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Topic: 21st Century 'Hilbert' problems
Replies: 25   Last Post: Feb 10, 1999 1:11 AM

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 Stephen Montgomery-Smith Posts: 97 Registered: 12/6/04
Re: 21st Century 'Hilbert' problems
Posted: Feb 6, 1999 9:05 AM

Simon Croall wrote:
>
> Hi,
>
> With the 21st century almost upon us;
>
> In a similar fashion to Hilbet's 23 problems but forward in 1900. What
> would be the problems that you think should be presented to challenge
> mathematicians of the 21st century?
>

Come on guy's - no one seems to be answering seriously !!!

OK, here is a problem. I am not quite sure that it attains to quite
the standards of Hilbert's problems, but I think that it is a fascinating
problem. I think that it goes back to Morrey. Basically it asks whether
rank-one convex functions are quasi-convex. Let me elaborate.

Suppose that U(x,y,z,w) is a function of four real numbers so that the
function

t mapsto U(x+ta,y+tb,z+tc,w+td)

is convex whenever the vector (a,b) is parallel to (c,d).

Show that if f(x,y) and g(x,y) are smooth functions with compact support
on R^2, and that U(0,0,0,0) = 0, then

int int U(f_x(x,y),f_y(x,y),g_x(x,y),g_y(x,y)) dx dy >= 0

where f_x(x,y) represents partial derivative of f(x,y) with respect to x, etc.

In a sense, this conjecture could be seen as extending Jensen's inequality.

There are counterexamples of higher dimensional analogues due to Sverak.
Well, if anyone is really interested, Al Baernstein and myself wrote a
paper with very partial results. Look at
http://math.missouri.edu/~stephen/preprints/
the paper called:Some conjectures about integral means of
$\partial f$ and $\overline{\partial} f$
It also has lots of references to the work of others.

--

Stephen Montgomery-Smith stephen@math.missouri.edu
307 Math Science Building stephen@showme.missouri.edu
Department of Mathematics stephen@missouri.edu
University of Missouri-Columbia
Columbia, MO 65211
USA

Phone (573) 882 4540
Fax (573) 882 1869

http://math.missouri.edu/~stephen