Hessian Equations Research Articles

Kiselman minimum principle and rooftop envelopes in complex Hessian equations

We initiate the study of m-subharmonic functions with respect to a semipositive (1, 1)-form in Euclidean domains, providing a significant element in understanding geodesics within the context of complex Hessian equations. Based on the foundational Perron envelope construction, we prove a decomposition of m-subharmonic solutions, and a general comparison principle that effectively manages singular Hessian measures. Additionally, we establish a rooftop equality and an analogue of the Kiselman minimum principle, which are crucial ingredients in establishing a criterion for geodesic connectivity among m-subharmonic functions, expressed in terms of their asymptotic envelopes.

Viscosity solutions of the augmented K -Hessian equations in exterior domains

Viscosity solutions of the augmented K -Hessian equations in exterior domains

An Obata-type formula and the Liouville-type theorem for a class of K-Hessian equations on the sphere

In this paper, we study a class of k k -Hessian equations, we can deduce an Obata-type formula and a Liouville-type theorem by integration by parts.

Asymptotic Behavior of Singular Solution to the k-Hessian Equation with a Matukuma-Type Source

This paper is concerned with radially positive solutions of the [see formula in PDF]-Hessian equation involving a Matukuma-type source [see formula in PDF], where [see formula in PDF] is the [see formula in PDF]-Hessian operator, [see formula in PDF], and [see formula in PDF] is a suitable bounded domain in [see formula in PDF]. It turns out that there are two different types of radially positive solutions for [see formula in PDF], i.e., M-solution (singular at [see formula in PDF]) and E-solution (regular at [see formula in PDF]), which is distinct from the case when [see formula in PDF]. For [see formula in PDF], we apply an iterative approach to improve accuracy of asymptotic expansions of M-solution step by step to the desired extend. In contrast to the case [see formula in PDF], we require a more precise range of parameters due to repeated application of Taylor expansions, which also makes asymptotic expansions need more delicate investigation.

Classification the elements of the twisted Hessian curves in the ring $\mathbb{F}_{q}[e], e^{3}=e^2$

Let $\mathbb{F}_{q}$ denote the finite field of $q$ elements, where $q$ is a prime power. In this paper, we study the twisted Hessian curves denoted $H_{a,d}(\mathbb{F}_{q}[e])$ over the ring $\mathbb{F}_{q}[e]$, where $e^{3}=e^2$ and $(a,d)\in (\mathbb{F}_{q}[e])^{2}$. More precisely, we study some arithmetical properties of this ring and using the Twisted Hessian equation, we define the twisted Hessian curves $H_{a,d}(\mathbb{F}_{q}[e])$. This work study the twisted Hessian curve helped us to define two twisted Hessian over the finite field $\mathbb{F}_{q}$. We end this paper by giving the classification of the elements in $H_{a,d}(\mathbb{F}_{q}[e])$.

Solution to Hessian type equation with prescribed singularities on compact Kähler manifolds

Let (X,ω) be a compact Kähler manifold of dimension n and fix an integer m such that 1≤m≤n. We reformulate Darvas-Nezza-Lu's latest survey [12] into the Hessian setting. Namely, we characterize the relative full mass class Eϕ(X,ω,m) and prove the integration by parts formula of Hessian type. Given a model potential ϕ, we study degenerate complex Hessian equations of the form (ω+ddcφ)m∧ωn−m=F(x,φ)ωn. Under some natural conditions on F, we prove that this equation has a unique solution (up to a constant) which has the same singularity type as ϕ.

On L∞ estimates for Monge–Ampère and Hessian equations on nef classes

On L∞ estimates for Monge–Ampère and Hessian equations on nef classes

The weak solutions to complex Hessian equations

The weak solutions to complex Hessian equations

Existence of Positive Solutions for a Singular Hessian Equation with a Negative Augmented Term

In this paper, we focus on the existence of positive solutions for a singular Hessian equation with a negative augmented term. By finding more appropriate upper and lower solutions, we not only overcome the difficulty due to the negative augmented term but also remove a critical condition required in the existing work and establish new results for the existence of positive solutions of the equations under study. Our results improve and complement many existing works.

Boundary estimate of large solution to the [formula omitted]-Hessian equation

This paper investigates boundary blow-up problem of the k-Hessian equation σk(μ(D2u))=b(z)h(u) in a uniformly (k−1)-convex domain D⊂RN, where b(z) grows like δ−α(z) as the distance function δ(z)→0 and h(u) grows like uk(lnu)β as u→∞. Boundary estimate of large solution u is obtained by the method of sub- and super- solutions. Compared to previous research (Zhang et al., 2022), this paper provides a more accurate estimate of the large solution when 0<α<1,β>2k+1.

