EQUATIONS AUX DERIVEES PARTIELLES. Sur des soluions globales disconinues des équaions d Hamilon-Jacobi, par Gui-Qiang Chen e Bo Su Résumé. On éabli l unicié des soluions de viscosié semiconinues classiques du problème de Cauchy des équaions d Hamilon-Jacobi possèdan des Hamilonien H = H(Du) convexe e Lipschiz coninue globale, si la foncion iniiale disconinue ϕ(x) es coninue à l exérieur de l ensemble Γ de mesure zéro e saisfai (*). On monre la régularié des soluions disconinues des équaions d Hamilon-Jacobi possédan des Hamiloniens localemen sricemen convexes: Les soluions disconinues possédan les données iniiales coninues presque parou e saisfaisan (*) deviennen Lipschiz coninues après un emps fini. On prouve la L 1 -accessibilié des données iniiales e un principe de comparaison. On clarifie aussi l équivalence des soluions de viscosié semiconinues, des soluions bi-laérales, des L-soluions, des soluions minimax, e des L -soluions. PARTIAL DIFFERENTIAL EQUATIONS. On global disconinuous soluions of Hamilon-Jacobi equaions. Absrac. The uniqueness of classical semiconinuous viscosiy soluions of he Cauchy problem for Hamilon-Jacobi equaions is esablished for globally Lipschiz coninuous and convex Hamilonian H = H(Du), provided he disconinuous iniial value funcion ϕ(x) is coninuous ouside a se Γ of measure zero and saisfies ϕ(x) ϕ (x) := lim inf ϕ(y). (*) y x,y RI d \Γ We prove ha he disconinuous soluions wih almos everywhere coninuous iniial daa saisfying ( ) become Lipschiz coninuous afer finie ime for locally sricly convex Hamilonians. The L 1 -accessibiliy of iniial daa and a comparison principle for disconinuous soluions are shown for a general Hamilonian. The equivalence of semiconinuous viscosiy soluions, bi-laeral soluions, L-soluions, minimax soluions, and L -soluions is clarified. Version Abrégée en Français. La héorie des soluions de viscosié coninues des équaions d Hamilon-Jacobi a éé éablie (voir [6]) quand Crandall e Lions on inrodui la noion de soluions de viscosié dans [5]. Dans cee Noe, on s inéresse aux soluions globales disconinues du problème de Cauchy des équaions d Hamilon-Jacobi (1). Les soluions disconinues des équaions d Hamilon-Jacobi surviennen dans bien des siuaions imporanes, par example les mouvemens basés sur la géoméie, la héorie du conrôle, e la héorie des jeux différeniels, où les héories sandard de soluions de viscosié ne s appliquen pas. Dans Ishii [10], les soluions de viscosié semiconinues classiques on éé inroduies, e la méhode de Perron a éé appliquée pour monrer l exisence de possibles soluions semiconinues classiques. On ne sai pas si la soluion d Ishii es unique, même dans le cas pariculer imporan où le Hamilonien es convexe. Aussi, voir [1,2,3,8,15] pour plusieurs noions différenes de soluions disconinues de (1), e pour des résulas apparenés sur les soluions disconinues. On éabli l unicié de la soluion de viscosié semiconinue classique du problème de Cauchy (2). Théorème 1. Supposons que H : RI d RI es convexe e possède une conane de Lipschiz uniforme, e que ϕ(x) es coninue presque parou avec ϕ(x) ϕ (x),x RI d. Alors la soluion semiconinue classique de (2) es unique e coninue presque parou avec u(, x) u (, x). De plus, la soluion es déerminée par la formule de Lax (3). Un aure problème es de savoir, quand les données iniiales disconinues son presque parou coninues avec ( ), si les soluions semiconinues classiques deviennen Lipschiz coninues après un emps fini pour les Hamiloniens localemen sricemen convexes. Nous disons qu un Hamilonien H(p) es localemen sricemen convexe si D 2 H(p) > 0 pour ou p RI d (la srice convexié peu se déériorer quand p ). Un Hamilonien localemen sricemen convexe ypique es H(p) =( p 2 +1) 1 2. Des exemples monren qu on ne peu pas s aendre à des soluions insananémen régulière pour de els Hamiloniens. Gui-Qiang Chen: Deparmen of Mahemaics, Norhwesern Universiy, Evanson, Illinois 606037-2730, USA. Adresse élecronique: gqchen@mah.norhwesern.edu; Téléphone: 847-491-5553; Télécopie: 847-491-8906. Bo Su: Deparmen of Mahemaics, Universiy of Wisconsin-Madison, Madison, Wisconsin 53706-1380, USA. Adresse élecronique: su@mah.wisc.edu; Téléphone: 608-263-4219; Télécopie: 608-263-8891. 1
