L. C. Evans: In pde, a breakthrough occurs when things that were

Caricatured from the preface of Ian Stewart s The Problems of Mathematics, via Drawbridge Up by H. M. Enzenberger: Mathematician: It s one of the most important breakthroughs of the last decade! Normal Person: Can you explain it to me? Mathematician: How can I talk about this advance without mentioning that the theorems only work if the manifolds are finite dimensional para-compact Hausdorff with empty boundary? Normal Person: LIE A LITTLE BIT? L. C. Evans: In pde, a breakthrough occurs when things that were formerly impossible become merely difficult and as well as when things that were formerly difficult become easy. M. B. Maple: In superconductivity we find three things that fascinate the human mind: zero, perfection and infinity. M. G. Crandall: Thank goodness that Gunnar Aronsson invented the infinity Laplacian. It s perfect, has infinity in its name, and it s particularly fascinating when the gradient is zero. 1 PROLOGUE: THE FOLLOWING ARE EQUIVALENT! Second order version: u 0 in U in the viscosity sense, aka, u is infinity subharmonic. First order version: Du(x) max { w x =r} u(w) u(x) r aka, the gradient estimate. for Br(x) U Zero order version #1: ( ) max{ w x =r} u(w) u(y) u(x) u(y)+ x y for x Br(y) U r aka, comparison with cones from above. Zero order version #2 (almost the same as above): ( u(w) u(y) u(w) u(x) max w Br(y) Convexity version: r max w Br(x) max w Br(y) ) r r x y for x Br(y) U aka, the Harnack inequality. u(w) isconvexinr, 0 r<r= dist (x, U). Variational Property: If V U, v C(V ), u v in V and u = v on V, then Du L (V ) Dv L (V ) aka, u is sub-absolutely minimizing for F (u) = Du L (or some such). 2

THE CONTINUING SAGA OF THE INFINITY-LAPLACE EQUATION It all began with GUNNAR ARONSSON s paper: Extension of functions satisfying Lipschitz conditions, Arkiv för Matematik, 1967. To explain, let us suppose U IR n is open and bounded and b : U IR ( b is a mnemonic for boundary data ) and we have a functional F : C = {functions v on U such that v = b on U} [, ] and seek to solve the standard problem: Minimum Does there exist u Csatisfying F (u) F (v) for v in C, and if so, can there be more than one? Aronsson considered three choices of F (v): the first was the least Lipschitz constant for v Lip(v, U) = inf{l [0, ] : v(x) v(y) L x y for x, y U}, and the second was the L norm of the gradient F (v, U) = ess sup U 3 Dv. Here we put F (v, U) =+ if v is not continuous on U or the distributional gradient Dv =(vx1,...,v xn ) is not in L. Aronsson s third F we set aside for now. Note that above is used for the absolute value on IR as well as the Euclidean norm on IR n. Note that we have indexed these F s by the sets over which they are computed for later convenience. Note that these F s are convex, but they are not strictly convex. Aronsson observed that Lip = F if U is convex, but this is not generally the case if U is not convex. In the case F = Lip, the problem Minimum was known to have a largest and a smallest solution, via the works of McShane and Whitney: E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc., 1934. H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 1934. These McShane-Whitney solutions are: Largest solution = min y U Smallest solution = max y U (b(y) + Lip(b, U) x y ) (b(y) Lip(b, U) x y ) 4

