THREE SINGULAR VARIATIONAL PROBLEMS. Lawrence C. Evans Department of Mathematics University of California Berkeley, CA 94720

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1 THREE SINGLAR VARIATIONAL PROBLEMS By Lawrence C. Evans Department of Mathematics niversity of California Berkeley, CA 9470 Some of the means I use are trivial and some are quadrivial. J. Joyce Abstract. We discuss here a variational viewpoint common to three problems in nonlinear PDE: the construction of optimal Lipschitz extensions, the Monge Kantorovich problem, and weak KAM theory for Hamiltonian dynamics. We establish also some interesting analytic estimates. 1. Overview. This expository paper discusses some viewpoints and estimates common to three related singular variational problems, which turn out in asymptotic limits to have quite different interpretations. These are: (I) the construction of optimal Lipschitz extensions of given boundary data, (II) the Monge Kantorovich optimal mass transfer problem, and (III) a form of weak KAM theory for Hamiltonian dynamics. The above quotation from Joyce (cited in the book [J]) depends upon the Latin derivation of the word trivial, from tri (= three) and via (= road or way). It is interesting that there is a trivial (three way) variational principle behind these apparently rather different problems. Perhaps a fourth application remains to be found, so that our method would then be quadrivial. I would like to reemphasize here that this is an expository paper. I have not yet written up fully detailed proofs of some of the assertions herein, and consequently this should be regarded as an informal research announcement. Supported in part by NSF Grant DMS and by the Miller Institute for Basic Research in Science, C Berkeley 1

2 I presented some of these results at an meeting at RIMS in Kyoto, during September, 00. I thank the organizers, and especially Professor Hitoshi Ishii, for their hospitality. I also thank G. Aronsson for correcting an error in my definition of the term L for Problem I. 1.1 Three variational problems. First of all, fix a parameter k>1, which we will later send to infinity. Problem I: Optimal Lipschitz Extensions. Assume for our first problem that we are given a bounded, smooth domain R n and a Lipschitz continuous function g : R n R. Then define (1.1) I 1 [w] := e k Dw dx for functions w in the admissible class (1.) A 1 := {w : Ū R w is Lipschitz continuous, w = g on }. We minimize I 1 [ ] over A 1. Problem II: Optimal Mass Transfer. Let = B(0,R) denote the ball in R n with center 0 and (large) radius R. Assume f : R n R is summable and has compact support, lying within B(0,R). We write f = f + f and suppose the mass balance condition that f + dx = f dy =1. Then define (1.3) I [w] := for w belonging to 1 k e k ( Dw 1) wf dx (1.4) A := {w : R w is Lipschitz continuous,w =0on }. We minimize I [ ] over A. Problem III: Weak KAM Theory. For our last example, let denote the flat unit torus in R n, that is, the unit cube with opposite faces identified. Suppose P R n is fixed and V : R n R is a smooth, -periodic potential. Define P +Dw (1.5) I 3 [w] := +V ) dx on the admissible set e k( (1.6) A 3 := {w : R n R w is Lipschitz continuous and -periodic}. Once again, we minimize I 3 [ ] over A 3.

3 1. Euler Lagrange equations. For Problems I and II, let u k denote a minimizer and for Problem III, write u k := P x + v k ; where v k is a minimizer. The Euler Lagrange equations then take these forms: Problem I: (1.7) div(σ k Du k )=0 in where (1.8) σ k := e k( Du k L k ) for (1.9) L k := Problem II: ( 1/ k log e k Du k dx). (1.10) div(σ k Du k )=f in (1.11) σ k := e k ( Du k 1) Problem III: (1.1) div(σ k Du k )=0 in for (1.13) σ k := e k( Du k +V H k (P )) and (1.14) Hk (P ):= 1 k log e k( Du k +V ) dx. Remark. We call (1.7), (1.10), (1.1) continuity (or transport) equations. Notice that for Problems I and III we have normalized so that σ k 0 satisfies (1.15) σ k dx =1, σ k dx =1 respectively, but have not done so for Problem II. Our goal is to understand for each of our problems what happens in the limit as the parameter k goes to infinity. 3

