Semiconcave Functions, Hamilton Jacobi Equations, and Optimal Control. Piermarco Cannarsa & Carlo Sinestrari

Transcription

1 Semiconcave Funcions, Hamilon Jacobi Equaions, and Opimal Conrol Piermarco Cannarsa & Carlo Sinesrari May 16, 2003

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3 Preface A gifed Briish crime novelis 1 once wroe ha mahemaics is like one of hose languages ha are simple, sraighforward and logical in he early sages, bu which rapidly spiral ou of conrol in a frenzy of idioms, oddiies, idiosyncrasies and excepions o he rule which even naive speakers canno always ge righ, never mind explain. In fac, providing evidence o conradic such a saemen has been one of our guidelines in wriing his monograph. I may hen be recommendable o describe, righ from he beginning, he essenial objec of our ineres, ha is, semiconcaviy, a propery ha plays a cenral role in opimizaion. There are differen possible ways of inroducing semiconcaviy. For insance, one can say ha a funcion u is semiconcave if i can be represened, locally, as he sum of a concave funcion plus a smooh one. Thus, semiconcave funcions share many regulariy properies wih concave funcions, bu include many oher significan examples. Roughly speaking, semiconcave funcions can be obained as envelopes of smooh funcions, jus in he same way as concave funcions are obained as envelopes of linear funcions. Typical examples of semiconcave funcions are: he disance funcion from a closed se S IR n, he leas eigenvalue of a symmeric marix depending smoohly on parameers, and he so-called inf-convoluions. Anoher example we are paricularly ineresed in because of applicaions o conrol heory, are he generalized soluions of Hamilon-Jacobi-Bellman equaions. A his poin, he reader may wonder why we concenrae on semiconcaviy raher han on he equivalen ye more usual noion of semiconvexiy. The answer is ha his choice beer fis minimizaion, as opposed o maximizaion, and his is he formulaion we have adoped for he opimizaion problems of ineres here. Ineres in semiconcave funcions was iniially moivaed by research on nonlinear parial differenial equaions. In fac, i was exacly in classes of semiconcave funcions ha he firs global exisence and uniqueness resuls were obained for Hamilon-Jacobi-Bellman equaions, see Douglis [67] and Kruzhkov [97, 98, 100]. Aferwards, more powerful uniqueness heo- 1 M. Dibdin, Blood rain, Faber and Faber, London, iii

4 iv ries, such as viscosiy soluions and minimax soluions, were developed. Neverheless, semiconcaviy mainains is imporance even in modern PDE heory, as he maximal ype of regulariy ha can be expeced for cerain nonlinear problems. As such, i has been invesigaed in he modern exbooks on Hamilon-Jacobi equaions by Lions [107], Bardi and Capuzzo Dolcea [20], Fleming and Soner [79], and Li and Yong [106]. In he conex of nonsmooh analysis and opimizaion, semiconcave funcions have also received aenion under he name of lower-c k funcions, see, e.g., Rockafellar [120]. We adop, here, a differen perspecive. Firs, in Chapers 2,3 and 4, we develop he heory of semiconcave funcions wihou aiming a one specific applicaion, bu as a opic in convex analysis wih is own ineres. The exposiion ranges from well-known properies for he expers in he field ha are exposed here in a comprehensive way for he firs ime o recen resuls, like he laes developmens in he analysis of singulariies. Then, in Chapers 5 o 8, we discuss some ypical problems for which semiconcaviy plays an imporan role, such as Hamilon-Jacobi equaions and conrol heory. Moreover, he book opens wih an inroducory chaper analyzing a model problem from calculus of variaions: his allows us o presen in a simple conex some of he main ideas of he res of he work. An aracive feaure of he presen exposiion is ha he proofs even of he mos advanced resuls require lile more han a sandard background in real analysis and PDE s. We use noions and echniques from several differen fields, like conrol heory, nonsmooh analysis, geomeric measure heory and viscosiy soluions; however, we have ried o keep he book as self-conained as possible, by providing all definiions and proofs of he basic resuls, and giving deailed references for he more advanced ones. Thus, we hope ha he book can be of ineres for differen kinds of readers. Researchers in opimal conrol heory and Hamilon-Jacobi equaions will find he recen progress of his heory as well as a sysemaic collecion of classical resuls, for which a precise ciaion is a imes hard o find in he lieraure. For readers a graduae level, learning he basic properies of semiconcave funcions can also be an occasion o become familiar wih imporan fields of modern analysis like he ones we menioned above. A deailed descripion of he conens is provided a he beginning of each chaper. Here, we would like o propose shorcus for readers wih specific ineress. The firs secion of chaper 2 and mos of chaper 3 are essenial for he comprehension of anyhing ha follows. Chaper 4, on singulariies, could be omied on firs reading. A his poin, he PDEoriened reader could move on o chaper 5 on HJB equaions and, hen, o chaper 6 on calculus of variaions, where sharp regulariy resuls are obained for cerain classes of HJB equaions. On he oher hand, a direc

5 pah o dynamic opimizaion would go direcly o chaper 6 and/or o chapers 7 and 8 dealing wih finie horizon opimal conrol problems and opimal exi ime problems, respecively. v

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7 Conens 1 A model problem Semiconcave funcions A problem in calculus of variaions The Hopf formula Hamilon Jacobi equaions The mehod of characerisics Semiconcaviy of Hopf s soluion Semiconcaviy and enropy soluions Semiconcave funcions Definiion and basic properies Examples Special properies of SCL (A) A differenial Harnack inequaliy A generalized semiconcaviy esimae Generalized gradiens and semiconcaviy Generalized differenials Direcional derivaives The superdifferenial of a semiconcave funcion Marginal funcions Inf convoluions Proximal analysis and semiconcaviy Singulariies of semiconcave funcions Recifiabiliy of he singular ses Propagaion along Lipschiz arcs Singular ses of higher dimension Applicaion o he disance funcion vii

8 viii 5 Hamilon Jacobi equaions The mehod of characerisics Viscosiy soluions Semiconcaviy and viscosiy Propagaion of singulariies Generalized characerisics Examples Calculus of variaions Exisence of minimizers Necessary condiions and regulariy The problem wih one free endpoin The value funcion The singular se of u Recifiabiliy of Σ Opimal conrol problems The Mayer problem The value funcion Opimaliy condiions The Bolza problem Conrol problems wih exi ime Opimal conrol problems wih exi ime Lipschiz coninuiy and semiconcaviy Semiconvexiy resuls in he linear case Opimaliy condiions A 297 A.1 Convex ses and convex funcions A.2 The Legendre ransform A.3 Hausdorff measure and recifiable ses A.4 Ordinary differenial equaions A.5 Se-valued analysis A.6 BV funcions

