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Extra info for Directions in Partial Differential Equations. Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin–Madison, October 28–30, 1985

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Caffarelli +1 ΔΐιΎ = C(y)ut which we normalize to m Au = u t. This interpretation indicates that if one wants to study the mechanics of the free boundary displacement one m - 1 should concentrate his attention on ρ = u , since the speed of the free boundary is expected to be IVp|. That is, under ideal circumstances the free boundary 3(supp p(·,t)) should be a smooth surface, ρ should be smooth up to it, and the free surface should move with speed |Vp|. Of course, these ideal circumstances should not necessarily hold from time t = 0, since supp φ may be topologically complicated and advancing free boundaries may hit each other.

And lu |P where u of and the nonzero critical points of F = max(u,0). - c e for every u = 0 is a critical point precisely to the solutions of Problem F (I). correspond The proof of Theorem 6 is by contradiction and henceforth we shall assume that u = 0 is the only critical point of It is rather easy to see that the c = kz at the levels n2 (1/N)S / . Σ = sequence F ( u ) + kz n where F . (PS) k = 1,2,3,... g. [14]). 0 weakly kZ; and It is a striking fact - which is much more difficult to prove - that the fails only at the levels in (PS) C condition the argument relies on a blowup technique originally due to J.

To prove it we look at the problem this way: First we think of property b) as saying "for any small translation εμ, with μ in the cone r ( a n , e n) = { μ | α ( μ , β η) < an> we have v(x + ey),t) > v ( x , t ) " . Now for some translations μ in the cone r ( a Q , e n) it may well be that this inequality cannot is not strict, or at least its "strictness" cannot be estimated. Luis A. Caffarelli 40 For instance if ν is a linear front with a free boundary tangent to the cone actually be an equality for direction of tangency.

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