It is a simple exercise to show that these functions are solutions and bound all solutions from above and below. They are finite iff Lip(b, U) <, which we are assuming. Aronsson derived, among other things, interesting information about the set on which these two functions coincide (which may be only U) and the derivatives of any solution on this contact set. In particular, he established that solutions of Minimum with F = Lip are unique iff there is a function u C 1 (U) C(U) which satisfies Du Lip(b, U) inu, u = b on U, which is then the one and only solution. This is a very special circumstance. The following question naturally arises: is it possible to find a canonical Lipschitz extension that would enjoy comparison and stability properties? And furthermore, could this special extension be unique once the boundary data is fixed? We explain Aronsson s successful proposal in this regard. Thinking in terms of F = F = Lip on convex sets, Aronsson was led to the now famous pde: A(u) = u = n i,j=1 uxi u xj u xixj =0. 5 This was discovered by the heuristic reasoning: u is a solution of Minimum for F = Fp, where ( )1 p Fp(v, U) = Dv L p (U) = Dv p iff u = b on U and U 1 p 2 Du 2 u + u =0. Letting p yields u =0. Moreover, Fp(u, U) F (u, U) if Du L (U). He further observed that for u C 2, u = 0 amounts to if Ẋ(t) = Du(X(t)), then d dt Du(X(t) 2 = D( Du(x) 2 ),Du(x) x=x(t) = 2( u)(x(t))= 0; that is, Du is constant on the trajectories of the gradient flow of u. Moreover, Aronsson generalized a notion appearing in works of his in one dimension to introduce the notion of an absolutely minimizing function. 6

We formulate this in the generality of the problem Minimum, relating it to the functional F : u is an absolute minimzer for F in U if for V U and v satisfying v = u on V we have F (u, V ) F (v, V ). He then went on to prove, for F = Lip, that: (#1) If u C 2 (U), then u is absolutely minimizing in U iff u =0inU. (#2) If Lip(b, U) <, then u C(U) which is absolutely minimizing in U and satisfies u = b on U. With the technology of the times, this is about all anyone could have proved. The gaps between (#1) above, which required u C 2 and (#2), which produced, by a Perron method, a function only known to be Lipschitz continuous, could not be closed at that time. In particular, Aronsson already knew that solutions of the eikonal equation Du = constant, which might not be C 2, are absolutely minimizing. However, he had no way to interpret them as solutions of u = 0. Moreover, the question of uniqueness of the function whose existence Aronsson proved would be unsettled for 26 years. 7 Aronsson himself made the gap more evident in the paper On the partial differential equation u 2 xuxx +2uxuyuxy + u 2 yuyy =0, Arkiv för Matematik, 1968. in which he produced examples of U, b for which the problem of (#2) has no C 2 solution. This work also contained a penetrating analysis of classical solutions of the pde. However, all of these results are false in the generality of viscosity solutions of the equation, which appear below as the perfecting instrument of the theory. The best known explicit irregular absolutely minimizing function - outside of solutions of the eikonal equation - was exhibited again by Aronsson, who showed in On certain singular solutions of the partial differential equation u 2 xuxx +2uxuy + u 2 yuyy =0,Manuscripta Math., 1984. that u(x, y) = x 4/3 y 4/3 is absolutely minimizing in R 2 for F = Lip AND F = F. Most of the interesting results for classical solutions in 2 dimensions are falsified by this example. These include: Du is constant on trajectories of the gradient flow, global absolutely minimizing functions are linear, and Du cannot vanish unless u is locally constant. A rich supply of other solutions was provided as well. 8