9 Chaper 1 A model problem The purpose of his chaper is o ouline some of he main opics of he book hrough he analysis of a simple problem in calculus of variaions. The sudy of his model problem allows us o inroduce he dynamic programming approach and o show how he class of semiconcave funcions naurally appears in his conex. In Secion 1.1 we inroduce semiconcave funcions and give some equivalen definiions. Then, in Secion 1.2 we sae our variaional problem, give he dynamic programming principle and define he value funcion associaed wih he problem. In Secion 1.3, we resric our aenion o he case where he inegrand has no explici (, x) dependence; in his case he value funcion admis a simple represenaion formula due o Hopf. In Secion 1.4 we observe ha he value funcion is a soluion of a specific parial differenial equaion, called he Hamilon Jacobi (or someimes Hamilon Jacobi Bellman) equaion. However, he equaion is no saisfied in a classical sense. In fac, he value funcion in general is no differeniable everywhere, bu only Lipschiz coninuous, and he equaion holds a he poins of differeniabiliy. Such a propery is no sufficien o characerize he value funcion, since a Hamilon Jacobi equaion may have infiniely many Lipschiz coninuous soluions aking he same iniial daa. Before seeing how o handle his difficuly, we give in Secion 1.5 an accoun of he classical mehod of characerisics for Hamilon Jacobi equaions. This echnique gives in an elemenary way a local exisence resul for smooh soluions, and a he same ime shows ha no global smooh soluion exiss in general. Alhough he mehod is compleely independen of he conrol heoreic inerpreaion of he equaion, here is an ineresing connecion beween he soluions of he characerisic sysem and he opimal rajecories of he corresponding problem in conrol or calculus of variaions. In Secion 1.6 we use he semiconcaviy propery o characerize he 1

10 2 A model problem value funcion among he many possible soluions of he Hamilon Jacobi equaion. In fac, we prove ha he value funcion is semiconcave, and ha semiconcave Lipschiz coninuous soluions of Hamilon Jacobi equaions are unique. We conclude he chaper by describing, in Secion 1.7, he connecion beween Hamilon-Jacobi equaions and anoher class of parial differenial equaions, called hyperbolic conservaion laws. In he one-dimensional case he wo classes of equaions are sricly relaed; in paricular, we show ha semiconcaviy corresponds o a well-known esimae for soluions of conservaion laws due o Oleinik. Le us menion ha a more general reamen of he problem in calculus of variaions inroduced here, including a deailed analysis of he singulariies of he value funcion, will be given in Chaper Semiconcave funcions Before saring he analysis of our variaional problem, le us inroduce semiconcave funcions, which are he cenral opic in his monograph and will play an imporan role laer in his chaper. I is convenien o consider, firs, a special class of semiconcave funcions, while he general definiion will be given in Chaper 2. Here and in he following we wrie [x, y] o denoe he segmen wih endpoins x, y, for any x, y IR n. Definiion Le A IR n be an open se. We say ha a funcion u : A IR is semiconcave wih linear modulus if i is coninuous in A and here exiss C 0 such ha u(x + h) + u(x h) 2u(x) C h 2, (1.1) for all x, h IR n such ha [x h, x + h] A. The consan C above is called a semiconcaviy consan for u in S. Remark The above definiion is ofen aken in he lieraure as he definiion of a semiconcave funcion. For us, insead, i is a paricular case of Definiion 2.1.1, where he righ-hand side of (1.1) is replaced by a erm of he form h ω( h ) for some funcion ω( ) such ha ω(ρ) 0 as ρ 0. The funcion ω is called modulus of semiconcaviy, and herefore we say ha a funcion which saisfies (1.1) is semiconcave wih a linear modulus. Semiconcave funcions wih a linear modulus admi some ineresing characerizaions, as he nex resul shows.

11 Chaper 1 3 Proposiion Given u : A IR, wih A IR n open convex, and given C 0, he following properies are equivalen: (a) u is semiconcave wih a linear modulus in A wih semiconcaviy consan C; (b) u saisfies λu(x) + (1 λ)u(y) u(λx + (1 λ)y) C λ(1 λ) 2 for all x, y such ha [x, y] A and for all λ [0, 1]; (c) he funcion x u(x) C 2 x 2 is concave in A; x y 2, (1.2) (d) here exis wo funcions u 1, u 2 : A IR such ha u = u 1 + u 2, u 1 is concave, u 2 C 2 (A) and saisfies D 2 u 2 C; (e) for any ν IR n such ha ν = 1 we have 2 u ν 2 C in A in he sense of disribuions, ha is A u(x) 2 φ ν (x) dx C φ(x) dx, φ C A 2 0 (A), φ 0; (f) u can be represened as u(x) = inf i I u i (x), where {u i } i I is a family of funcions of C 2 (A) such ha D 2 u i C for all i I. Proof Le us se v(x) = u(x) C 2 x 2. Using he ideniy we see ha (1.1) is equivalen o x + h 2 + x h 2 2 x 2 = 2 h 2, v(x + h) + v(x h) 2v(x) 0 for all x, h such ha [x h, x + h] A. I is well known (see Proposiion A.1.2) ha such a propery, ogeher wih coninuiy, is equivalen o he concaviy of v, and so (a) and (c) are equivalen. The equivalence of (b) and (c) is proved analogously. In fac, using he ideniy λ x 2 + (1 λ) y 2 λx + (1 λ)y 2 = λ(1 λ) x y 2, we see ha inequaliy (1.2) is equivalen o λv(x) + (1 λ)v(y) v(λx + (1 λ)y) 0

12 4 A model problem for all x, y such ha [x, y] A and for all λ [0, 1]. Now le us show he equivalence of (c) and (d). If (c) holds, hen (d) immediaely follows aking u 1 (x) = u(x) C 2 x 2 and u 2 (x) = C 2 x 2. Conversely, if (d) holds, hen for any uni vecor ν we have 2 (u ν 2 2 C ) 2 x 2 = 2 u 2 ν C 0, 2 which implies ha u 2 (x) C 2 x 2 is concave. Thus, u(x) C 2 x 2 is concave since i is he sum of he wo concave funcions u 1 (x) and u 2 (x) C 2 x 2. The equivalence beween (c) and (e) is an easy consequence of he characerizaion of concave funcions as he funcions having nonposiive disribuional hessian. Finally, le us prove he equivalence of (c) and (f). We recall ha any concave funcion can be wrien as he infimum of linear funcions (see Corollary A.1.14). Thus, if (b) holds, we have ha u(x) C 2 x 2 = inf i I v i (x), where he v i s are linear. Therefore, u(x) = inf i I u i (x), where u i (x) = v i (x) + C 2 x 2, and his proves (e). Conversely, assume ha (e) is saisfied. Then, seing v i (x) = u i (x) C 2 x 2, we see ha 2 ννv i 0 for all ν IR n, and so v i is concave. Therefore u(x) C 2 x 2 is concave, being he infimum of concave funcions, and his proves (b). From he previous proposiion one can have an inuiive idea of he behavior of semiconcave funcions wih a linear modulus. Propery (e) shows ha hey are he funcions whose second derivaives are bounded above, in conras wih concave funcions whose second derivaives are nonposiive. Propery (d) shows ha a semiconcave funcion can be regarded as a smooh perurbaion of a concave funcion: hus, is graph can have a nonconcave shape in he smooh pars, bu any corner poins upwards, as for concave funcions. Propery (f) gives a firs explanaion of why semiconcave funcions naurally occur in minimizaion problems. Examples of semiconcave funcions will be given hroughou he book and in paricular in Chaper 2. We conclude he secion, insead, wih a ypical example of a funcion which is no semiconcave. Example The funcion u(x) = x is no semiconcave in any open se conaining 0. In fac, inequaliy (1.1) is violaed for any C > 0 if one akes x = 0 and h small enough. More generally, we find ha u(x) = x α, is no semiconcave wih a linear modulus if α < 2; we will see, however, ha, if α > 1, i is semiconcave according o he general definiion which will be given in Chaper 2.

