# MATH 521: Advanced Topics in Real Analysis, Fall 2008

### Description:

Classically the length or total variation of a (not necessarily continuous) curve  g : [a,b] X  is simply

L(g)  =  limd0 inf { ∑dist(g(ai-1),g(ai)) :   a = a0 < a1 < . . .  < aj = b ,   max i | ai - ai-1|  <  d } .

If  L(g) < ∞ , then  g  is called BV (of bounded variation) .  Then the left and right limits exist at every point of  [a,b] , and coincide except at an at most countable set.  Thus  g  is a finite length curve with a countable number of jumps, the total jumping being finite.  For  X  being a normed vector space,  g  will also be differentiable almost everywhere, but these ideas also make sense in a general metric space  X

Remarkably, there is a notion of total variation for a  function g  of  n  variables with values in  X, and there are interesting generalizations of most of the properties.  In modern lingo, this is an integrable function whose distribution partial derivatives are finite signed measures. Classically, it corresponds to having each  restriction of  g   to almost any line in  Rn  becoming a finite-length curve, and to having the integral over all such lines of the lengths of these curves being finite.

There is a measure-theoretic approximate continuity off of a  n-1 dimensional jump set  J  which is rectifiable.  This means that  J  is, except for an (n - 1) dimensional measure zero set, contained a countable union of  C1 manifolds.  Even for most points on  J ,  there are also one-sided approximate limits. These jumps are useful to control for BV functions satisfying some variational condition.   In image processing applications with  n = 2 and  X = [0,1], representing the range of light to dark shades, the jump set may represent the outline of a figure which is desired in the enhancement of a noisy image.   For a general real-valued BV function  g  on  Rn , one also has numerous uses for the level sets

At = ∂{ x Rn :  g(x) > t } which start with the Co-Area Formula:

Total Variation of  g      Rn  | Dg(x) |   =   R  [ (n-1)-measure of At ]   dt  .

In particular, if  n = 3  and g  is a minimizer of the total variation, then almost all its level sets At  turn out to be area-minimizing minimal surfaces.

Fortunately there are now several accessible books*  treating the basic development of BV functions, a subject which is about 50 years old.

We will use Evans-Gariepy the most.  Our study of functions of bounded variation in several variables will involve introductions to several broader subjects:

• Partial Differential Equations with weak derivatives and Sobolev spaces
• Geometric Measure Theory with Hausdorff measures and rectifiability for n-1 dimensional subsets of Rn.
• Image Segmentation with Total Variation Minimization and the Mumford-Shah Functional

Prerequesites for the course include some knowledge of basic analysis and measure theory as in Math 425.

### *Some References:

Ambrosio, Fusco, Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford, 2000.

L.C. Evans and R.Gariepy, Measure Theory and the Fine Properties of Functions, CRC Press, 1992.

H. Federer, Geometric Measure Theory, Springer-Verlag, 1970.

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhauser, 1984.

W. Ziemer, Weakly Differentiable Functions, Springer, 1989.