ENTER BOB JENSEN AND VISCOSITY SOLUTIONS Let u USC(U) (the upper semicontinuous functions on U). Then u is a viscosity subsolution of u = 0 (equivalently, a viscosity solution of u 0) in U if: whenever φ C 2 (U) and u φ has a local maximum at ˆx U, then φ(ˆx) 0. Let u LSC(U). Then u is a viscosity supersolution of u =0 (equivalently, a viscosity solution of u 0) in U if whenever φ C 2 (U) and u φ has a local minimum at ˆx U, then φ(ˆx) 0. The impetus for this definition arises from the standard maximum principle argument at a point ˆx where u φ has a local maximum. Let D 2 u = ( uxi,xj) be the Hessian matrix of the second order partial derivatives of u. Then u = D 2 udu, Du 0, Du(ˆx) =Dφ(ˆx) and D 2 u(ˆx) D 2 φ(ˆx) = D 2 φ(ˆx)dφ(ˆx),dφ(ˆx) 0. This puts the derivatives on the test function φ, a device first used by L. C. EVANS in On solving certain nonlinear partial differential equations by accretive operator methods, Israel J. Mathematics (1980). The theory of viscosity solutions of much more general equations, born in the 1980 s, and to which Jensen made major contributions, contained strong results of the form: 9 Comparison Theorem Let u, v USC(U) and G(x, u, Du, D 2 u) 0 and G(x, v, Dv, D 2 v) 0inU in the viscosity sense, and u v on U. Then u v in U. The developers of these results were Jensen, Lions, Souganidis, Ishii,. Of course, there are some structure conditions needed on G, and these typically imply that the solutions of G 0 perturb to a solution of G > 0 or that some change of variable u = g(w) produces a new problem with this property. This has not been not accomplished in any simple way for, except at points where Du 0, when it is achieved, for example, through a change of variable u = w λ 2 w2. Then u 0 = w λ 1 λw Dw 2. Moreover, Hitoshi Ishii proved existence via uniqueness, that is, he proved, using the Perron Method, Existence Theorem When the comparison theorem is true and for each ɛ>0 there exist u, v satisfying its assumptions as well as b ɛ u b v b + ɛ on U, then there exists a (unique) viscosity solution u C(U) of 0 G(x, u, Du, D 2 u) 0inU and u = b on U. 10

We wrote 0 G 0 above to highlight that a viscosity solution of G =0is exactly a function which is a viscosity solution of both G 0 AND G 0; there is no other notion of G = 0 in the viscosity sense. NOTE: we have used the sign normalization which admits G = as an example, as opposed to G =. REALLY NOTE: hereafter all references to solutions, subsolutions, u 0, etc, are meant in the viscosity sense! (Unless otherwise said.) Jensen proved, in Uniqueness of Lipschitz Extensions: Minimizing the Sup Norm of the Gradient, Arch. Rational Mech. Anal. (1993), that: (J1) An absolute minimizer u for F = F is a viscosity solution of u = 0 and conversely. (J2) The comparison theorem holds for G =. Jensen s proof of the comparison theorem was remarkable. In order to deal with the difficulties associated with points where Du = 0, he used approximations via the obstacle problems max{ɛ Du +, u + } =0, min{ Du ɛ, u } =0! 11 These he solved by approximation with modifications of Fp, although they are amenable to the theory of Comparison Theorem and Existence Theorem. It is easy enough to show that u u u + when u = 0 and u = u + = u = b on U. Comparison then followed from an estimate, involving Sobolev inequalities, which established u + u γ(ɛ) where γ(0+) = 0. The first assertion of (J1) was proved directly, via a modification of Aronsson s original proof, while the conversely was a consequence of the uniqueness. The relation between absolutely minimizing relative to Fp and relative to Lip had become even more murky. Jensen referred as well to another Lip, as had Aronsson earlier, namely the Lipschitz constant relative to the interior distance between points: dist U(x, y) = infimum of the lengths of paths in U joining x and y and the ordinary Lipschitz constant did not play a role in his work. Thus, after 26 years, the existence of absolutely minimizing functions assuming given boundary values was known (Aronsson and Jensen), and, at last, the uniqueness (Jensen). Jensen s work generated considerable interest in the theory. Among other contributions was an important new uniqueness proof by 12