13 Chaper A problem in calculus of variaions We now sar he analysis of our model problem. Given 0 < T +, we se Q T =]0, T[ IR n. We suppose ha wo coninuous funcions L : Q T IR n IR, u 0 : IR n IR are given. The funcion L will be called he running cos, or lagrangian, while u 0 is called he iniial cos. We assume ha boh funcions are bounded from below. For fixed (, x) Q T, we inroduce he se of admissible arcs and he cos funcional A(, x) = {y W 1,1 ([0, ]; IR n ) : y() = x} J [y] = Then we consider he following problem: 0 L(s, y(s), ẏ(s)) ds + u 0 (y(0)). minimize J [y] over all arcs y A(, x). (1.3) Problems of his form are classical in calculus of variaions. In he case we are considering he iniial endpoin of he admissible rajecories is free, and he erminal one is fixed. Cases where he endpoins are boh fixed or boh free are also ineresing and could be sudied by similar echniques, bu will no be considered here. The firs sep in he dynamic programming approach o he above problem is he inroducion of he value funcion. Definiion The funcion u : Q T IR defined as u(, x) = inf J [y] (1.4) y A(,x) is called he value funcion of he minimizaion problem (1.3). By our assumpions u is finie everywhere. In addiion we have u(0, x) = u 0 (x). (1.5) The basic idea of he approach is o show ha u admis an alernaive characerizaion as he soluion of a suiable parial differenial equaion, and hus i can be obained wihou referring direcly o he definiion. Once u is known, he minimizaion problem is subsanially simplified. The following resul is called Bellman s opimaliy principle or dynamic programming principle and is he saring poin for he sudy of he properies of u.

14 6 A model problem Theorem Le (, x) Q T and y A(, x). Then, for all [0, ], u(, x) L(s, y(s), ẏ(s)) ds + u(, y( )). (1.6) In addiion, he arc y is a minimizer for problem (1.3) if and only if equaliy holds in (1.6) for all [0, ]. Proof For fixed [0, ], le z be any arc in W 1,1 ([0, ]; IR n ) such ha z( ) = y( ). If we se { z(s), s [0, ξ(s) = ], y(s), s [, ], we have ha ξ A(, x) and herefore u(, x) J [ξ] = L(s, y, ẏ) ds + L(s, z, ż) ds + u 0 (z(0)). 0 Taking he infimum over all z A(, y( )) we obain (1.6). If (1.6) holds as an equaliy for all [0, ], hen choosing = 0 yields ha y is a minimizer for J. Conversely, if y is a minimizer we find, by he definiion of u and by inequaliy (1.6), L(s, y, ẏ) ds + u 0 (y(0)) = u(, x) L(s, y, ẏ) ds + u(, y( )) (1.7) 0 for any given [0, ]. This implies ha J [y] u(, y( )). Since by definiion J [y] u(, y( )), we mus have equaliy in (1.7), and herefore also in (1.6). We can give a sharper formulaion of he dynamic programming principle, as in following resul. Theorem Le (, x) Q T. Then, for all [0, ], { } u(, x) = inf L(s, y(s), ẏ(s)) ds + u(, y( )) y A(,x). (1.8) Proof Given ε > 0, le y A(, x) be such ha Then u(, x) u(, x) + ε 0 L(s, y, ẏ) ds + L(s, y, ẏ) ds + u 0 (y(0)). 0 L(s, y, ẏ) ds + u 0 (y(0)) ε L(s, y, ẏ) ds + u(, y( )) ε. By he arbirariness of ε we deduce ha u(, x) is greaer han or equal o he righ hand side of (1.8). The converse inequaliy follows from Theorem

15 Chaper The Hopf formula From now on we consider he special case of L(, x, q) = L(q) and T = +. We assume ha L(q) (i) L is convex and lim = + q q (ii) u 0 Lip (IR n ). (1.9) Then we can show ha he value funcion of our problem admis a simple represenaion formula, called Hopf s formula. Theorem Under hypoheses (1.9) he value funcion u saisfies for all (, x) Q T. [ u(, x) = min L z IR n ( ) x z ] + u 0 (z) (1.10) Proof Observe ha he minimum in (1.10) exiss hanks o hypoheses (1.9). Le us denoe by v(, x) he lef hand side of (1.10). For fixed (, x) Q T and z IR n, le us se Then y A(, x) and herefore y(s) = z + s (x z), 0 s. ( ) x z u(, x) J [y] = L + u 0 (z). Taking he infimum over z we obain ha u(, x) v(, x). To prove he opposie inequaliy, le us ake ζ A(, x). From Jensen s inequaliy i follows ha L ( ) x ζ(0) ( 1 = L 0 ) ζ(s) ds 1 L( ζ(s)) ds 0 and herefore ( ) x ζ(0) v(, x) u 0 (ζ(0)) + L J [ζ]. Taking he infimum over ζ A(, x) we conclude ha v(, x) u(, x). Using Hopf s formula we can prove a firs regulariy propery of u.

16 8 A model problem Theorem Under he assumpions (1.9) he value funcion u is Lipschiz coninuous in Q T. More precisely, we have u(, x ) u(, x) L 0 x x + L 1, (1.11) where L 0 = Lip (u 0 ) and L 1 0 is a suiable consan. Proof Le us firs observe ha (1.10) implies u(, x) u 0 (x) L(0) (1.12) for all (, x) wih > 0. Le us now ake (, x), (, x ) Q T and le y IR n be such ha ( ) x y u(, x) = L + u 0 (y). Then we find, using (1.12), ( ) x y L = u(, x) u 0(x) x y L(0) + L 0. + u 0(x) u 0 (y) Since L is superlinear, we can find a consan C 1 > 0 depending on L 0 and L(0) such ha L(q) L(0)+L 0 q if q > C 1. Then he previous inequaliy implies Le us now se Then we have x y C 1. (1.13) y = x x y. x y = x y, y y = x x + ( ) x y, and so Hopf s formula (1.10), ogeher wih (1.13), yields ( ) x y u(, x ) u(, x) ( )L + u 0 (y ) u 0 (y) max L(q) + L 0 y y q C 1 ( ) L 0 C 1 + max q C 1 L(q) + L 0 x x.

