Some properties of semiconcave functions with general modulus
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 \(D^+u(x_0)\).
scientific article; zbMATH DE number 1836749