u)^p$ for $p \in (0,2)$, proving convexity of $v = -(-u)^{(2-p)/4}$.
• Uses the structure of $\mathcal{C}_1^+$ domains (convex sets with $C^{3,1}$ boundary and positive mean curvature) as the natural class for existence and regularity of solutions.

Experimental results

Research questions

• RQ1 Does the solution $u$ to $S_2(D^2u) = \Lambda_2(\Omega)(-u)^2$ in a $\mathcal{C}_1^+$ domain in $\mathbb{R}^3$ yield a convex transformation $v = -\log(-u)$?
• RQ2 Is the solution $u$ to $S_2(D^2u) = 1$ in a $\mathcal{C}_1^+$ domain in $\mathbb{R}^3$ such that $v = -\sqrt{-u}$ is convex?
• RQ3 Can a Brunn-Minkowski inequality be established for the $2$-torsional rigidity functional $\tau_2(\Omega)$ in $\mathbb{R}^3$?
• RQ4 Does the Rogers-Shephard inequality hold for $\tau_2(D\Omega)$, i.e., is $\tau_2(D\Omega) \leq C \tau_2(\Omega)$ for some universal constant $C$?
• RQ5 Can a Minkowski problem be formulated for the $2$-torsional rigidity functional $\tau_2(\Omega)$ using the derived representation formula?

Key findings

• The function $v = -\log(-u)$ is convex for the solution $u$ of $S_2(D^2u) = \Lambda_2(\Omega)(-u)^2$ in a $\mathcal{C}_1^+$ domain $\Omega \subset \mathbb{R}^3$.
• The function $v = -\sqrt{-u}$ is convex for the solution $u$ of $S_2(D^2u) = 1$ in a $\mathcal{C}_1^+$ domain $\Omega \subset \mathbb{R}^3$.
• For $p \in (0,2)$, the function $v = -(-u)^{(2-p)/4}$ is convex for solutions to $S_2(D^2u) = \lambda (-u)^p$ in $\mathcal{C}_1^+$ domains in $\mathbb{R}^3$.
• A Brunn-Minkowski inequality holds for the $2$-torsional rigidity functional $\tau_2(\Omega)$, i.e., $\tau_2(\Omega_0 + \Omega_1)^{1/10} \geq \tau_2(\Omega_0)^{1/10} + \tau_2(\Omega_1)^{1/10}$ for $\Omega_0, \Omega_1 \in \mathcal{C}_1^+$.
• A representation formula for $\tau_2(\Omega)$ is derived: $\tau_2(\Omega) = \frac{1}{10} \int_{S^2} h_\Omega(X) |Du(\nu_\Omega^{-1}(X))|^3 d\sigma^\Omega_1(X)$, linking it to boundary data and curvature measures.