17 Chaper 1 9 By inerchanging he role of (x, ) and (x, ), we obain he reverse inequaliy. This proves he conclusion in he case of, > 0. If we have, for insance, = 0, we can esimae ( ) u(, x) u(0, x ) = x y L + u 0 (y) u 0 (x ) max L(q) + L 0 ( y x + x x ) q C 1 ( ) max q C 1 L(q) + L 0 C 1 + L 0 x x. A well-known heorem due o Rademacher (see e.g. [69, 70, 14]) assers ha a Lipschiz coninuous funcion is differeniable almos everywhere, and so he previous resul immediaely implies Corollary Under hypoheses (1.9), he value funcion u is differeniable a.e. in Q T. On he oher hand, i is easy o see ha, in general, u fails o be everywhere differeniable, even if u 0 and L are differeniable. Example Le us consider problem (1.4) wih n = 1, L(q) = q 2 /2 and he iniial cos given by z 2 if z < 1 u 0 (z) = 1 2 z if z 1. Observe ha u 0 is of class C 1 bu no C 2. This is no essenial for he behavior we are going o describe; one can build examples wih C daa exhibiing similar properies (see Example 6.3.5). Le us compue he value funcion using Hopf s formula. We find ha he minimum in (1.10) is aained a z = x 1 2 if < 1/2 and x < 1 2 z = x + sgn(x)2 if x or if 1/2, whence x2 if < 1/2 and x < 1 2 u(, x) = ( x + ) if x or if 1/2. Therefore u(, x) is no differeniable a he poins of he form (,0) wih 1/2.

18 10 A model problem 1.4 Hamilon Jacobi equaions In his secion we inroduce a parial differenial equaion which is solved by he value funcion of our variaional problem. We assume hroughou ha hypoheses (1.9) are saisfied. We use he noaion u = u, u = ( u x 1,..., u x n Theorem Le u be differeniable a a poin (, x) Q T. Then u (, x) + H( u(, x)) = 0, (1.14) ). where H(p) = sup q IR n [p q L(q)]. (1.15) Equaion (1.14) is called he Hamilon Jacobi equaion of our problem of calculus of variaions. In he erminology of conrol heory, such an equaion is also called Bellman s equaion or dynamic programming equaion. The funcion H is called he hamilonian. In general, a funcion defined as in (1.15) is called he Legendre ransform of L (see Appendix A.1). Proof Le (, x) be a poin a which u is differeniable. Given q IR n, s > 0, le us se y(τ) = x + (τ )q. Then we obain from (1.6) u( + s, x + sq) +s L(ẏ(τ))dτ + u(, x), which implies u( + s, x + sq) u(, x) L(q). s Leing s 0 we obain u (, x) + q u(, x) L(q) 0, and we deduce, by he arbirariness of q, u (, x) + H( u(, x)) 0. To prove he converse inequaliy, le us ake ˆx IR n such ha ( ) x ˆx u(, x) = L + u 0 (ˆx). Such an ˆx exiss by Hopf s formula. We hen se, for s [0, [, x s = s ( x + 1 s ) ˆx.

19 Chaper 1 11 Since (x ˆx)/ = (x s ˆx)/s, we deduce from Hopf s formula ( ) ( ) x ˆx xs ˆx u(, x) u(s, x s ) L + u 0 (ˆx) sl + u 0 (ˆx) s ( ) x ˆx = ( s)l. Dividing by s and leing s we obain u (, x) + x ˆx ( ) x ˆx u(, x) L 0, and we conclude ha u (, x) + H( u(, x)) 0. From Corollary and Theorem i follows ha he value funcion u given from (1.10) saisfies u + H( u) = 0 (, x) IR + IR n a.e. (1.16) u(0, x) = u 0 (x) x IR n. The involuive characer of he Legendre ransform implies ha he correspondence beween calculus of variaions and Hamilon Jacobi equaions is valid in boh direcions. More precisely, le H, u 0 be given such ha H(p) (i) H is convex and lim = + p p (ii) u 0 Lip (IR n ) (1.17) and suppose ha we wan o solve problem (1.16). To his purpose, we can define L o be he Legendre ransform of H: L(q) = max p IR n[p q H(p)]. Then (see Theorem A.2.3) L saisfies propery (1.9)-(i) and H is he Legendre ransform of L, i.e. (1.15) holds. Therefore, Hopf s formula (1.10) yields a Lipschiz funcion u ha solves (1.16). The propery of solving problem (1.16) almos everywhere, however, is no enough o characerize he value funcion u. Indeed, such a problem can have more han one soluion in he class Lip(IR + IR n ), as he nex example shows. Example The problem u + u 2 x = 0 u(0, x) = 0 (, x) IR + IR a.e. x IR (1.18)

20 12 A model problem admis he soluion u 0. However, for any a > 0, he funcion u a defined as 0 if x a u a (, x) = a x a 2 if x < a is a Lipschiz funcion saisfying he equaion almos everywhere ogeher wih is iniial condiion. Simple examples like he one above show ha he propery of solving he equaion almos everywhere is oo weak and does no suffice o provide a saisfacory noion of generalized soluion. I is herefore desirable o find addiional condiions o ensure uniqueness and characerize he value funcion among he Lipschiz coninuous soluions of he equaion. A possible way of doing his relies on he semiconcaviy propery and will be pursued in Secion The mehod of characerisics We describe in his secion he mehod of characerisics, which is a classical approach o he sudy of firs order parial differenial equaions like he Hamilon-Jacobi equaion (1.16). This mehod explains why such equaions do no possess in general smooh soluions for all imes, and has some ineresing connecions wih he variaional problem associaed o he equaion. A more general reamen of hese opics will be given in Secion 5.1. Suppose ha H, u 0 are in C 2 (IR n ), and suppose ha we already know ha problem (1.16) has a soluion u of class C 2 in some srip Q T. For fixed z IR n, le us denoe by X(; z) he soluion of he ordinary differenial equaion (here he do denoes differeniaion wih respec o ) Ẋ = DH( u(, X)), X(0) = z. (1.19) Such a soluion is defined in some maximal inerval [0, T z [ (alhough i will laer urn ou ha T z = T for all z). The curve (, X(; z)) is called he characerisic curve associaed wih u and saring from he poin (0, z). Le us now se U(; z) = u(, X(; z)), P(; z) = u(, X(; z)). (1.20) Then, using he fac ha u solves problem (1.16) we find ha U = u (, X) + u(, X) Ẋ = H(P) + DH(P) P, P = u (, X) + 2 u(, X)Ẋ = (u + H( u))(, X) = 0.

21 Chaper 1 13 Therefore P is consan, and so he righ hand side of (1.19) is also consan. Thus, X is defined in [0, T[ and we can compue explicily X, U, P obaining P(; z) = Du 0 (z) X(; z) = z + DH(Du 0 (z)) U(; z) = u 0 (z) + [DH(Du 0 (z)) Du 0 (z) H(Du 0 (z))]. (1.21) Observe ha he righ hand side of (1.21) is no longer defined in erms of he soluion u, bu only depends on he iniial value u 0. This suggess ha, even wihou assuming in advance he exisence of a soluion, one can use hese formulas o define one. As we are now going o show, such a consrucion can be in general carried ou only locally in ime. We need he following classical resul abou he global inveribiliy of maps (see e.g. [11, Th ]). Theorem Le F : IR n IR n be of class C 1 and proper (ha is, F 1 (K) is compac whenever K is compac). If de DF(x) 0 for all x IR n, hen F is a global C 1 -diffeomorphism from IR n ono iself. As a consequence, we obain he following inveribiliy resul for maps depending on a parameer. Theorem Le Φ : [a, b] IR n IR n be of class C 1. Suppose ha here exiss M > 0 such ha Φ(, z) z M for all (, z) [a, b] IR n. Suppose also ha he jacobian wih respec o he z-variable D z F(, z) has nonzero deerminan for all (, z). Then here exiss a unique map Ψ : [a, b] IR n IR n of class C 1 such ha Φ(,Ψ(, x)) = x for all (, x) [a, b] IR n. Proof For fixed [a, b], define F(z) = Φ(, z). Then F : IR n IR n is of class C 1 and de F(z) 0 for all z by our assumpions. To show ha F is proper, le us ake any compac se K IR n. Then F 1 (K) is closed by he coninuiy of F. To prove ha i is also bounded, le us denoe by d he diameer of K and ake any z 1, z 2 in F 1 (K). Then we have, by our assumpions on Φ, z 2 z 1 z 2 F(z 2 ) + z 1 F(z 1 ) + F(z 2 ) F(z 1 ) 2M + d, showing ha F 1 (K) is bounded. Hence F is proper. Now we can apply Theorem o obain ha F has a global inverse. Since he same holds for all, we obain ha here exiss Ψ(, x) such ha Φ(, Ψ(, x)) = x for all (, x) [a, b] IR n. I remains o prove ha Ψ is regular wih respec o boh argumens. For his purpose, we se Φ(, z) = (, Φ(, z)) and Ψ(, x) = (, Φ(, x)).