Sharp conditions for the existence of infinitely many positive solutions to $ q $-$ k $-Hessian equation and systems

&lt;p&gt;In this paper, only under the $ q $-$ k $-Keller–Osserman conditions, we consider the existence and global estimates of innumerable radial $ q $-$ k $-convex positive solutions to the $ q $-$ k $-Hessian equation and systems. Our conditions are strictly weaker than those in previous papers.&lt;/p&gt;

Parabolic Hessian equations outside a bowl-shaped domain

Parabolic Hessian equations outside a bowl-shaped domain

Curvature estimates for semi-convex solutions of Hessian equations in hyperbolic space

Curvature estimates for semi-convex solutions of Hessian equations in hyperbolic space

Augmented Hessian equations on Riemannian manifolds: from integral to pointwise local second derivative estimates

We obtain a priori local pointwise second derivative estimates for solutions u to a class of augmented Hessian equations on Riemannian manifolds, in terms of the C1 norm and certain W2,p norms of u. We consider the case that no structural assumptions are imposed on either the augmenting term or the right hand side of the equation, and the case where these terms are convex in the gradient variable. In the latter case, under an additional ellipticity condition we prove that the dependence on any W2,p norm can be dropped. Our results are derived using integral estimates.

A characterization of the degenerate complex Hessian equations for functions with bounded (p,m)-energy

A characterization of the degenerate complex Hessian equations for functions with bounded (p,m)-energy

Continuous Solutions to Complex Hessian Equations on Hermitian Manifolds

Continuous Solutions to Complex Hessian Equations on Hermitian Manifolds

Trace inequalities, isocapacitary inequalities, and regularity of the complex Hessian equations

Trace inequalities, isocapacitary inequalities, and regularity of the complex Hessian equations

The role of leading-edge serrations in controlling the flow over owls’ wing

We studied the effects of leading-edge serrations on the flow dynamics developed over an owl wing model. Owls are predatory birds. Most owl species are nocturnal, with some active during the day. The nocturnal ones feature stealth capabilities that are partially attributed to their wing microfeatures. One of these microfeatures is small rigid combs (i.e. serrations) aligned at an angle with respect to the incoming flow located at the wings’ leading-edge region of the primaries. These serrations are essentially passive flow control devices that enhance some of the owls’ flight characteristics, such as aeroacoustics and, potentially, aerodynamics. We performed a comparative study between serrated and non-serrated owl wing models and investigated how the boundary layer over these wings changes in the presence of serrations over a range of angles of attack. Using particle image velocimetry, we measured the mean and turbulent flow characteristics and analyzed the flow patterns within the boundary layer region. Our experimental study suggests that leading-edge serrations modify the boundary layer over the wing at all angles of attack, but not in a similar manner. At low angles of attack (<20°), the serrations amplified the turbulence activity over the wing planform without causing any significant change in the mean flow. At 20° angle of attack, the serrations act to suppress existing turbulence conditions, presumably by causing an earlier separation closer to the leading-edge region, thus enabling the flow to reattach prior to shedding downstream into the wake. Following the pressure Hessian equation, turbulence suppression reduces the pressure fluctuations gradients. This reduction over the wing would weaken, to some extent, the scattering of aerodynamic noise in the near wake region.

Hessian Equations of Krylov Type on Kähler Manifolds

Hessian Equations of Krylov Type on Kähler Manifolds

Space-Time Mixed System Formulation of Phase-Field Fracture Optimal Control Problems

In this work, space-time formulations and Galerkin discretizations for phase-field fracture optimal control problems are considered. The fracture irreversibility constraint is formulated on the time-continuous level and is regularized by means of penalization. The optimization scheme is formulated in terms of the reduced approach and then solved with a Newton method. To this end, the state, adjoint, tangent, and adjoint Hessian equations are derived. The key focus is on the design of appropriate function spaces and the rigorous justification of all Fréchet derivatives that require fourth-order regularizations. Therein, a second-order time derivative on the phase-field variable appears, which is reformulated as a mixed first-order-in-time system. These derivations are carefully established for all four equations. Finally, the corresponding time-stepping schemes are derived by employing a dG(r) discretization in time.