2 CHEN AND SU Théorème 2. Supposons que le Hamilonien H(p) es C 2, globalemen Lipschiz, localemen sricemen convexe, e que lim p + H(p) =+. Alors oue soluion semiconinue classique de (2) devien Lipschiz après un emps fini, à condiion que les données iniiales disconinues soien uniformémen bornées, preque parou coninues, e saisfassen ( ). On monre la L 1 -accessibilié des donées iniiales e un principe de comparaison. On clarifie aussi l équivalence des soluions de viscosié semiconinues, des soluions bi-laérales, des L-soluions, des soluions minimax, e des L -soluions. English Version. 1. Inroducion. The heory of coninuous viscosiy soluions for Hamilon-Jacobi equaions has been esablished (see [6]) since Crandall-Lions inroduced he viscosiy soluions in [5]. In his Noe, we are concerned wih global disconinuous soluions of he Cauchy problem for Hamilon-Jacobi equaions: { u + H(, x, u, Du) =0, x RI d, > 0, (1) u(0,x)=ϕ(x). The disconinuous soluions of Hamilon-Jacobi equaions arise in many imporan siuaions. The sudy of geomerically based moions demands deep undersanding of disconinuous soluions of Hamilon-Jacobi equaions. Many examples in he conrol heory and he differenial game heory do no have coninuous soluions. The convenional heories of viscosiy soluions do no apply. In Ishii [11], he classical semiconinuous viscosiy soluions were inroduced and he Perron mehod was applied o show he exisence of possible classical semiconinuous soluions. The classical semiconinuous soluions well fi he applicaions. I is unclear wheher he Ishii soluion is unique even in he imporan special case ha he Hamilonian is convex. Also see [1,2,3,8,15] for several differen noions of disconinuous soluions for (1) and relaed resuls on he disconinuous soluions. In his Noe, we esablish he uniqueness and regulariy of he disconinuous soluions of (2) wih globally convex Hamilonians for disconinuous funcion ϕ(x) ha is coninuous ouside a se Γ of measure zero and saisfies ( ). More precisely, we prove he classical semiconinuous soluion of (1) is unique when ϕ(x) is almos everywhere coninuous (a.e.-coninuous, for shor) and saisfies ( ) wih he aid of a comparison principle. We show ha he disconinuous soluions preserve he a.e.-coninuiy, which implies ha he space of a.e.-coninuous funcions is well-posed for disconinuous soluions. We also conclude ha classical semiconinuous soluions [11], bi-laeral soluions [2], L-soluions [8], and L -soluions [3] are he same a.e.. Then we esablish he Lipschiz regulariy of he disconinuous soluions afer finie ime for (2) wih globally Lipschiz coninuous, locally sricly convex Hamilonians. This regularizaion effec is a nonlinear feaure and is new even for convenional coninuous viscosiy soluions. We also clarify he connecions among he disconinuous soluions from he differen noions in he semiconinuous seing and show he L 1 loc-accessibiliy of iniial daa of hese soluions for a general Hamilonian. The comparison principle beween coninuous soluions and disconinuous soluions wih a.e.-coninuous iniial daa are exended o general Hamilonians in he L seing. 2. Uniqueness and A.E.-Coninuiy. Consider he following Cauchy problem: { u + H(Du) =0, x RI d,>0, (2) u =0 = ϕ(x), where H(p) is a convex funcion wih uniform Lipschiz consan. A firs, we inroduce he following sense of a.e.-coninuiy. Definiion. A measurable funcion f : RI N RI is said o be a.e.-coninuous if here is a se Γ saisfying m N (Γ) = 0 such ha f(x) is coninuous a every poin X RI N \Γ. Tha is, lim Y X f(y )=f(x),for any X RI N \Γ. Denoe f (X) := lim sup f(y ), Y X f (X) := lim inf f(y ); Y X ha is, f (X) is he upper envelope of f(x) and f (X) he lower envelope of f(x). Remark 1. From he definiion of a.e.-coninuiy, i follows ha he space of a.e.-coninuous funcions wih he L -norm is a closed subspace in L and conains he well-known space of piecewise coninuous funcions wih he L -norm.