22 14 A model problem Then Φ and Ψ are reciprocal inverse. Since Φ is of class C 1, we have ha Ψ (and herefore Ψ) is also of class C 1 provided he Jacobian of Φ has nonzero deerminan. Bu his follows from our hypoheses on Φ, since we have D Φ(, z) = ( 1 0 Φ(, z) D z Φ(, z) We can now give he local exisence resul for classical soluions based on he mehod of characerisics. Theorem Le u 0, H C 2 (IR n ) be given, and suppose ha Du 0 and D 2 u 0 are bounded. Le us se T = sup{ > 0 : I + D 2 H(Du 0 (z))d 2 u 0 (z) is inverible for all z IR n }. Then problem (1.16) has a unique soluion u C 2 ([0, T [ IR n ). Proof Le X, U, P be defined as in (1.21). Then, for any T < T, we can apply Theorem o he map X in [0, T] IR n. In fac, he jacobian X z (; z) = I + D 2 H(Du 0 (z))d 2 u 0 (z) is inverible by hypohesis, and he quaniy X(; z) z = DH(Du 0 (z)) is uniformly bounded. Therefore we can find a map Z(; x) of class C 1 such ha X(; Z(; x)) = x for all (, x) [0, T] IR n. Afer seing u(, x) = U(; Z(; x)), (, x) [0, T] IR n, le us prove ha u is a C 2 soluion of problem (1.16). To check ha he iniial condiion is saisfied, i suffices o observe ha X(0; z) = z and U(0; z) = u 0 (z) for all z; herefore Z(0; x) = x and u(x, 0) = u 0 (x) for all x. The verificaion ha u is of class C 2 and saisfies he equaion is echnical bu sraighforward, as we now show. From he definiion i is clear ha u is of class a leas C 1. Le us compue is derivaives. By (1.21) we have U z (; z) = Du 0 (z) + Du 0 (z) D 2 H(Du 0 (z)) D 2 u 0 (z) = P(; z) X z (; z), where he subscrip denoes parial differeniaion. This implies, by he definiion of Z, We also find u(, x) = U z (; Z(; x)) Z x (; x) = P(; Z(, x)). (1.22) u (, x) = U (; Z(, x)) + U z (; Z(, x)) Z (; x) = DH(P) P H(P) + PX z Z, ).

23 Chaper 1 15 where we have wrien for simpliciy P, Z, ec. insead of P(; Z(, x)), Z = Z(; x) respecively. Keeping ino accoun ha X(; Z(; x)) x we obain X (; Z) + X z (; Z)Z = 0. Therefore, since X = DH(P) by (1.21), u (, x) = X P H(P) + P X z Z = H(P). This equaliy, ogeher wih (1.22), implies ha u C 2 and saisfies problem (1.16). Uniqueness follows from he remarks a he beginning of his secion. Indeed, if we have anoher soluion v, we can define he characerisic curves associaed o v, which are also given by (1.21), since hey only depend on H and u 0. In paricular, we have v(, X(; z)) = U(, z) for all, z. Therefore, v(, x) = v(, X(; Z(; x))) = U(; Z(, x)) = u(, x) for all (, x) [0, T] IR n, showing ha u and v coincide. Le us recall an elemenary propery from linear algebra. Lemma Le B be a symmeric posiive semidefinie n n-marix. Then, for any v IR n we have Bv, v = 0 if and only if Bv = 0. Proof Le e 1,..., e n be an orhonormal basis of eigenvecors of B, and le λ 1,..., λ n be he corresponding eigenvalues. If we se v i = v e i, we have n Bv, v = λ i vi 2. i=1 Since B is posiive semidefinie, we have λ i 0 for all i and so all erms in he above sum are nonnegaive. If Bv, v = 0, hen λ i vi 2 = 0 for all i = 1,..., n. Bu hen also λ 2 i v2 i = 0 for all i, and we deduce n Bv 2 = Bv, Bv = λ 2 i v2 i = 0. i=1 Corollary Le u 0 and H be as in Theorem Se M 0 = sup z IR n Du 0 (z), M 1 = sup z IR n D 2 u 0 (z), M 2 = sup p M 0 D 2 H(p). Then problem (1.16) has a C 2 soluion a leas for ime [0, (M 1 M 2 ) 1 [. If H and u 0 are boh convex (or boh concave) hen he problem has a C 2 soluion for all posiive imes.

24 16 A model problem Proof If M 1, M 2 are defined as above, he marix D 2 H(Du 0 (z))d 2 u 0 (z) has norm less han M 1 M 2. Thus, if < (M 1 M 2 ) 1 we have D 2 H(Du 0 (z))d 2 u 0 (z)v < v for all v IR n, v 0, showing ha he marix I + D 2 H(Du 0 (z))d 2 u 0 (z) is inverible. Le us now prove he las par of he saemen in he case when H, u 0 are convex (he concave case is compleely analogous). I suffices o show ha he marix I + D 2 H(Du 0 (z))d 2 u 0 (z) is inverible for all, z. Le us argue by conradicion and suppose ha here exis, z and a nonzero vecor v such ha v + D 2 H(Du 0 (z))d 2 u 0 (z)v = 0. Le us se for simpliciy A = D 2 H(Du 0 (z)), B = D 2 u 0 (z). Then A, B are boh posiive semidefinie, and we have v = ABv. Bu hen 0 ABv, Bv = v, Bv 0, which implies ha v, Bv = 0. By Lemma 1.5.4, we deduce ha Bv = 0, in conradicion wih he propery ha v = ABv. The ime T given by Theorem is opimal, as shown by he nex resul. Theorem Le H, u 0 and T be as in Theorem If T > T, hen no C 2 soluion of problem (1.16) exiss in [0, T [ IR n. We give wo differen proofs of his saemen; he argumen of he second one is more inuiive, bu can be applied only in he case n = 1. Firs proof We argue by conradicion and suppose ha here exiss a soluion u C 2 ([0, T [ IR n ), wih T > T. By definiion of T, here exis ]T, T [, z IR n and θ IR n \ {0} such ha X z ( ; z ) θ = θ + D 2 H(Du 0 (z )) D 2 u 0 (z ) θ = 0. This implies in paricular ha P z ( ; z ) θ = D 2 u 0 (z ) θ 0. Differeniaing (1.20) wih respec o z we obain 2 u(, X(, z )) X z (, z ) = P z (, z ).