4 1.3 Limits as k. We describe next the limiting behaviors of u k and σ k : Problem I: Optimal Lipschitz Extensions. For our first problem, we assert this asymptotic behavior: (i) As k, u k u uniformly on Ū and u is the unique viscosity solution of (1.16) { uxi u xj u xi x j =0 in u = g in. (ii) Furthermore, L k L for { g(x) g(y) L := sup dist(x, y) } x, y, x y, where dist(x, y) denotes the distance from x to y within. That is, dist(x, y) :=inf{ length of γ γ is a Lipschitzcurve in Ū connecting x and y}. (iii) Also, σ k σweakly as measures, where σ is a probability measure on Ū such that Du = L σ-a.e. in. (iv) We have (1.17) div(σdu) =0 in. The idea here is to construct an optimal Lipschitz extension into the domain of the given boundary values g, following Aronsson s variational principle that for each subdomain V we should have Du L (V ) Dv L (V ) for each Lipschitz function v satisfying u v on V. The PDE in (1.16) is in effect the Euler Lagrange equation for this sup norm minimization problem. We sometimes write u xi u xj u xi x j =: u, the so called infinity Laplacian. See for instance Aronsson [A], Barron [B], Barron Jensen Wang [B-J-W] for more detailed explanations. Problem II: Optimal Mass Transfer. We examine next Problem II as k : 4

5 (i) As k, u k u locally uniformly on, where (1.18) Du 1 a.e. (ii) Furthermore σ k σweakly as measures and (1.19) div(σdu) =f in. (iii) σ() is the Monge Kantorovich cost of optimally rearranging the probability measure dµ + = f + dx to dµ = f dy. (iv) We also have u xi u xj u xi x j =0 in spt(f). The basic Monge Kantorovich problem asks us to find a mapping s to minimize the cost functional C[r] := x r(x) dµ + (x) R n among one to one mappings r : R n R n that push forward µ + into µ. As explained in Ambrosio [Am], Caffarelli Feldman McCann [C-F-M], [E1], [E-G1], etc., the potential function u can be employed to design an optimal mass allocation plan s. The measure σ is called the transport measure (or the transport density, when it has a density with respect to Lebesgue measure). Problem III: Weak KAM Theory. Finally, we address the asymptotic limit of Problem III: (i) As k, u k u uniformly on and u is a viscosity solution of (1.0) u xi u xj u xi x j = V xj u xj in. (ii) Furthermore H k (P ) H(P ), where H is the effective Hamiltonian in the sense of Lions Papanicolaou Varadhan [L-P-V]. (iii) We have σ k σweakly as measures and (1.1) div(σdu) =0 in. (iv) In addition, (1.) Du + V = H(P ) σ-a.e. A full proof can be found in [E]. As explained in Evans Gomes [E-G], we can regard (1.) as the generalized eikonal equation and (1.1) as the continuity equation corresponding to the dynamics ẍ(t) = DV (x(t)). The support of the measure σ is the projection of the Mather set onto. See also Fathi [F], Mather [Mt], Mather Forni [M-F] for other viewpoints. 5

6 . L and L p bounds. An advantage of the common viewpoint set forth above is that we can at least hope to find analytic methods applicable to several of the problems at once. In this and the next two sections we illustrate some common PDE methods, which apply variously to Problems I III, for deriving useful estimates..1 An L -estimate (Problem III). Consider from Problem III the Euler Lagrange equation (.1) div(σ k Du k )=0 in for (.) σ k = e k( Du k +V H) and u k = P x + v k, v k periodic. Write (.3) h k := Du k + V H. Lemma.1. We have the identity (.4) D u k + k Dh k dσ k = Vdσ k, and consequently (.5) D u k + k Dh k dσ k C, for a constant C independent of k. Here we write dσ k for σ k dx. Proof. To simplify notation, we henceforth drop the subscript k. Owing to (.1) we have 0= (σu xi ) xi u xj x j dx = T n (σu xi ) xj u xi x j dx = σu xi x j u xi x j + σ xj u xi u xi x j dx. Now σ xj = k(u xi u xi x j + V xj )σ = kh xj σ. Therefore 0= σ D u + kσdh (Dh DV ) dx; and this gives D u + k Dh dσ = Dσ DV dx = Vdσ. 6