G. BARLES and J. BUSCA: Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. Partial Diff. Equations, 2001. Roughly speaking, this proof couples some penetrating observations to the standard machinery of viscosity solutions to reach the same conclusions as Jensen, but without obstacle problems or integral estimates. After existence and uniqueness, one wants to know about REGULARITY! Aronsson s example, u(x, y) =x 4/3 y 4/3, sets limits on what might be true. The first derivatives of u are Hölder continuous with exponent 1/3; second derivatives do not exist on the lines x =0 and y =0. Lipschitz Continuity and the Harnack Inequality Jensen also gave arguments of comparison with cones type (see below and the supplement), which can be used to show that an infinity subharmonic function is locally Lipschitz continuous. Aronsson s original derivation of u = 0 as the Euler equation corresponding to the property of being absolutely minimizing and Jensen s existence and uniqueness proofs closely linked letting 13 p in the problem pup = Du ( p 4 Du 2 u +(p 2) u ) =0inU and up = b on U. with our problem u =0inU and u = b on U. With this connection in mind and using estimates learned from the theory of p, Linqdvist and Manfredi (1995) proved that if u 0 is a variational solution of u 0 (i.e, a limit of solutions of pup 0), then one has the Harnack Inequality u(x) e x y R r u(y) for x, y Br( ) x 0 BR( ) x 0 U. They derived this from the elegant Du(x) Gradient Estimate 1 dist (x, U) (u(x) inf u), U valid at points where Du(x) exists. The appropriate Harnack Inequality is closely related to regularity issues for classes of elliptic and parabolic equations, which is one of the reasons to be interested in it. However, so far, it has not played a similar role in the theory of. The same authors extended this result, a generalization of an earlier result of Evans (1993) for smooth functions, to all - superharmonic functions, ie, solutions of u 0 in 14

Note on -superharmonic functions, Rev. Mat. Univ. Complut. Madrid 10 (1997) by showing that ALL -superharmonic functions are variational. This perfected the relationship between p,fp and,f. BTW, the original observation that if solutions of pup = 0 have a limit up u as p, then u = 0 is due to Bhattacharya, Di Benedetto and Manfredi, (1989). This sort of observation is a routine matter in the viscosity solution theory; there is much else in that paper. One point of concern is the relationship between the notions of viscosity solutions and solutions in the sense of distributions for the p-laplace equation. See also Juutinen, Lindqvist and Manfredi, (2001) and Ishii, (1995). Lindqvist and Manfredi also showed that (LM1) If u(x) v(x) min (u v) when v =0, V and x V U, then u 0. (LM2) If u =0inIR n and u is bounded below, then u is constant. Subsequently, C., Gariepy and Evans, (2001), showed that it suffices to take functions of the form v(x) = b x z, aka, a cone function, in (LM1), and introduced the terminology comparison with cones. 15 The assumptions of (LM1) with v a cone function as above is called comparison with cones from below; the corresponding relation for u 0 is called comparison with cones from above. When u enjoys both of these, it enjoys comparison with cones. All the information contained in u 0, u 0, u =0 is contained in the corresponding comparison with cones property. CEG also proved, with a 2-cone argument, the generalization u 0 and u(x) a + p, x in IR n = u(x) =u(0) + p, x. of (LM2). With this generality, it follows that If u 0, φ C 1 and ˆx is a local minimum of u φ, then u is differentiable at ˆx. At last! A result asserting the existence of a derivative at a particular point. But, more importantly, it had become clear that approximation by p is not necessarily the most efficient path to deriving properties of -sub and super harmonic and -harmonic functions. Of course, comparison with cones had already been used by Jensen to derive Lipschitz continuity, and was used contemporaneously by Bhattacharya, (2001), etc., but it was now understood that this approach was made use of all the available information. Recall that in Jensen s organization, he showed that if u =0 fails, then u is not absolutely minimizing for F ; equivalently, if u is absolutely minimizing, then u = 0. Then he used existence/uniqueness in order to establish the converse: if u = 0, then u is absolutely minimizing. CEG proved directly - without 16

