Abstract
According to C. Kuratovsky, the use of the distance
function to define metrics on sets is due to the Romanian
mathematician D. Pompéju in a paper in the Annales de
Toulouse in 1905. The notion was popularized by F. Hausdorff in his
book Grundzüge der Mengenlehre published in 1914 in
Leipzig and is often referred to as the Hausdorff metric. This
function provides an analytical description of a geometrical object,
but is difficult to use in investigating the smoothness of the
associated set. In their book in 1977 D. Gilbard and N.S. Trudinger
provided the connection between the derivatives of the distance
function to the boundary of a set and the curvatures of its boundary
for sets of class C2. This was further formalized in 1981 by
Krantz
and Parks by using the signed distance function.
In 1994 M. Delfour and J.-P. Zolésio significantly sharpened and
expanded the theory of distance functions and their relationship to the
geometry of the underlying set. The notion of oriented distance
function was introduced. This new terminology arises from the fact
that, for smooth sets, the oriented distance function specifies the
orientation of the exterior unit normal to the boundary and makes the
curvature of the unit ball positive. The oriented distance function of
a set coincides with its signed distance function, but it offers many
technical advantages and avoids the frequent confusion with the
properties of the distance function to a smooth submanifold.
New metrics and new compactness theorems were introduced to deal with
sets ranging from sets with a non empty boundary to sets of class
C∞. At the rough end of the spectrum, the new cracked sets
were
recently introduced for image processing problems. At the smooth end,
the characterization of C1,1 sets by the C1,1
smoothness of the
oriented distance function in a neighborhood of its boundary led to an
intrinsic differential calculus on C1,1 hypersurfaces
without local
bases and Christoffel symbols and an equally intrinsic theory of
Sobolev spaces on C1,1 hypersurfaces. This also resulted in
significant breakthroughs and considerable simplifications in the
theory of thin and asymptotic shells.
In this lecture we review the basic definitions and results on distance
and oriented distance functions and introduce the associated
differential calculus on an hypersurface. If time permits we shall look
at some applications such as the Laplace-Beltrami operator.
References
M.C. Delfour, Tangential differential calculus and functional analysis
on a C1,1- submanifold, in "Differential-geometric methods
in the
control of partial differential equations", R. Gulliver, W. Littman and
R. Triggiani, eds., pp. 83--115, Contemp. Math, Vol. 268, AMS
Publications, 2000.
M. C. Delfour and J.-P. Zolésio, Shape analysis via oriented
distance functions, J. Funct. Anal. 123 (1994), no. 1, 129--201.
M. C. Delfour and J.-P. Zolésio, Differential equations for
linear shells: comparison between intrinsic and classical models, in
"Advances in the Mathematical Sciences - CRM's 25 years", Luc
Vinet,ed., pp. 42-124, CRM Proc. Lecture Notes, vol. 11, Amer. Math.
Soc., Providence, RI, 1997.
M. C. Delfour and J.-P. Zolésio, Shapes and Geometries:
Analysis, Differential Calculus and Optimization, SIAM series on
Advances in Design and Control, Society for Industrial and Applied
Mathematics, Philadelphia, USA 2001.
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations
of second order, Springer-Verlag, Berlin, New York, 1977 and 1983
S.G. Krantz and H.R. Parks, Distance to Ck hypersurfaces, J.
Differential Equations 40 (1981), no. 1, 116-120.