7 . An L p estimate for the transport density (Problem II). We turn now to Problem II, for which (.6) div(σ k Du k )=f in and (.7) σ k = e k ( Du k 1). We assume that u k has compact support, and u k M for some constant M. The following estimate is from the forthcoming paper [D-E-P]. Lemma.. For each p<, there exists a constant C, depending on p but not k, such that (.8) σ p k dx C( f p dx +1). Proof. 1. We again omit the subscripts k. Let q = p 1 1. Multiply (.6) by σ q u to discover σu xi (σ q u) xi dx = fσ q u dx. Hence σ q+1 Du + qσ q Du Dσu dx = fσ q u dx. Since Du 1ifσ 1, owing to (.7), we can deduce that (.9) σ q+1 dx C f q+1 dx + C σ q Du Dσ dx + C.. We must control the second term on the right-hand side of (.9). To do so, we next multiply our PDE (.6) by div(σ q Du) and integrate by parts: (.10) (σu xi ) xj (σ q u xj ) xi dx = f(σ q u xj ) xj dx. (We are ignoring here a boundary term, which turns out to have a good sign: see [D-E-P] for details.) The term on the left equals (σu xi x j + σ xj u xi )(σ q u xi x j + qσ q 1 σ xi u xj ) dx (.11) = σ q+1 D u + qσ q 1 Du Dσ +(q +1)σ q σ xi u xj u xi x j dx = σ q+1 D u + qσ q 1 Du Dσ (q +1) + σ q 1 Dσ dx, k since (.7) implies σ xi = ku xj x i u xj σ. The term on the right-hand side of (.10) is (.1) f(σ q u xj x j +qσ q 1 σ xj u xj ) dx 1 σ q+1 D u + qσ q 1 Du Dσ dx + C f σ q 1 dx. We combine (.10) (.1) and perform elementary estimates to arrive at (.8). 7

8 3. Bounds and formulas involving Du. 3.1 Detailed mass balance (Problem II). Interesting identities sometime result if we multiply the various transport equations by Φ(Du), where Φ:R n R is an arbitary smooth function. For example, turn again to Problem II: (3.1) div(σ k Du k )=f in, where (3.) σ k = e k ( Du k 1). Lemma 3.1. We have the identity 1 (3.3) Dσ k DΦ(Du k ) dx k Proof. We calculate σu xi Φ(Du) xi dx and the first term on the left is σu xi Φ pj (Du)u xj x i dx = R n u k σ k ν Φ(Du k) dh n 1 = fφ(du k ) dx. σ u ν Φ(Du) dhn 1 = fφ(du) dx, R n σ xj k Φ p j (Du) dx. Remark. This formula suggests that in the limit k, we should have Φ(Du)f dx=0, R n if σ k goes to zero on. Consequently, (3.4) Φ(Du)f + dx = Φ(Du)f dy R n R n for all smooth Φ : R n R. This is a form of detailed mass balance for the Monge Kantorovich problem: see [E-G1]. The basic insight is that the mass of the measure dµ + = f + dx is optimally rearranged into dµ = f dy by moving each mass point in the direction Du. If we formally take Φ=χ B, where B is some set of directions in the unit sphere, the identity (3.4) reads f + dx = f dy A for A := {x R n Du(x) B}, and this is consistent with the foregoing interpretation. 8 A

9 3. Gradient bounds (Problem III). For Problem III we can as in [E] bound the term in the exponential: Lemma 3.. We have the estimate (3.5) Du k + V H k (P )+ C log k. k Proof. Somewhat as in our proof of Lemma., we multiply the PDE by div(σ q Du) and integrate by parts: (σu xi ) xj (σ q u xj ) xi dx =0. As before the term on the left is σ q+1 D u + qσ q Du Dσ +(q +1)σ q σ xi u xj u xi x j dx. Du k( Now σ = e +V H) and so σ xi = k(u xi u xi x j + V xi )σ. Hence 1 σ q 1 Dσ dx σ q Dσ DV dx k and therefore (3.6) σ q 1 Dσ dx Ck σ q+1 dx. sing Sobolev s inequality, we deduce ( σ (q+1)(1+θ) dx) 1 1+θ C A standard Moser iteration implies that Dσ q+1 + σ q+1 dx C(q +1) k T σ q+1 dx. n σ L Ck α for some power α>0. But then k( Du and estimate (3.5) follows. + V H) = log σ C + α log k This estimate, combined with a minimax formula explained in [E], implies H k (P ) H(P ) H k (P )+ C log k. k Therefore the normalization factor (1.14) provides an approximation to the effective Hamiltonian H. 9