reference to existence or uniqueness - that comparison with cones, and hence u = 0, implies that u is absolutely minimizing for F = F. The next piece of evidence in the regularity mystery was provided by C. and Evans, (2001), using tools from CEG and some new arguments, all rather simple. They proved that if u = 0 near x 0 and rj 0,vj(x) = u(x0 + rjx) u(x 0 ) v(x) = lim v j(x), j then v(x) = p, x for some p IR n and all x. Note that since u is Lipschitz in each ball BR( ) x 0 in its domain, each vj is Lipschitz with the same constant in B R/r (0). Thus any j sequence rj 0 has a subsequence along which the vj converge locally uniformly in IR n. In consequence, if x 0 is a point for which rj, zr x 0 = r, u(zr) = max Br(x 0 ) zr x 0 u = lim r 0 zr x 0 exists, then u is differentiable at x 0. However, no one has been able to show that the maximum points have a limiting direction, except Ovidiu Savin, in the case n =2: C 1 regularity of infinity harmonic functions in two dimensions, preprint He doesn t work on the directions directly, and starts from a nice reformulation of the v(x) = p, x result above, which reads 17 lim min max r 0 p =L x r where L = lim Lip(u, Br r 0 u(x 0 + x) u(x 0 ) p, x r ( ) x 0 ), aka, =0, the Lipschitz constant of u at x 0. Savin proves that if n = 2 and u =0,then u C 1! He employs two cone arguments, and we look forward to hearing his talk here. SOME HIGHLIGHTS (A) u is AM for Lip iff it is AM for F iff u = 0 iff it enjoys comparison with cones. (B) If b C( U), then u C(U) u =0. in U and u = b on U. (C) v 0, u 0 = u v max U (D) x 4/3 y 4/3 is AM in IR 2. (u v). (E) u 0, u 0 = u(x) e x y R r u(y). (F) u =0,v(x) = lim rj 0 u(x 0 + rjx) u(x 0 ) (G) If n = 2 and u = 0, then u C 1. rj = v linear. 18

OPEN (A) If u is AM and Lipschitz continuous on IR n, is it linear? (True if n = 2, Savin.) (B) If u is AM, is it C 1? (True if n = 2, Savin.) (C) Can anything, perhaps generic, be said about the structure of AM functions? NOT YET MENTIONED (A) u 0iffr min u is concave. Br(x 0 ) (B) If u 0, u =0inx1 > 0, u x1=0 = 0, then u = kx1. Bhattacharya, to appear. (C) Unique continuation and the strong comparison principle fail for. Both are illustrated taking u(x) = x,v(x) = xn. Both are infinity harmonic on IR n \{0}. They agree only on the ray IR + xn (0,...,xn), while u(x) >v(x) elsewhere. Moreover, - well watch my hand waving if I get this far! OTHER DIRECTIONS There is a lot of interesting work we did not refer to, having restricted our attention to highlights in the setting of the Euclidean distance on IR n. In spirit, this work extends the theory known in this case to other settings. Absolutely minimizing make sense 19 even in metric spaces and existence results are available in this generality (Juutinen, Milman, ). One can take F in more general forms ess inf f(x, u, Du), derive the Aronsson equation (Wu, Juutinen, Barron, Jensen, Wang,..), see if it characterizes the functions sought (Yu, Champion, De Pascale, ), consider geometrical structures (Bieske, Manfredi, Wang,... ), there is an associated eigenvalue problem (Juutinen, Lindqvist, Manfredi, lectures here, ), the topics to be discussed by Le Gruyer,. The field is still in its youth. A TOUR OF THE THEORY Gunnar, Mike and Petri offer you the resource A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Society, just appeared, available online. It develops the basic theory in the generality in which is any norm on IR n, primarily from the point of view of comparison with cones. Tidbits it contains: (i) a simple proof of the most general existence result - U need not be bounded and b grows at most linearly, (ii) a derivation of the Aronsson equation for general norms via comparison with cones, (iii) the proof of (A) above, (iv) the fact that an absolutely minimizing function is constant near a local maximum point, the analogue of x 4/3 y 4/3 for the norm (x, y) = max( x, y ) isx 2 y 2, (v) absolutely minimizing for Lip and F and comparison with cones coincide in general norms,. 20

OPEN Uniqueness in the case of general norms (lots of cases closed 10/5/04!). More generally, any example of nonuniqueness in conditions where it is not obvious whether or not there is uniqueness. (Eg, any linear function vanishing on the line x = 0 is absolutely minimizing in R 2, but it is not uniquely identified by its values on the line. But do two solutions of this problem coincide if their difference is bounded?) 21