HAMILTON-JACOBI EQUATIONS 3 We recall he noion of he classical semiconinuous super(sub)-soluions of Ishii [11] in he conex of he Cauchy problem (2). Definiion. A funcion u(, x) is called a sub-soluion of (2) if ϕ (, x)+h(dϕ(, x)) 0 holds for all (, x) [0, ) RI d, ϕ C 1 ([0, ) RI d ) saisfying max (u ϕ) =(u ϕ)(, x); [0, ) RI d A funcion u(, x) is called a super-soluion of (2) if ϕ (, x)+h(dϕ(, x)) 0 holds for all (, x) [0, ) RI d, ϕ C 1 ([0, ) RI d ) saisfying min (u ϕ) =(u ϕ)(, x). [0, ) RI d The proof of he following comparison principle can be found in [5] and [10], in which he iniial daa are locally bounded measurable wihou resricion of he a.e-coninuiy. Theorem 1. Le ϕ ± (x) saisfy (ϕ + ) (x) (ϕ ) (x). Le u ± (, x) be Ishii s semiconinuous super(sub)- soluions of (2) wih iniial daa ϕ ± (x) Then (u + ) (, x) (u ) (, x). Observe ha any d-dimensional convex body Ω in RI d has he following inerior cone propery: For any x Ω, here exiss a cone C x wih exreme poin a x such ha C x \{x} in(ω), where C x = B(x, r 1 (x)) {x+ λ(y x) y B(z,r 2 (x)),λ>0,x / B(z,r 2 (x))} for some r j (x) > 0,j =1,2.Then we use his fac o show he following uniqueness heorem in he measure sense. Theorem 2. Suppose ha H : RI d RI is convex and ϕ(x) is a.e.-coninuous wih ϕ(x) ϕ (x),x RI d. Then he classical semiconinuous soluion of (2) is unique a.e. and a.e.-coninuous wih u(, x) u (, x); and he soluion is deermined by he Lax formula: u(, x) = inf {ϕ(y)+l( x y )}, (3) y RI d where L(q) := sup p RI d{ q, p H(p)} is he Legendre ransform of H(p). Proof. We can consruc a monoone decreasing (or increasing) sequence of coninuous funcions {ϕ ± k } k=1 saisfying (ϕ k (x) ϕ(x)) 0, lim k m({ ϕ k (x) ϕ(x) > 1 } B(0,r)) = 0, (4) k for r>0 and, for x Γ (he se of disconinuiy poins of ϕ(x)), lim r 0 (ϕ ) (x, r) =ϕ (x), lim(ϕ + ) (x, r) =ϕ (x), r 0 lim(ϕ ) (x, r) ϕ (x), (5) r 0 where (ϕ ) (x, r) = lim k inf y B(x,r) ϕ k (y) and (ϕ ) (x, r) = lim k sup y B(x,r) ϕ k (y). Le {u ± k } k=1 be he sequences of coninuous viscosiy soluions o (2) wih iniial daa {ϕ± k } k=1. By Theorem 1, for every k, u k (, x) u (, x) u(, x) u (, x) u + k (, x), (6) where u(, x) is any classical semiconinuous soluion wih iniial daa ϕ(x). I is clear ha {u ± k } k=1 are monoone decreasing and increasing sequences of coninuous funcions, respecively. Denoe u = lim k u + k (, x) and u = lim k u k (, x). Then hese funcions ū and u are measurable in RI + RI d and in {} RI d for every >0, and u(, x) u(, x). From (6), we have u(, x) u (, x) u(, x) u (, x) u(, x). (7) On he oher hand, we can prove u(, x) =ū(, x) a almos everywhere Lebesgue poin of u(, x) and ū(, x); is proof is achieved by a conradicion o (5) by using he Lax formula for u k (, x) and ū k (, x) for large k wih he aid of he inerior cone properies of Dom(L) if we assume m({ū(, x) >u(, x)}) > 0. Then heorem follows by (7). For he deails, see [4]. 3. Regulariy. We say a Hamilonian H is locally sricly convex if D 2 H(p) > 0 for any p RI d (he sric convexiy may fail as p ). A ypical locally sricly convex Hamilonian is H(p) =( p 2 +1) 1 2.
