scholarly journals Generalised solutions of Hessian equations

1997 ◽  
Vol 56 (3) ◽  
pp. 459-466 ◽  
Author(s):  
Andrea Colesanti ◽  
Paolo Salani
Keyword(s):  
Weak Solutions ◽  
Convex Functions ◽  
Symmetric Function ◽  
Convex Set ◽  
Hessian Equation ◽  
Hessian Equations ◽  
Definition Of ◽  
Monge Ampere

We introduce a definition of generalised solutions of the Hessian equation Sm(D2u) = f in a convex set ω ⊂ ℝn, where Sm(D2u) denotes the m-th symmetric function of the eigenvalues of D2u, f ∈ Lp(ω), p ≥ 1, and m ∈ {1, …, n}. Such a definition is given in the class of semi-convex functions, and it extends the definition of convex generalised solutions for the Monge-Ampère equation. We prove that semiconvex weak solutions are solutions in the sense of the present paper.

scholarly journals The Dirichlet Problem of Hessian Equation in Exterior Domains

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 666
Author(s):  
Hongfei Li ◽  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Dirichlet Problem ◽  
Viscosity Solutions ◽  
Exterior Domains ◽  
Hessian Equation ◽  
Hessian Equations ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

In this paper, we will obtain the existence of viscosity solutions to the exterior Dirichlet problem for Hessian equations with prescribed asymptotic behavior at infinity by the Perron’s method. This extends the Ju–Bao results on Monge–Ampère equations det D 2 u = f ( x ) .

scholarly journals Hermite–Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Dafang Zhao ◽  
Muhammad Aamir Ali ◽  
Artion Kashuri ◽  
Hüseyin Budak ◽  
Mehmet Zeki Sarikaya
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Special Cases ◽  
Definition Of ◽  
The Given ◽  
Class Of Functions ◽  
Interval Valued ◽  
New Inequalities

Abstract In this paper, we present a new definition of interval-valued convex functions depending on the given function which is called “interval-valued approximately h-convex functions”. We establish some inequalities of Hermite–Hadamard type for a newly defined class of functions by using generalized fractional integrals. Our new inequalities are the extensions of previously obtained results like (D.F. Zhao et al. in J. Inequal. Appl. 2018(1):302, 2018 and H. Budak et al. in Proc. Am. Math. Soc., 2019). We also discussed some special cases from our main results.

Weak solutions of the Monge–Ampère equation on compact Hermitian manifolds

2017 ◽  
Vol 28 (09) ◽  
pp. 1740002
Author(s):  
Sławomir Kołodziej
Keyword(s):  
Weak Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Stability Of Solutions ◽  
Hermitian Manifolds ◽  
Priori Estimates ◽  
Existence And Stability ◽  
Monge Ampere

In this paper, we describe how pluripotential methods can be applied to study weak solutions of the complex Monge–Ampère equation on compact Hermitian manifolds. We indicate the differences between Kähler and non-Kähler setting. The results include a priori estimates, existence and stability of solutions.

Some Elements of Functional Analysis

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Functional Analysis ◽  
Weak Solutions ◽  
Vector Fields ◽  
Stokes Equations ◽  
Stokes Operator ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Dual Spaces ◽  
Type Spaces ◽  
Definition Of

Before introducing the concept of Leray’s weak solutions to the incompressible Navier–Stokes equations, classical definitions of Sobolev spaces are required. In particular, when it comes to the analysis of the Stokes operator, suitable functional spaces of incompressible vector fields have to be defined. Several issues regarding the associated dual spaces, embedding properties, and the mathematical way of considering the pressure field are also discussed. Let us first recall the definition of some functional spaces that we shall use throughout this book. In the framework of weak solutions of the Navier– Stokes equations, incompressible vector fields with finite viscous dissipation and the no-slip property on the boundary are considered. Such H1-type spaces of incompressible vector fields, and the corresponding dual spaces, are important ingredients in the analysis of the Stokes operator.

CAPITAL ALLOCATION FOR SET-VALUED RISK MEASURES

2020 ◽  
Vol 23 (01) ◽  
pp. 2050009
Author(s):  
FRANCESCA CENTRONE ◽  
EMANUELA ROSAZZA GIANIN
Keyword(s):  
Convex Set ◽  
Risk Measures ◽  
Risk Measure ◽  
Capital Allocation ◽  
Allocation Rule ◽  
Scalar Case ◽  
Representation Theorems ◽  
Allocation Rules ◽  
The One ◽  
Definition Of

We introduce the definition of set-valued capital allocation rule, in the context of set-valued risk measures. In analogy to some well known methods for the scalar case based on the idea of marginal contribution and hence on the notion of gradient and sub-gradient of a risk measure, and under some reasonable assumptions, we define some set-valued capital allocation rules relying on the representation theorems for coherent and convex set-valued risk measures and investigate their link with the notion of sub-differential for set-valued functions. We also introduce and study the set-valued analogous of some properties of classical capital allocation rules, such as the one of no undercut. Furthermore, we compare these rules with some of those mostly used for univariate (single-valued) risk measures. Examples and comparisons with the scalar case are provided at the end.

Strict convexity, strong ellipticity, and regularity in the calculus of variations

1980 ◽  
Vol 87 (3) ◽  
pp. 501-513 ◽  
Author(s):  
J. M. Ball
Keyword(s):  
Nonlinear Systems ◽  
Calculus Of Variations ◽  
Weak Solutions ◽  
Convex Functions ◽  
Strict Convexity ◽  
Mathematical Tool ◽  
Strong Ellipticity ◽  
Regularity Of Weak Solutions ◽  
Nonlinear Elastostatics

In this paper we investigate the connection between strong ellipticity and the regularity of weak solutions to the equations of nonlinear elastostatics and other nonlinear systems arising from the calculus of variations. The main mathematical tool is a new characterization of continuously differentiable strictly convex functions. We first describe this characterization, and then explain how it can be applied to the calculus of variations and to elastostatics.

scholarly journals Valuations on convex functions and convex sets and Monge–Ampère operators

2019 ◽  
Vol 19 (3) ◽  
pp. 313-322 ◽  
Author(s):  
Semyon Alesker
Keyword(s):  
Convex Functions ◽  
Convex Sets ◽  
Convex Bodies ◽  
Linear Functionals ◽  
Translation Invariant ◽  
Infinite Dimensional ◽  
Dense Image ◽  
Monge Ampere ◽  
Explicit Relation ◽  
Class Of Functions

Abstract The notion of a valuation on convex bodies is very classical; valuations on a class of functions have been introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translation invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite-dimensional kernel. The proof uses the author’s irreducibility theorem and properties of the real Monge–Ampère operators due to A.D. Alexandrov and Z. Blocki. Furthermore we show how to use complex, quaternionic, and octonionic Monge–Ampère operators to construct more examples of continuous valuations on convex functions in an analogous way.

scholarly journals Hermite-Hadamard Type Inequalities Associated with Coordinated((s,m),QC)-Convex Functions

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Yu-Mei Bai ◽  
Shan-He Wu ◽  
Ying Wu
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Integral Inequalities ◽  
Type Integral ◽  
Definition Of

In this paper, we introduce the definition of coordinated((s,m),QC)-convex function and establish some Hermite-Hadamard type integral inequalities for coordinated((s,m),QC)-convex functions.

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