Caricatured from the preface of Ian Stewart’s The Problems of PROLOGUE: THE FOLLOWING ARE EQUIVALENT! Mathematics , via Drawbridge Up by H. M. Enzenberger: Second order version: ∆ ∞ u ≥ 0 in U in the viscosity sense , aka, u is “infinity subharmonic”. Mathematician: “It’s one of the most important breakthroughs of First order version: the last decade!” u ( w ) − u ( x ) | Du ( x ) | ≤ max for B r ( x ) ⊂⊂ U r {| w − x | = r } Normal Person: “Can you explain it to me?” aka, “the gradient estimate”. Mathematician: “How can I talk about this advance without men- Zero order version #1: tioning that the theorems only work if the manifolds are finite dimensional para-compact Hausdorff with empty boundary?” � max {| w − x | = r } u ( w ) − u ( y ) � u ( x ) ≤ u ( y ) + | x − y | for x ∈ B r ( y ) ⊂⊂ U r Normal Person: “LIE A LITTLE BIT?” aka, “comparison with cones from above”. L. C. Evans: “In pde, a breakthrough occurs when things that were Zero order version #2 (almost the same as above): formerly impossible become merely difficult and as well as when � � r w ∈ B r ( y ) u ( w ) − u ( y ) ≤ max w ∈ B r ( y ) u ( w ) − u ( x ) max r − | x − y | things that were formerly difficult become easy.” for x ∈ B r ( y ) ⊂⊂ U aka, “the Harnack inequality”. Convexity version: M. B. Maple: “In superconductivity we find three things that fas- r �→ max w ∈ B r ( x ) u ( w ) is convex in r, 0 ≤ r < R = dist ( x, ∂U ) . cinate the human mind: zero, perfection and infinity.” Variational Property: M. G. Crandall: “Thank goodness that Gunnar Aronsson invented the infinity Laplacian. It’s perfect, has “infinity” in its name, and If V ⊂⊂ U , v ∈ C ( V ), u ≥ v in V and u = v on ∂V , then it’s particularly fascinating when the gradient is zero.” �| Du |� L ∞ ( V ) ≤ �| Dv |� L ∞ ( V ) aka, “ u is sub-absolutely minimizing for F ( u ) = �| Du |� L ∞ ” (or some such). 1 2
Here we put F ∞ ( v, U ) = + ∞ if v is not continuous on U or the distributional gradient Dv = ( v x 1 , . . . , v x n ) is not in L ∞ . THE CONTINUING SAGA OF THE INFINITY-LAPLACE Aronsson’s third F we set aside for now. EQUATION Note that above | · | is used for the absolute value on IR as well as the Euclidean norm on IR n . It all began with GUNNAR ARONSSON’s paper: Extension of functions satisfying Lipschitz conditions, Arkiv f¨ or Note that we have indexed these F ′ s by the sets over which they Matematik , 1967. are computed for later convenience. To explain, let us suppose Note that these F ’s are convex, but they are not strictly convex. U ⊂ IR n is open and bounded and b : ∂U → IR Aronsson observed that Lip = F ∞ if U is convex, but this is not generally the case if U is not convex. (“ b ” is a mnemonic for “boundary data”) and we have a functional In the case F = Lip, the problem Minimum was known to have a largest and a smallest solution, via the works of McShane and F : C = { functions v on U such that v = b on ∂U } → [ −∞ , ∞ ] Whitney: and seek to solve the standard problem: Minimum E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. , 1934. Does there exist u ∈ C satisfying F ( u ) ≤ F ( v ) for v in C , and if so, can there be more than one? H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. , 1934. Aronsson considered three choices of F ( v ): the first was the least Lipschitz constant for v These “McShane-Whitney” solutions are: Lip( v, U ) = inf { L ∈ [0 , ∞ ] : | v ( x ) − v ( y ) | ≤ L | x − y | for x, y ∈ U } , Largest solution = min y ∈ ∂U ( b ( y ) + Lip( b, ∂U ) | x − y | ) and the second was the L ∞ norm of the gradient Smallest solution = max y ∈ ∂U ( b ( y ) − Lip( b, ∂U ) | x − y | ) F ∞ ( v, U ) = ess sup | Dv | . U 3 4