4 CHEN AND SU Theorem 3. Suppose ha globally Lipschiz coninuous Hamilonian H(p) is C 2, locally sricly convex, and lim p + H(p) =+. Then any classical semiconinuous soluion of (2) becomes Lipschiz coninuous for T 0 > 0 wih T 0 depending on sup x RI d ϕ(x) and H(p), provided ha he disconinuous iniial daa are uniformly bounded, a.e.-coninuous, and saisfy ( ). Proof: We firs observe ha he Legendre ransform L(q) = sup p RI d{ q, p H(p)}of H is a convex funcion; Ω:=Dom(L) is a bounded convex body in RI d conaining he origin as is inerior poin; Ω α = {q RI d L(q) α} is a bounded closed convex subse in Ω; and L aains is minimum in in(ω). Se K = sup x RI d ϕ(x), M = inf q Ω L(q), and m = inf q Ω L(q). Since L aains is minimum in in(ω), M m = a>0. I is easy o see ha Ω m+ ia is a bounded convex body in RI d for i =1,2,3. Se d = 4 dis(ω m+ a, Ω\Ω 2 m+ 3a ) > 0. For any q Ω 4 m+ 3a, L(q) saisfies L(q 1) L(q 4 2 ) C 1 q 1 q 2,where C 1 is a consan depending only on C. For> 8K a, we have u(, x) = inf {ϕ(z)+l( x z )} = inf {ϕ(z)+l( x z )}. z RI d z x Ω m+ a 2 For any wo poins x 1 and x 2 wih x 1 x 2 4Kb a saisfying ϕ(z ɛ )+L( x1 zɛ ) ɛ x 1 x 2 u(, x 1 ), for > 8K a. Hence, u(, x 2 ) u(, x 1 ) (L( x 2 z ɛ and posiive ɛ<1, here is a poin z ɛ x 1 Ω m+ a 2 ) L( x 1 z ɛ )) + ɛ x 1 x 2 C 1 x 1 x 2 + ɛ x 1 x 2, for > 8K a. Swiching he roles of x 1 and x 2 and leing ɛ 0, we finally obain u(, x 1 ) u(, x 2 ) C 1 x 1 x 2,which implies he Lipschiz coninuiy of u(, x) wih respec o x for > 8K a. Using he equaion, he Lipschiz coninuiy of u(, x) wih respec o immediaely follows. Remark 2. In general, one can no expec he insananeous regulariy in Theorem 3 since he speed of propagaion is finie when he soluion experiences a jump. Consider a one-dimensional example of (2): H(p) = ( p 2 +1) 1 2 and ϕ(x) =0,ifx<0; ϕ(x) =1,ifx 0. Easy calculaion shows ha L(q) = (1 q 2 ) 1 2, u(, x) = inf z RI 1{ϕ(z) ( 2 (x z) 2 ) 1 2 }. Then, if x 0, u(, x) = ( 2 x 2 ) 1 2, if 0 <x when 1, or 0 <x (2 1) 1 2 when >1, 1, if x>, when 1, or x>(2 1) 1 2 when >1. which is disconinuous on he line x = for 0 1, and, a =1,u x (1,x) as x goes o 1 from he lef side. However, u(, x) is Lipschiz coninuous afer = 1. Remark 3. When he Hamilonian H(Du) is sricly convex, he viscosiy soluions insananeously become Lipschiz coninuous afer iniial ime as shown by Kruzhkov [12]. The regulariy resul was generalized o superlinear convex Hamilonians