25 Chaper 1 17 Taking he scalar produc wih θ of boh sides of he equaliy yields a conradicion. Second proof Here we resric ourselves o he case n = 1. As in he firs proof, we argue by conradicion and suppose ha here exiss a soluion u up o some ime T > T. By definiion of T we can find < T and z IR such ha 1 + H (u 0 (z ))u 0 (z ) = 0. Then H (u 0 (z ))u 0 (z ) < 0 and so, if we fix any ˆ ], T [, we have 1 + ˆH (u 0 (z ))u 0 (z ) < 1 + H (u 0 (z ))u 0 (z ) = 0. This shows ha he funcion z X(ˆ; z) has a negaive derivaive a z = z and so is decreasing in some neighborhood of z. Thus, here exis z 1 < z 2 such ha X(ˆ; z 1 ) > X(ˆ; z 2 ). On he oher hand, we have X(0; z 1 ) = z 1 < z 2 = X(0; z 2 ). By coninuiy, here exiss s ]0, ˆ[ such ha X(s; z 1 ) = X(s; z 2 ). I follows ha u 0 (z i) = P(s; z i ) = u x (s, X(s; z i )), i = 1, 2, which implies u 0(z 1 ) = u 0(z 2 ). Bu hen we deduce from (1.21) ha X(s; z 2 ) X(s; z 1 ) = z 2 z 1 0, in conradicion wih our choice of s. Remark As i is clear from he second proof, he criical ime T is he ime a which he characerisic lines sar o cross. Inuiively speaking, since u x is consan along any of hese lines, he crossing of characerisics corresponds o he formaion of disconinuiies in he gradien of u. Le us do some observaions abou his behavior. From he compuaions of he second proof we see ha he map z X(; z) is sricly increasing if < T and no monoone for > T. Thus, he map z X(T ; z) is increasing; i is eiher sricly increasing or i is consan on some inerval. In he firs case we can inuiively say ha he characerisics are no ye crossing a he criical ime = T, bu sar crossing immediaely aferwards. In he laer case here is a family of characerisics which are all converging o he same poin a = T, like in Example However, i is easily seen ha his second behavior is nongeneric, in he sense ha i is no sable under small perurbaions of he iniial value. In fac, i only occurs if he derivaive of z X(T ; z) vanishes idenically on an inerval. Le us now focus on he case of a Hamilon-Jacobi equaion coming from a variaional problem of he form considered in he previous secions. We have seen ha he value funcion is Lipschiz coninuous and solves

26 18 A model problem he equaion almos everywhere; we now sudy he relaionship beween he value funcion and he smooh soluion obained by he mehod of characerisics. We will show ha he wo funcions agree as long as he laer exiss, and ha here is a connecion beween he value funcion and he characerisics which is valid for all imes. We make some more regulariy assumpions on L, u 0 wih respec o he previous secions; namely we assume, in addiion o (1.9), ha L is sricly convex and ha u 0 is differeniable. Then, if we denoe by H he Legendre ransform of L, we have ha H is differeniable (see Theorem A.2.4 in he Appendix). Observe ha such properies of H and u 0 are weaker han he ones which are needed in Theorem 1.5.3, bu are enough o ensure ha he funcions P, X, U in (1.21) are well defined. Our firs remark is ha minimizers for he problem in calculus of variaions are also characerisic curves for he Hamilon Jacobi equaion. Proposiion Le L, u 0 : IR n IR saisfy assumpions (1.9); suppose in addiion ha L is sricly convex and ha u 0 is of class C 1. Given (, x) [0, [ IR n, le y( ) be a minimizer for problem (1.3). Then y( ) = X(, z) for some z IR n. Proof Le y( ) be a minimizer for problem (1.3) and le z = y(0). In he proof of Hopf s formula (Theorem 1.3.1) we have seen ha y has he form y(s) = z + s(x z)/ for some z IR n. In addiion, z is a minimizer for he righ hand side of (1.10), which implies ha DL((x z)/) = Du 0 (z). By well known properies of he Legendre ransform (see (A.19)), his implies ha DH(Du 0 (z)) = (x z)/, and so y(s) = X(s; z). Le us se for (, x) [0, + [ IR n, Z(; x) = {z IR n : X(; z) = x}. (1.23) If he hypoheses of Theorem are saisfied hen such a se is a singleon for small imes; in general i is a compac se, as a consequence of he definiion of X and of he properies of u 0, H. Proposiion Given L, u 0 as in he previous proposiion, le u(, x) be he value funcion of problem (1.4). Then u(, x) = min U(; z). (1.24) z Z(;x) Proof As we have seen in he proof of he previous proposiion, when one akes he minimum in Hopf s formula (1.10) i is enough o consider he values of z such ha x = X(; z). Therefore u(, x) = min z Z(;x) [ u 0 (z) + L ( x z )].

27 Chaper 1 19 The condiion z Z(; x) is equivalen o DH(Du 0 (z)) = (x z)/. Recalling ideniy (A.21), we have, for z Z(; x), ( ) x z U(; z) = u 0 (z) + L and so he wo sides of (1.24) coincide. In oher words, he soluion given by Hopf s formula is a suiable selecion of he mulivalued funcion obained by he mehod of characerisics. Le us poin ou ha such a behavior is peculiar of Hamilon Jacobi equaions wih convex hamilonian: if H is neiher convex nor concave here is in general no way of finding a weak soluion by aking such a selecion. Corollary Le L, u 0 C 2 (IR n ) saisfy assumpions (1.9). Suppose in addiion ha D 2 L is posiive definie everywhere and ha D 2 u 0 is uniformly bounded. Then here exiss T > 0 such ha he value funcion of problem (1.4) is of class C 2 in [0, T [ IR n. If u 0 is convex, hen T = + and so he value funcion is C 2 everywhere. In addiion, given (, x) wih < T, here exiss a unique minimizer for problem (1.3), which is given by y(s) = x + (s )DH( u(, x)). (1.25) Proof By Theorem A.2.5 he Legendre ransform of L is of class C 2 and sricly convex. Hence, Theorem ensures he exisence of T > 0 such ha Z(, x) is a singleon for < T and such ha he mehod of characerisics yields a smooh soluion of he Hamilon Jacobi equaion in [0, T [ IR n. By he previous heorem, his smooh soluion coincides wih he value funcion. As observed in Corollary 1.5.5, T = + if u 0 is convex. Finally, (1.25) follows from Proposiion 1.5.8, from (1.22) and (1.21). The previous heorem shows how he dynamic programming approach can provide a soluion o he problem of calculus of variaions we are considering. Such a resul, however, relies on he smoohness of he value funcion and we have seen ha such a propery does no hold in general for all imes. When one ries o exend his approach o he cases when he value funcion is only Lipschiz, several difficulies arise. We have already menioned he non uniqueness of Lipschiz coninuous soluions of he Hamilon Jacobi equaion. I is also no obvious how o resae equaion (1.25) a he poins where u is no differeniable. These issues will be discussed in deail in he remainder of he book.