It is a simple exercise to show that these functions are solu- This was discovered by the heuristic reasoning: tions and bound all solutions from above and below. They are u is a solution of Minimum for F = F p , where finite iff Lip( b, ∂U ) < ∞ , which we are assuming. Aronsson de- � 1 �� p rived, among other things, interesting information about the set on | Dv | p F p ( v, U ) = � Dv � L p ( U ) = iff which these two functions coincide (which may be only ∂U ) and U 1 the derivatives of any solution on this “contact set”. In particular, p − 2 | Du | 2 ∆ u + ∆ ∞ u = 0 . u = b on ∂U and he established that solutions of Minimum with F = Lip are unique Letting p → ∞ yields ∆ ∞ u = 0. Moreover, F p ( u, U ) → F ∞ ( u, U ) iff there is a function u ∈ C 1 ( U ) ∩ C ( U ) which satisfies if | Du | ∈ L ∞ ( U ). | Du | ≡ Lip( b, ∂U ) in U, u = b on ∂U, He further observed that for u ∈ C 2 , ∆ ∞ u = 0 amounts to which is then the one and only solution. This is a very special if ˙ X ( t ) = Du ( X ( t )), then circumstance. d dt | Du ( X ( t ) | 2 = � D ( | Du ( x ) | 2 ) , Du ( x ) � � The following question naturally arises: is it possible to find a � x = X ( t ) “canonical” Lipschitz extension that would enjoy comparison and = 2(∆ ∞ u )( X ( t )) = 0; stability properties? And furthermore, could this special extension that is, | Du | is constant on the trajectories of the gradient flow of be unique once the boundary data is fixed? We explain Aronsson’s u . successful proposal in this regard. Moreover, Aronsson generalized a notion appearing in works of Thinking in terms of F = F ∞ = Lip on convex sets, Aronsson his in one dimension to introduce the notion of an absolutely min- was led to the now famous pde: imizing function. n � A ( u ) = ∆ ∞ u = u x i u x j u x i x j = 0 . i,j =1 5 6
We formulate this in the generality of the problem Minimum, Aronsson himself made the gap more evident in the paper relating it to the functional F : On the partial differential equation u 2 x u xx +2 u x u y u xy + u 2 y u yy = 0, Arkiv f¨ or Matematik , 1968. u is an absolute minimzer for F in U if for ∀ V ⊂⊂ U in which he produced examples of U , b for which the problem of and v satisfying v = u on ∂V we have F ( u, V ) ≤ F ( v, V ). (#2) has no C 2 solution. This work also contained a penetrating He then went on to prove, for F = Lip, that: analysis of classical solutions of the pde. However, all of these re- sults are false in the generality of viscosity solutions of the equation, (#1) If u ∈ C 2 ( U ), then u is absolutely minimizing in U which appear below as the perfecting instrument of the theory. iff ∆ ∞ u = 0 in U. The best known explicit irregular absolutely minimizing function (#2) If Lip( b, ∂U ) < ∞ , then ∃ u ∈ C ( U ) which is absolutely - outside of solutions of the eikonal equation - was exhibited again by Aronsson, who showed in minimizing in U and satisfies u = b on ∂U . On certain singular solutions of the partial differential equation With the technology of the times, this is about all anyone could u 2 x u xx + 2 u x u y + u 2 have proved. The gaps between (#1) above, which required u ∈ C 2 y u yy = 0, Manuscripta Math., 1984. that u ( x, y ) = x 4 / 3 − y 4 / 3 is absolutely minimizing in R 2 for and (#2), which produced, by a Perron method, a function only known to be Lipschitz continuous, could not be closed at that time. F = Lip AND F = F ∞ . Most of the interesting results for classi- In particular, Aronsson already knew that solutions of the eikonal cal solutions in 2 dimensions are falsified by this example. These equation | Du | = constant, which might not be C 2 , are absolutely include: | Du | is constant on trajectories of the gradient flow, global minimizing. However, he had no way to interpret them as solutions absolutely minimizing functions are linear, and Du cannot vanish of ∆ ∞ u = 0. Moreover, the question of uniqueness of the function unless u is locally constant. A rich supply of other solutions was whose existence Aronsson proved would be unsettled for 26 years. provided as well. 7 8
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