by Lions [13]. In he conex of one-dimensional scalar conservaion laws wih sricly convex flux funcions, Liu-Pierre [14] showed ha he measure soluions in he Kruzhkov sense insananeously become L funcions wih iniial daa involving δ-measures. Demengel-Serre [7] showed ha, when he flux funcions are sricly convex wih linear growh a infiniy, here are measure soluions wih iniial daa of posiive measures which become L 1 funcions afer finie ime. The example in [4] shows ha, even when u 0 L q for some q>1, one can no expec ha he corresponding soluion insananeously become an L funcion for >0. In paricular, he proof of Theorem 3 implies he following heorem. Theorem 4. Le he C 2 Hamilonian H(p) be locally sricly convex, and lim p + H(p) = +. Then any coninuous soluion of (2) wih uniformly bounded and a.e.-coninuous iniial daa ϕ(x) becomes Lipschiz coninuous for T 0 > 0 wih T 0 depending on sup x RI d ϕ(x) and H(p). 4. Comparison Principle, Iniial Daa Accessibiliy, and Connecions. As usual, we consider he Hamilonians saisfying:
HAMILTON-JACOBI EQUATIONS 5 (A1). H(, x, z, p) is coninuous in (, x, z, p) and increasing in z; (A2). H(, x, z, p 1 ) H(, x, z, p 2 ) C 0 (1 + x ) p 1 p 2, and H(, x, z, 0) C 0 (1 + x + z ), for all (0,T]; (A3). H(, x 1,z,p) H(, x 2,z,p) λ(l 0 )(1 + p ) x 1 x 2, for x 1, x 2 L 0 ; (A4). H(, x, z 1,p) H(, x, z 2,p) C 0 (1 + x + p ) z 1 z 2. The global exisence of L soluions was esablished for general Lipschiz Hamilonians for arbirary large L iniial daa in [3] under he condiions (A1) (A4). I was also shown ha L soluions consis wih viscosiy soluions inroduced by Crandall-Lions [5] and minimax soluions inroduced by Subboin [15]. In he semiconinuous seing, i is sraighforward o verify ha he definiion of L soluions is equivalen o Subboin s definiion of semiconinuous super-soluions and sub-soluions. The noion of L soluions becomes he noion of semiconinuous soluions when he essenial infimum and superemum in he definiion (see [3, 4]) are replaced by he sandard infimum and superemum. The exisence of disconinuous soluions can be shown by adoping he proof in [3] o he semiconinuous seing. In [15], i is shown ha Subboin s definiion is equivalen o Ishii s definiion for semiconinuous super-soluions and sub-soluions. Therefore, in he semiconinuous seing, Ishii s semiconinuous soluions are he same as L soluions. In [8], i is shown ha he definiion of bi-laeral soluions is equivalen o ha of L-soluions when he Hamilonian is convex, and L-soluions are exacly he maximal sup-semiconinuous sub-soluions in he sense of Ishii [11]. Theorem 5. Assume ha ϕ ± (x) are a.e.