28 20 A model problem 1.6 Semiconcaviy of Hopf s soluion In his secion we show ha he semiconcaviy propery characerizes he value funcion among all possible Lipschiz coninuous soluions of he Hamilon-Jacobi equaion (1.16). Theorem Le L, u 0 saisfy assumpions (1.9). Suppose in addiion ha (i) L C 2 (IR n ), D 2 L(q) 2 α I q IRn (1.26) (ii) u 0 (x + h) + u 0 (x h) 2u 0 (x) C 0 h 2, x, h IR n for suiable consans α > 0, C 0 0. Then here exiss a consan C 1 0 such ha u( + s, x + h) + u( s, x h) 2u(, x) 2C α( 2 s 2 )C 0 ( h + C 1 s ) 2 (1.27) for all > 0, s ], [, x, h IR n. Proof For fixed, s, x, h as in he saemen of he heorem, le us choose ˆx IR n such ha ( ) x ˆx u(, x) = L + u 0 (ˆx). (1.28) Such a ˆx exiss by Hopf s formula; in addiion, by (1.13), here exiss C 1, depending only on L, such ha We se, for λ 0, x ˆx C 1. (1.29) ( x + λ = ˆx + λ h s x ˆx ) (, x λ = ˆx λ h s x ˆx ). Then we have x + λ + x λ 2 = ˆx, x + λ x λ 2 = λ ( h s x ˆx ). (1.30) By (1.29) we have x + λ x λ 2 λ( h + C 1 s ). (1.31)

29 Chaper 1 21 By Hopf s formula (1.10) we have u( ± s, x ± h) ( ± s)l ( x ± h x ± λ ± s Thus, keeping ino accoun (1.28), we can esimae u( + s, x + h) + u( s, x h) 2u(, x) [ ( ) + s x + h x L λ + s ( x h x L λ + s 2 s ) + u 0 (x ± λ ). ) L ( )] x ˆx +u 0 (x + λ ) + u 0(x λ ) 2u 0(ˆx). (1.32) From (1.31) and (1.26)(ii) i follows ha u 0 (x + λ ) + u 0(x λ ) 2u 0(ˆx) C 0 λ 2 ( h + C 1 s ) 2. (1.33) We now ake q 0, q 1 IR n, θ [0, 1] and se q θ = θq 1 + (1 θ)q 0. Using assumpion (1.26)(i) and he equivalence beween (b) and (e) in Proposiion 1.1.3, we have θl(q 1 ) + (1 θ)l(q 0 ) L(q θ ) Therefore, observing ha θ(1 θ) q 1 q 0 2. (1.34) α + s 2 x + h x + λ + s + s 2 we obain from (1.34) ( ) + s x + h x + 2 L λ + s L + s 2 2 s 2 α(2) 2 In addiion x h x λ s x + h x + λ + s ( x h x λ s = x ˆx, x h x λ s ) ( ) x ˆx L 2. (1.35) x + h x + λ x h x λ + s s ( s)(x + h) ( + s)(x h) = ( s)x+ λ ( + s)x λ 2 s 2 2 s 2 h sx + sˆx = 2 2λ ( h s x ˆx ) 2 s 2 2 s 2 ( ) = 2(1 λ) 2 s 2 h s x ˆx,

30 22 A model problem which implies, by (1.29) x + h x + λ x h x λ + s s 2(1 λ) 2 s 2 ( h + C 1 s ). (1.36) From (1.32), (1.33), (1.35) and (1.36) we obain, for all λ 0, where u( + s, x + h) + u( s, x h) 2u(, x) C(λ) = I is easily checked ha and his proves (1.27). C(λ)[ h + C 1 s ] 2, 2 α( 2 s 2 ) (1 λ)2 + C 0 λ 2. min C(λ) = 2C 0 λ αc 0 ( 2 s 2 ), (1.37) Esimae (1.30) of he previous heorem easily implies ha u is semiconcave wih a linear modulus. Acually, o have a semiconcaviy esimae for u i is no necessary ha boh assumpions in (1.26) are saisfied, bu i is enough o assume one of he wo, as shown in he nex corollary. Corollary Le assumpions (1.9) be saisfied and le u be he funcion defined by Hopf s formula (1.10). (i) If L saisfies propery (1.26)(i) for some α > 0, hen here exiss C 1 such ha u( + s, x + h) + u( s, x h) 2u(, x) 2 α( 2 s 2 ) ( h + C 1 s ) 2. (ii) If u 0 saisfies (1.26)(ii) for some C 0 0, hen here exiss C 1 such ha u( + s, x + h) + u( s, x h) 2u(, x) C 0 ( h + C 1 s ) 2. (iii) If u 0 is concave, hen u is concave (joinly in, x). Proof Formally, saemens (i) and (ii) are obained by leing respecively C 0 and α 0 in esimae (1.27). A precise moivaion can be obained by a suiable adapaion of he proof of he previous heorem. To prove (i), we apply inequaliy (1.32) wih λ = 0. Wih his choice of λ we have x + λ = x λ = ˆx; hus, he erms wih u 0 drop ou while he erms

31 Chaper 1 23 involving L can be esimaed as before. To prove (ii) we choose insead λ = 1; hen i is easily checked ha he oal conribuion of he erms wih L in (1.32) is zero, while he remaining erms can be esimaed as before. Finally, saemen (iii) follows from (ii) aking C 0 = 0. If boh hypoheses in (1.26) are violaed, i.e. L is no wice differeniable and u 0 is no semiconcave, hen u may fail o be semiconcave, as shown by he nex example. Example Consider a one-dimensional problem wih lagrangian and iniial cos given respecively by L(q) = q2 2 + q, u 0(x) = x 2. Clearly, hypoheses (1.9) are saisfied. On he oher hand, assumpions (1.26) are boh violaed, since L is no wice differeniable a q = 0 and u 0 is no semiconcave (see Example 1.1.4). By Hopf s formula he value funcion is given by u(, x) = min z IR { z 2 + (x z)2 2 + x z I is easily found ha he quaniy o minimize is a decreasing funcion of z for z < x and increasing for z > x. Hence he minimum is aained for z = x, i.e. u(, x) = x, 0, x IR. 2 Thus, he value funcion is no semiconcave. }. We now show ha he semiconcaviy propery singles ou Hopf s soluion among all Lipschiz coninuous soluions of he Hamilon Jacobi equaion. Theorem Le H C 2 (IR n ) be convex and le u 1, u 2 Lip (Q T ) be soluions of (1.16) such ha, for any > 0 and i = 1, 2, u i (, x+h)+u i (, x h) 2u i (, x) C for some C > 0. Then u 1 = u 2 everywhere in Q T. We firs recall an elemenary algebraic propery. ( ) h 2, x, h IR n (1.38)