10 4. Monotonicity formulas. 4.1 Monotonicity (Problem I). We write Problem I in the form (4.1) div(σ k Du k )=0 in, for (4.) σ k = e k Du k. Lemma 4.1. For each ball B(y, r) we have the identity (4.3) B(y,r) ( Du k n k )σ k dx = r (( u k B(y,r) ν ) 1 k )σ k dh n 1. Proof. The Euler-Lagrange equation (1.7) says (e k Du u xi ) xi = 0, and therefore (4.4) (e k Du (δ ij ku xi u xj )) xi =0 for j =1,...,n. Assume y = 0 and the ball B(0,r) lies within. Multiply (4.4) by x j φ( x ) where φ 1on[0,r ε], φ 0on[r, ) and φ is linear on [r ε, r]. We find 0= B(0,r) e k Du (δ ij ku xi u xj )(δ ij φ + x jx i x φ )dx. Hence (4.5) B(0,r) σ(n k Du )φdx= 1 ε B(0,r)\B(0,r ε) σ( x k Du x )dx. x Let ε 0 to derive (4.3). We can formally interpret this by first renormalizing so that σ k (B(y, r)) 1 and letting σ k σ. Then (4.3) should imply B(y,r) Du dσ = r ( u B(y,r) ν ) dτ, where τ denotes the restriction of σ to the sphere B(y, r). If for instance σ(b 0 (y, r))=0, then our passing to limits in (4.3) as k, before sending ɛ 0, allows us to guess that u ν = 0 almost everywhere on B(y, r) with respect to the measure τ. 10

11 4. Monotonicity (Problem II). The Euler Lagrange PDE for Problem II reads (4.6) div(σ k Du k )=f in, for (4.7) σ k = e k ( Du k 1). Lemma 4.. For each ball B(y, r), (4.8) B(y,r) ( Du k n k )σ k dx = r (( u k B(y,r) ν ) 1 k )σ k dh n 1 + fx Du k dx. B(x,r) Proof. We have (e k ( Du 1) u xi ) xi = f and so (e k ( Du 1) (δ ij ku xi u xj )) xi = kfu xj for j =1,...,n. Again suppose y = 0 and take φ as above. Then k fx Duφ dx = σ(k Du n)φdx+ 1 Du x σ(k x ) dx. B(0,r) B(0,r) ε B(0,r) B(0,r ε) x Let ε 0 and divide by k. At least formally, this identity in the limit k provides some analytic control over the transport density, although we do not here attempt to provide any details. 11

12 References [Am] L. Ambrosio, Lecture notes on optimal transport problems, Preprint: Scuola Normale Superiore (000). [A] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), [B] E. N. Barron, Viscosity solutions and analysis in L, in Nonlinear Analysis, Differential Equations and Control (1999), Dordrecht, [B-J-W] E. N. Barron, R. Jensen and C. Y. Wang, The Euler equation and absolute minimizers of L functionals, Arch. Ration. Mech. Analysis 157 (001), [C-F-M] L. Caffarelli, M. Feldman and R. McCann, Constructing optimal maps for Monge s transport problem as a limit of strictly convex costs, to appear. [D-E-P] L. De Pascale, L. C. Evans and A. Pratelli, PDE estimates for transport densities, forthcoming.. [E1] L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer (survey paper), available at the website of LCE, at math.berkeley.edu. [E] L. C. Evans, Some new PDE methods for weak KAM theory, to appear in Calculus of Variations. [E-G1] L. C. Evans and W. Gangbo, Differential equations methods in the Monge Kantorovich mass transfer problem, Memoirs American Math. Society # (1999). [E-G] L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics I, Archive Rational Mech and Analysis 157 (001), [F] A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sr. I Math. 34 (1997), [J] K. Jackson, Invisible Forms, Picador, [L-P-V] P.-L. Lions, G. Papanicolaou, and S. R. S. Varadhan, Homogenization of Hamilton Jacobi equations, unpublished, circa [Mt] J. Mather, Minimal measures, Comment. Math Helvetici 64 (1989), [M-F] J. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems, Transition to Chaos in Classical and Quantum Mechanics, Lecture Notes in Math 1589 (S. Graffi, ed.), Sringer,