From: tao@sonia.math.ucla.edu (Terence Tao)
Subject: Re: Generalized harmonic maps
Date: 15 Jan 2000 00:37:14 GMT
Newsgroups: sci.math.research
Summary: [missing]

In article <85nlid$58c$1@nnrp1.deja.com>,
Marco de Innocentis  <mdi11@hotmail.com> wrote:
>Harmonic maps are those maps between manifolds which minimize a
>certain energy functional, usually given by
>
>   E = Int (e1 + e2 + e3)   (1)
>
>where e1, e2 and e3 are the eigenvalues of the "stress tensor" D
>of the map, given by
>
>   D = J J^t   (2)
>
>J^t denoting the transpose of J, J being the Jacobian matrix.
>
>The properties of harmonic maps have being studied for a long time
>(see for instance Eels and Sampson: "Harmonic mappings of Riemannian
>manifolds", Am. J. Math 86, 109, 1964).
>Now, has anyone studied what happens when the energy depends on
>e1, e2 and e3 in a more general way than (1)? For instance we could
>have terms depending on (e1e2e3) or (e1e2 + e2e3 + e1e3) under the
>integral sign in (1). How do the theorems and results on harmonic
>maps generalize to these cases?

There is some literature on p-harmonic maps, where one integrates
(e1 + e2 + e3)^{p/2} for some 1 < p < \infty.  See

R. Hardt, F. Lin, "Maps minimizing the L^p norm of the gradient",
CPAM (1987) 555-588.

J. Lewis, "Smoothness of certain degenerate elliptic equations",
Proc. AMS 80 (1980) 259-265

K. Uhlenbeck, "Regularity of a class of non-linear elliptic systems",
Acta Math. V. 138 (1970), 219-240.

These equations are still quite elliptic, and so one has most of the
basic elliptic regularity theory, but I believe the more difficult low 
regularity problems (e.g. are weak p-harmonic maps regular) are only
partially understood at present.

Terry