Anatomy of the shape Hessian via lie brackets

The goal of this paper is to study the anatomy of the shape Hessian for some classes of smooth shape functionals. A structure theorem gives a decomposition of the shape Hessian in three additive bilinear forms acting on the two fields: the first one acting on the normal components at the boundary, the second one being symmetrical and the third one involving a half of the Lie bracket of the pair of fields at which the shape Hessian is computed. Applications to the commutation of the mixed derivatives and the symmetry of the linear operator which appears in the structure theorem are given.     

On Computation of the Shape Hessian of the Cost Functional Without Shape Sensitivity of the State Variable

A framework for calculating the shape Hessian for the domain optimization problem, with a partial differential equation as the constraint, is presented. First and second order approximations of the cost with respect to geometry perturbations are arranged in an efficient manner that allows the computation of the shape derivative and Hessian of the cost without the necessity to involve the shape derivative of the state variable. In doing so, the state and adjoint variables are only required to be Hölder continuous with respect to geometry perturbations.     

Positivity of the Shape Hessian and Instability of some Equilibrium Shapes

We study the positivity of the second shape derivative around an equilibrium for a 2-dimensional functional involving the perimeter of the shape and its the Dirichlet energy under volume constraint. We prove that, generally, convex equilibria lead to strictly positive second derivatives. We also exhibit some examples where strict positivity of the second order derivative holds at an equilibrium while existence of a minimum does not.     

Preconditioning the Pressure Tracking in Fluid Dynamics by Shape Hessian Information

Potential flow pressure matching is a classical inverse design aerodynamic problem. The resulting loss of regularity during the optimization poses challenges for shape optimization with normal perturbation of the surface mesh nodes. Smoothness is not enforced by the parameterization but by a proper choice of the scalar product based on the shape Hessian, which is derived in local coordinates for starshaped domains. Significant parts of the Hessian are identified and combined with an aerodynamic panel solver. The resulting shape Hessian preconditioner is shown to lead to superior convergence properties of the resulting optimization method. Additionally, preconditioning gives the potential for level independent convergence.     

Shape Hessian for generalized Oseen flow by differentiability of a minimax: A Lagrangian approach

The goal of this paper is to compute the shape Hessian for a generalized Oseen problem with nonhomogeneous Dirichlet boundary condition by the velocity method. The incompressibility will be treated by penalty approach. The structure of the shape gradient and shape Hessian with respect to the shape of the variable domain for a given cost functional are established by an application of the Lagrangian method with function space embedding technique. This work was supported by the National Natural Science Fund of China (No. 10371096) for ZM Gao and YC Ma.     

Hessian estimates for convex solutions to quadratic Hessian equation

We derive Hessian estimates for convex solutions to quadratic Hessian equation by compactness argument.     

Hessian Measures III

In this paper, we continue previous investigations into the theory of Hessian measures. We extend our weak continuity result to the case of mixed k-Hessian measures associated with k-tuples of k-convex functions, on domains in Euclidean n-space, k=1,2,…,n. Applications are given to capacity, quasicontinuity, and the Dirichlet problem, with inhomogeneous terms, continuous with respect to capacity or combinations of Dirac measures.     

Curvature of Hessian manifolds

We prove that, in dimensions greater than 2, the generic metric is not a Hessian metric and find a curvature condition on Hessian metrics in dimensions greater than 3. In particular we prove that the forms used to define the Pontryagin classes in terms of the curvature vanish on a Hessian manifold. By contrast all analytic Riemannian 2-metrics are Hessian metrics.     

Adjoint-based exact Hessian computation

BIT Numerical Mathematics - We consider a scalar function depending on a numerical solution of an initial value problem, and its second-derivative (Hessian) matrix for the initial value. The need...     

A variational theory of the Hessian equation

By studying a negative gradient flow of certain Hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous Hessian equations. To achieve this we derive uniform, first‐ and second‐order a priori estimates for the elliptic and parabolic Hessian equations. Our results generalize well‐known results for semilinear elliptic equations and the Monge‐Ampère equation. © 2001 John Wiley & Sons, Inc.     

Moser-Trudinger type inequalities for the Hessian equation

The k-Hessian equation for k?2 is a class of fully nonlinear partial differential equation of divergence form. A Sobolev type inequality for the k-Hessian equation was proved by the second author in 1994. In this paper, we prove the Moser-Trudinger type inequality for the k-Hessian equation.     

Integral Inequalities in the Theory of Hessian Operators

Mathematical Notes - The paper discusses the influence of new geometric invariants of domains on Hessian integral inequalities and provides a new proof of the well-known Trudinger–Wang...     

Stability of the equator map for the Hessian energy

In this paper we show that the equator map is a minimizer of the Hessian energy in for and is unstable for

     

Upper bounds for the eigenvalues of Hessian equations

In this paper, we prove some upper bounds for the Dirichlet eigenvalues of a class of fully nonlinear elliptic equations, namely the Hessian equations.     

Hessian Potentials with Parallel Derivatives

Let ${U \subset \mathbb A^n}$ be an open subset of real affine space. We consider functions ${F: U \to \mathbb R}$ with non-degenerate Hessian such that the first or the third derivative of F is parallel with respect to the Levi-Civita connection defined by the Hessian metric ${F{^\prime{^\prime}}}$ . In the former case the solutions are given precisely by the logarithmically homogeneous functions, while the latter case is closely linked to metrised Jordan algebras. Both conditions together are related to unital metrised Jordan algebras. Both conditions combined with convexity provide a local characterization of canonical barriers on symmetric cones.     

Multi-valued solutions to Hessian equations

In this paper, we use the Perron method to prove the existence of bounded multi-valued viscosity solutions to Hessian equations and interior Lipschitz continuity of the multi-valued solutions.     

Hessian orders and multinormal distributions

Several well known integral stochastic orders (like the convex order, the supermodular order, etc.) can be defined in terms of the Hessian matrix of a class of functions. Here we consider a generic Hessian order, i.e., an integral stochastic order defined through a convex cone of Hessian matrices, and we prove that if two random vectors are ordered by the Hessian order, then their means are equal and the difference of their covariance matrices belongs to the dual of . Then we show that the same conditions are also sufficient for multinormal random vectors. We study several particular cases of this general result.     

Hessian nilpotent polynomials and the Jacobian conjecture

Let and let be the Laplace operator. The main goal of the paper is to show that the well-known Jacobian conjecture without any additional conditions is equivalent to what we call the vanishing conjecture: for any homogeneous polynomial of degree , if for all , then when , or equivalently, when . It is also shown in this paper that the condition () above is equivalent to the condition that is Hessian nilpotent, i.e. the Hessian matrix is nilpotent. The goal is achieved by using the recent breakthrough work of M. de Bondt, A. van den Essen and various results obtained in this paper on Hessian nilpotent polynomials. Some further results on Hessian nilpotent polynomials and the vanishing conjecture above are also derived.