Some properties of semiconcave functions with general modulus
type: Item
instance of: scholarly article
title: Some properties of semiconcave functions with general modulus
zbMATH Open document ID: 1019.49027
DOI: 10.1016/S0022-247X(02)00117-8
author: Paolo Albano
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-11-28T00:00:00Z
review text: The author extends to semiconcave functions with a general modulus \(\omega(.):\mathbb{R}^+\to \mathbb{R}^+\) (which is a nondecreasing upper semicontinuous function such that \(\lim_{r\to 0}\omega(r)=0\)) some results in [\textit{P. Albano} and \textit{P. Cannarsa}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., (4) 28, 719-740 (1999; Zbl 0957.26002)] proved for the case of linear modulus \(\omega(r)=C\cdot r\), \(C\geq 0\). A function \(u(.):A\subseteq \mathbb{R}^n\to \mathbb{R}\) is said to be semiconcave with modulus \(\omega(.)\) if: \[ \lambda u(x)+(1-\lambda)u(y)-u(\lambda x+(1-\lambda)y)\leq \lambda(1-\lambda) |x-y|\omega(|x-y|) \] \[ \forall \;\lambda \in [0,1], \quad x,y\in A, \quad [x,y]\subset A \] and is proved to be locally Lipschitz with several other remarkable properties. The main results in the paper concern the so-called ``propagation of singularities'' \(x_0\in\Sigma(u)\) (at which \(u(.)\) is not differentiable) in the sense that there exists a non-constant mapping \(x(.):[0,\sigma]\to \Sigma (u)\) such that \(x(0)=x_0, \;\lim_{s\to 0}x(s)=x_0\) and, possibly, with additional regularity properties in more particular cases. Moreover, in the case of semiconcave viscosity solutions of Hamilton-Jacobi equations of the form \(F(x,u(x),Du(x))=0, \;x\in \Omega\subseteq \mathbb{R}^n\), the mapping \(x(.)\) is proved to be related to the partial derivatives \(D_pF(x_0,u(x_0),p_0)\) for some \(p_0\) in the superdifferential \(D^+u(x_0)\).
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 49L20
zbMATH DE Number: 1836749
zbMATH Keywords: semiconcave function
zbMATH Keywords: superdifferential
zbMATH Keywords: semiconcavity modulus
zbMATH Keywords: propagation of singularities
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: viscosity solution
reviewed by: Ştefan Mirică
MaRDI profile type: Publication
cites work: On the singular set for solutions to a class of Hamilton-Jacobi-Bellman equations
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cites work: Q4487385
cites work: Q4762848
cites work: Propagation of singularities for solutions of nonlinear first order partial differential equations
cites work: On the singularities of convex functions
cites work: Q4296235
cites work: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations
cites work: Some Characterizations of Optimal Trajectories in Control Theory
cites work: Semiconcave functions, Hamilton-Jacobi equations, and optimal control
cites work: Q3753543
cites work: Generalized one-sided estimates for solutions of Hamilton-Jacobi equations and applications
cites work: Q3344757
cites work: Asymptotic behavior of some nonlinear heat equations
cites work: Semiconcavity of solutions of stationary Hamilton-Jacobi equations
cites work: Q4326169
cites work: Q4189690
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description: scientific article; zbMATH DE number 1836749