A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES

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1 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES ALESSIO FIGALLI, WILFRID GANGBO, AND TÜRKAY YOLCU Absrac. In is manuscrip we exend De Giorgi s inerpolaion meod o a class of parabolic equaions wic are no gradien flows bu possess an enropy funcional and an underlying Lagrangian. Te new fac in e sudy is a no only e Lagrangian may depend on spaial variables, bu i does no induce a meric. Assuming e iniial condiion o be a densiy funcion, no necessarily smoo, bu solely of bounded firs momens and finie enropy, we use a variaional sceme o discreize e equaion in ime and consruc approximae soluions. Ten De Giorgi s inerpolaion meod is revealed o be a powerful ool for proving convergence of our algorim. Finally we sow uniqueness and sabiliy in L 1 of our soluions. 1. Inroducion In e eory of exisence of soluions of ordinary differenial equaions on a meric space, curves of maximal slope and minimizing movemens play an imporan role. Te minimizing movemens in general are obained via a discree sceme. Tey ave e advanage of providing an approximae soluion of e differenial equaion by discreizing in ime wile no requiring e iniial condiion o be smoo. Ten a clever inerpolaion meod inroduced by De Giorgi [7, 6] ensures compacness for e family of approximae soluions. Many recen works [3, 14] ave used minimizing movemen meods as a powerful ool for proving exisence of soluions for some classes of parial differenial equaions PDEs. So far, mos of ese sudies concern PDEs wic can be inerpreed as gradien flow of an enropy funcional wi respec o a meric on e space of probabiliy measures. Tis paper exends e minimizing movemens and De Giorgi s inerpolaion meod o include PDEs wic are no gradien flows, bu possess an enropy funcional and an underlying Lagrangian wic may be dependen of e spaial variables. In e curren manuscrip R d is an open se wose boundary is of zero measure. We denoe by P1 ac e se of Borel probabiliy densiies on of bounded firs momens, endowed wi e 1-Wassersein disance W 1 cfr. subsecion 2.2. We consider disribuional soluions of a class of PDEs of e form 1.1 ϱ + divϱ V =, in D, T R d is implicily means a we ave imposed Neumann boundary condiion, wi ϱ V := ϱ p H x, ϱ 1 [P ϱ ] on, T and ϱ AC 1, T ; P1 ac C[, T ]; P1 ac. By abuse of noaion, ϱ will denoe a e same ime e soluion a ime and e funcion, x ϱ x defined over, T. I will be clear from e conex wic one we are referring Dae: January 28, 211. Key words: mass ransfer, Quasilinear Parabolic Ellipic Equaions, Wassersein meric. AMS code: 35, 49J4, 82C4 and 47J25. 1

2 2 A. FIGALLI, W. GANGBO, AND T. YOLCU o. We recall a e unknown ϱ is nonnegaive, and can be inerpreed as e densiy of a fluid, wose pressure is P ϱ. Here, e daa H, U and P saisfy specific properies, wic are saed in subsecion 2.1. We only consider soluions suc a [P ϱ ] L 1, T, and is absoluely coninuous wi respec o ϱ. If ϱ saisfies addiional condiions wic will soon commen on, en Uϱ := Uϱ dx is absoluely coninuous, monoone nonincreasing, and 1.2 d d Uϱ = [P ϱ ], V dx. Te space o wic e curve ϱ belongs ensures a ϱ converges o ϱ in P1 ac as. Soluions of our equaion can be viewed as curves of maximal slope on a meric space conained in P 1. Tey include e so-called minimizing movemens cfr. [3] for a precise definiion obained by many auors in case e Lagrangian does no depend on spaial variables e.g. [13] wen Hp = 1/2 p 2, [1, 3] wen Hx, p Hp. Tese sudies ave been very recenly exended o a special class of Lagrangian depending on spaial variables were e Hamilonian assume e form Hx, p = A xp, p [14]. In eir pioneering work Al and Luckaus [2] consider differenial equaions similar o 1.1, imposing some assumpions no very comparable o ours. Teir meod of proof is very differen from e ones used in e above cied references and is based on a Galerkin ype approximaion meod. Le us describe e sraegy of e proof of our resuls. Te firs sep is e exisence par. Le Lx, be e Legendre ransform of Hx,, o wic we refer as a Lagrangian. For a ime sep >, le c x, y, e cos for moving a uni mass from a poin x o a poin y, be e minimal acion min σ Lσ, σd. Here, e minimum is performed over e se of all pas no necessarily conained in suc a σ = x and σ = y. Te cos c provides a way of defining e minimal oal work C ϱ, ϱ cfr. 2.8 for moving a mass of disribuion ϱ o anoer mass of disribuion ϱ in ime. For measures wic are absoluely coninuous, e recen papers [4, 8, 9] give uniqueness of a minimizer in 2.8, wic is concenraed on e grap of a funcion T : R d R d. Furermore, C provides a naural way of inerpolaing beween ese measures: ere exiss a unique densiy ϱ s suc a C ϱ, ϱ = C s ϱ, ϱ s + C s ϱ s, ϱ for s,. Assume for a momen a is bounded. For a given iniial condiion ϱ P1 ac suc a Uϱ < + we inducively consruc {ϱ n } n in e following way: ϱ n+1 is e unique minimizer of C ϱ n, ϱ + Uϱ over Pac 1. We refer o is minimizaion problem as a primal problem. Under e addiional condiion a Lx, v > Lx, for all x, v R d suc a v, one as c x, x < c x, y for x y. As a consequence, under a condiion e following maximum principle olds: if ϱ M en ϱ n M for all n. We en sudy a problem, dual o e primal one, wic provides us wi a caracerizaion and some imporan regulariy properies of e minimizer ϱ n+1. Tese properies would ave been arder o obain sudying only e primal problem. Having deermined {ϱ n } n N, we consider wo inerpolaing pas. Te firs one is e pa ϱ suc a C ϱ n, ϱ n+1 = C sϱ n, ϱ n+s + C s ϱ n+s, ϱ n+1, < s <.