-coninuous and ϕ(x) is coninuous wih ±(ϕ ± (x) ϕ(x)) 0. Le u ± (, x) be L super(sub)-soluions of (1) wih iniial daa ϕ ± (x), and v(, x) he coninuous soluion wih iniial daa ϕ(x). Then ±(u ± (, x) v(, x)) 0 a.e. For Ishii s semiconinuous soluions, if he assumpion of ϕ ± (x) in Theorem 5 is replaced by (ϕ + ) (x) (ϕ ) (x), hen (u + ) (, x) (u ) (, x) holds everywhere. The L soluions of (1) access iniial daa in L 1 loc, provided he closure of he se of disconinuiy poins is measure zero. Theorem 6. Suppose ha u(, x) is an L soluion of (1) wih a.e.-coninuous iniial daa ϕ(x) and m d ( Γ) = 0. Then u(, ) ϕ( ), when 0, inhel 1 loc opology. Acknowledgemens. The auhors hank Takis Souganidis for simulaing conversaions. Gui-Qiang Chen s research was suppored in par by he Naional Science Foundaion hrough grans DMS-9971793, INT-9987378, and INT-9726215. References. [1] Barles, G. and Perhame, B., Disconinuous soluions of deerminisic opimal sopping problem, Mah. Model. Num. Anal. 2 (1987), 557-579. [2] Barron, E. N. and Jensen, R., Semiconinuous viscosiy soluions of Hamilon-Jacobi equaions wih convex Hamilonians, Commun. Parial Diff. Eqs. 15 (1990), 1713-1742. [3] Chen, G.-Q. and Su, B., Disconinuous soluions in L for Hamilon-Jacobi equaions, Chinese Annals Mah. 2 (2000), 165-186. [4] Chen, G.-Q. and Su, B., On Disconinuous soluions of Hamilon-Jacobi equaions, Preprin, June 2001. [5] Crandall, M. and Lions, P. L., Viscosiy soluions of Hamilon-Jacobi equaions, Trans. Amer. Mah. Soc. 277 (1983), 1-42. [6] Crandall, M., Ishii, H., and Lions, P. L., A user s guide o viscosiy soluions of second-order parial differenial equaions, Bullein Amer. Mah. Soc. 27 (1992), 1-67. [7] Demengel, F. and Serre, D., Nonvanishing singular pars of measure valued soluions for scalar hyperbolic equaions, Comm. Parial Diff. Eqs. 16 (1991), 221-254. [8] Giga, Y. and Sao, M. H., A level se approach o semiconinuous soluions for Cauchy problems, Preprin, 2001. [9] Glimm, J., Kranzer, H. C., Tan, D., and Tangerman, F. M., Wave frons for Hamilon-Jacobi equaions: he general heory for Riemann soluions in R n, Comm. Mah. Phys. 187 (1997), 647 677. [10] Ishii, H., Uniqueness of unbounded viscosiy soluion of Hamilon-Jacobi equaions, Indiana Univ. Mah. J. 33 (1984), 721-748. [11] Ishii, H., Perron s mehod for Hamilon-Jacobi equaions, Duke Mah. J. 55 (1987), 368-384. [12] Kruzhkov, S. N., Generalized soluions of nonlinear equaions of he firs order wih several independen variables, II, (Russian) Ma. Sb. (N.S.) 114 (1967), 108-134. [13] Lions, P. L., Generalized Soluions of Hamilon-Jacobi Equaions, Research Noes in Mah. 69, Piman, Boson-London- Melbourne, 1982. [14] Liu, T.-P. and Pierre, M., Source soluions and asympoic behavior in conservaion laws, J. Diff. Eqs. 51 (1984), 419-441.
6 CHEN AND SU [15] Subboin, A. I., Generalized Soluions of Firs Order PDEs, Birkhauser, Boson, 1995.