Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0874.49041
Ambrosio, L.; Cannarsa, P.; Soner, H.M.
On the propagation of singularities of semi-convex functions. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 20, No.4, 597-616 (1993). ISSN 0391-173X

In a recent paper, {\it G. Alberti}, {\it L. Ambrosio} and {\it P. Cannarsa} [Manuscr. Math. 76, No. 3/4, 421-435 (1992; Zbl 0784.49011)] gave upper bounds for the Hausdorff dimension of singular sets of a semi-convex function $u$. The present paper formulates the corresponding lower bounds. For the upper bounds, only semi-convexity properties of $u$ play a role, but additional information is needed to obtain the lower bounds, in the form of the set of so-called reachable subgradients at a point $x$. If this set is strictly contained in the ordinary subdifferential of $u$ at a singular point $x$, then the singularity is propagated. Moreover, if $x$ is an isolated singularity, then those two sets coincide. Using other concepts from nonsmooth analysis, the paper also provides more details about the directions along which singularities propagate, in terms of the geometry of the ordinary subdifferential. Finally, some applications are given to viscosity solutions of Hamilton-Jacobi-Bellman equations.
[E.J.Balder (MR 95b:49068)]

MSC 2000:: *49Q15 Geometric measure and integration theory, etc.
49L25 Viscosity solutions
35F20 General theory of first order nonlinear PDE
49J52 Nonsmooth analysis (other weak concepts of optimality)

Keywords: propagation of singularities; semi-convexity; reachable subgradients; viscosity solutions; Hamilton-Jacobi-Bellman equations

Citations: Zbl 0784.49011

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]