32 24 A model problem Lemma Le A, B be wo symmeric n n marices. Suppose ha 0 A ΛI and B ki for some Λ, k > 0. Then race(ab) nkλ. Proof Le {e 1,..., e n } be on orhonormal basis of eigenvecors of B and le {µ 1,..., µ n } be he corresponding eigenvalues. Then our assumpions imply µ i k, 0 e i, Ae i Λ, i = 1,..., n. I follows race(ab) = n n e i, ABe i = µ i e i, Ae i i=1 i=1 n k e i, Ae i nkλ. i=1 Proof of Theorem Seing ū(, x) = u 1 (, x) u 2 (, x) we have, for (, x) Q T a.e., where ū (, x) = H( u 2 (, x)) H( u 1 (, x)) = b(, x) ū(, x), b(, x) = 1 0 DH(r u 2 (, x) + (1 r) u 1 (, x))dr. Le φ : IR [0, [ be a C 1 funcion o be fixed laer. Seing v = φ(ū) we obain, for (, x) Q T a.e., v (, x) + b(, x) v(, x) = 0. (1.39) Le k C (IR n ) be a nonnegaive funcion whose suppor is conained in he uni sphere and whose inegral is 1. We define, for i = 1, 2, ε > 0 and (, x) Q T, u ε i (, x) = 1 ( ) x y u ε IR n i (, y)k dy. n ε By well-known properies of convoluions he funcions u ε i are of class C wih respec o x and saisfy (i) u ε i Lip (u i) (ii) for all > 0, u ε i (, ) u i(, ) a.e. as ε 0. (1.40) In addiion, u ε i saisfies he semiconcaviy esimae (1.38) for all ε > 0. Therefore, using propery (e) in Proposiion 1.1.3, ( 2 u ε i (, x) C ) I, (, x) Q T. (1.41)

33 Chaper 1 25 Seing b ε (, x) = 1 we can rewrie equaion (1.39) in he form Le us se 0 DH(r u ε 2 (, x) + (1 r) uε 1 (, x))dr, v + div (vb ε ) = (div b ε )v + (b ε b) v. R = max{ DH(p) : p max(lip(u 1 ), Lip (u 2 ))}, Λ = max{ D 2 H(p) : p max(lip(u 1 ), Lip (u 2 ))}. By (1.41) and Lemma i follows div b ε = = 1 0 n k,l=1 2 ( H (r u ε 2 p k p + (1 r) uε 1 ) r 2 u ε 2 + (1 r) 2 u ε ) 1 dr l x k x l x k x l nλc(1 + 1/). (1.42) This is he esimae where he semiconcaviy assumpion on he u i s is used. Given ( 0, x 0 ) Q T, le us inroduce he funcion E() = v(, x)dx, 0 0 and he cone where we use he noaion B(x 0,R( 0 )) C = {(, x) : 0 0, x B(x 0, R( 0 ))}, B(x 0, R( 0 )) = {x IR n : x x 0 < R( 0 )}. Then E is Lipschiz coninuous and saisfies, for a.e. > 0, E () = v dx R vds B(x 0,R( 0 )) B(x 0,R( 0 )) = { div(vb ε ) + (div b ε )v + (b ε b) v}dx B(x 0,R( 0 )) R vds B(x 0,R( 0 )) = v(b ε ν + R)dS B(x 0,R( 0 )) + {(div b ε )v + (b ε b) v}dx B(x 0,R( 0 )) {(div b ε )v + (b ε b) v}dx B(x 0,R( 0 )) nλc ( ) E() + (b ε b) v dx. B(x 0,R( 0 ))

34 26 A model problem Leing ε 0 we obain, by (1.40)(ii), ( E () nλc ) E(), ]0, 0 [ a.e. (1.43) We now choose he funcion φ in he definiion of v. We fix η > 0 and ake φ such ha φ(z) = 0 if z η[lip (u 1 ) + Lip (u 2 )] and φ(z) > 0 oherwise. Then he assumpion ha u 1 = u 2 for = 0 implies ha v(, x) = 0 if η. Thus we obain, from (1.43) and Gronwall s inequaliy, for any [0, 0 ]. Hence ( 0 E() E(η) exp nλc η ( s ) ds u 2 u 1 η [ Lip (u 1 ) + Lip (u 2 )] on C. ) = 0 By he arbirariness of η > 0, we deduce ha u 1 and u 2 coincide in C, and, in paricular, u 1 ( 0, x 0 ) = u 2 ( 0, x 0 ). We can summarize he resuls we have obained abou problem (1.16) as follows. Corollary Le H, u 0 saisfy hypoheses (1.17). Suppose in addiion ha H C 2 (IR n ) and ha eiher H is uniformly convex or u 0 is semiconcave wih a linear modulus. Then here exiss a unique u Lip ([0, [ IR n ) which solves problem (1.16) almos everywhere and which saisfies u(, x + h) + u(, x h) 2u(, x) C ( ) h 2, x, h IR n, > 0, (1.44) for a suiable C > 0. In addiion, u is given by Hopf s formula (1.10), aking as L he Legendre ransform of H. Proof We recall ha, if H is uniformly convex and L is he Legendre ransform of H, hen L saisfies propery (1.26)-(i) (see (A.20) in he appendix). The resul hen follows from Theorem 1.4.1, Corollary and Theorem Le us poin ou ha, nowadays, using semiconcaviy o obain uniqueness resuls is a procedure ha has mainly an hisorical ineres, since i has been laer incorporaed in he more general heory of viscosiy soluions. A sronger moivaion for he sudy of he semiconcaviy lies in he consequences for he regulariy of he value funcion. As we will see, he generalized differenial of a semiconcave funcion enjoys special properies, which play in imporan role in applicaions o calculus of variaions and opimal conrol heory.

35 Chaper Semiconcaviy and enropy soluions In his secion we discuss briefly he connecion beween Hamilon Jacobi equaions and anoher class of parial differenial equaions, called hyperbolic conservaion laws. Given H C 2 (IR) and T ]0, ], le u C 2 ([0, T[ IR) be a soluion of u + H(u x ) = 0. (1.45) Then, if we se v(, x) := u x (, x) and differeniae (1.45) wih respec o x, we see ha v solves v + H(v) x = 0. (1.46) Conversely, if v is a C 1 soluion of (1.46) and we se, for a given x 0 IR, u(, x) := x x 0 v(, y) dy 0 H(v(s, x 0 )) ds (1.47) hen u is a soluion of (1.45). Equaion (1.46) is called a hyperbolic conservaion law. We observe ha he above ransformaion can no longer be done when one considers equaions in several space dimensions or sysems of equaions. Jus like Hamilon Jacobi equaions, conservaion laws do no possess in general global smooh soluions. I is ineresing herefore o invesigae he relaion beween he wo equaions when dealing wih generalized soluions. Conservaion laws are in divergence form and i is naural o consider soluions in he sense of disribuions. More precisely, le us consider equaion (1.46) wih iniial daa v(0, x) = v 0 (x), x IR a.e., (1.48) where v 0 L (IR). Then we say ha v L ([0, T[ IR) is a weak soluion of problem (1.46) (1.48) if i saisfies T 0 [v(, x)φ (, x) + H(v(, x))φ x (, x)] dxd IR = v 0 (x)φ(0, x) dx, φ C0 ([0, T[ IR). (1.49) IR Here C0 denoes he class of C funcions wih compac suppor. I can be proved ha he weak soluions o hyperbolic conservaion laws defined above correspond o Lipschiz coninuous almos everywhere soluions o Hamilon-Jacobi equaions, as saed in he nex resul.