2. ESTIMATES ON THE p-LAPLACIAN 7

vanishes at the interior endpoints of transport rays. (Janfalk's notation differs from ours, as

does the physical setting.)

We make in §2 a number of assumptions on / + , / ~ . We in particular assume f+ and

/ " are Lipschitz continuous. We need this hypothesis mostly in order to ensure the ODE

(1.23) has a unique solution, but also to derive the sup-norm bound on a (Proposition 2.1).

In addition we suppose that X = supp(/ + ) and Y = supp(/~) are a positive distance

apart. The later assumption is useful in excluding certain bad behavior (e.g. a transport ray

entering V, then entering X, then entering Y, etc., infinitely many times), but is presumably

not essential. Wre hope to return to this point in future work.

Lastly we call the reader's attention to the many recent papers concerning the mass-

transport problem if the term \x — r(x)| in (1.4) is replaced by a general strictly convex cost

density c(x,r(x)). Brenier [B] employed convex analysis to construct an optimal mapping

s when c(x,y) = \x - y\2: see also Gangbo [G], Gangbo-McCann [G-M1,2], Caffarelli [Cl-

3], Wolfson [W], Urbas [UR], etc. for simplifications of Brenier's proof, generalizations, and

regularity theory. All of these papers exploit the fact that for c strictly convex an optimal s

can be built out of the gradient of a scalar potential, the analogue of our u. As noted above,

the primary difficulty for our nonstrictly convex cost density c(x, y) = \x — y\ is that u alone

does not contain enough information to build s.

The recent paper [E-F-G] employs a number of ideas from this paper, in constructing a

crude model for "collapsing sandpiles". The mathematical task is to study the limit p — oc

for a parabolic PDE governed by the p-Laplacian div(\Du\p~2Du), which we interpret as

forcing a Monge-Kantorovich optimal mass transfer on fast time scales. The derivation

then of certain equations of motion depends strongly on the observations in this paper that

detailed mass balance holds (Lemma 5.1) and that the density a vanishes at the endpoints

of rays (Proposition 7.1).

We are very grateful to H. Ishii and M. Feldman for pointing out several inaccuracies in

earlier drafts. The proofs to follow are quite involved. Consequently, although we have really

tried to put in all the details, we realistically expect that some errors will come to light. We

will as necessary post corrections in the first author's page at the math.berkeley.edu website.

This work was carried out while Wr.G. was a member of the Mathematical Sciences

Research Institute at Berkeley, whose financial support and hospitality are gratefully ac-

knowledged.

2. Uniform estimates on the p-Laplacian,

limits as p —• oo

In this section we set forth our hypotheses regarding the densities / + , / ~ and then

obtain estimates, independent of p, on solutions of the corresponding p-Laplacian equations.

Assumptions on the mass densities / + , / ~