3 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES 3 Te second pa ϱ is defined by ϱ n+s := arg min { C s ϱ n, ϱ + Uϱ }, < s <. Tis inerpolaion was inroduced by De Giorgi in e sudy of curves of maximal slopes wen Cs defines a meric. Te pa { ϱ } saisfies equaion 3.42, wic is a discree analogue of e differenial equaion 1.1. Ten we wrie a discree energy inequaliy in erms of bo pas { ϱ } and {ϱ }, and we prove a up o a subsequence bo pas converge in a sense o be made precise o e same pa ϱ. Furermore, ϱ saisfies e energy inequaliy T [ 1.3 Uϱ Uϱ T d L x, V + H x, ϱ 1 [P ϱ ] ] ϱ dx, wic anks o e assumpions on H cfr. subsecion 2.1 implies for insance a [P ϱ ] L 1, T. Te above inequaliy corresponds o wa can be considered as one alf of e cain rule: d d Uϱ V, [P ϱ ] dx. Here V is a velociy associaed o e pa ϱ, in e sense a equaion 1.1 olds wiou ye e knowledge a ϱ V = ϱ p H x, ϱ 1 [P ϱ ]. Te curren sae of e ar allows us o esablis e reverse inequaliy yielding o e wole cain rule only if we know a T T 1.4 d V α ϱ dx, d ϱ 1 [P ϱ ] α ϱ dx < + for some α 1, +, α = α/α 1. In a case, we can conclude a ϱ V = ϱ p H x, ϱ 1 [P ϱ ] and d d Uϱ = V, [P ϱ ] dx. In lig of e energy inequaliy 3.43, a sufficien condiion o ave e inequaliy 1.4 is a Lx, v v α. Tis is wa we laer impose in is work. Suppose now a may be unbounded. As poined ou in remark 3.18, by a simple scaling argumen we can solve equaion 1.1 for general nonnegaive densiies, no necessarily of uni mass. Lemma 4.1 sows a if we impose e bound 4.1 on e negaive par of U, en Uϱ is welldefined for ϱ P1 ac. We assume a e iniial condiion ϱ P1 ac and Uϱ dx is finie, and we sar our approximaion argumen by replacing by m := B m and ϱ by ϱ m := ϱ χ Bm. Here, B m is e open ball of radius m, cenered a e origin. Te previous argumen provides us wi a soluion of equaion 1.1, saring a ϱ m, for wic we sow a { } max x ϱ m dx + Uϱ m dx [,T ] m m is bounded by a consan independen of m. Using e fac a for eac m, ϱ m saisfies e energy inequaliy 1.3, we obain a a subsequence of {ϱ m } converges o a soluion of equaion 1.1 saring a ϱ. Moreover, as we will see, our approximaion argumen also allows o relax e regulariy assumpions on e Hamilonian H. Tis sows a remarkable feaure of e exisence sceme described before, as i allows o consruc soluions of a igly nonlinear PDE as 1.1 by

4 4 A. FIGALLI, W. GANGBO, AND T. YOLCU approximaing a e same ime e iniial daum and e Hamilonian and e same sraegy could also be applied o relax e assumpions on U, cfr. secion 4. Tis complees e exisence par. In order o prove uniqueness of soluion in equaion 1.1 we make several addiional assumpions on P and H. Firs of all, we assume a Lx, v > Lx, for all x, v R d suc a v o ensure a e maximum principle olds. Nex, le Q denoe e inverse of P and se u, := P ϱ. Ten equaion 1.1 is equivalen o 1.5 Qu = div ax, Qu, u in D, T, wic is a quasilinear ellipic-parabolic equaion. Here a is given by equaion 5.2. Te sudy in [15] addresses conracion properies of soluions of equaion 1.5 even wen Qu is no a bounded measure bu is merely a disribuion, as in our case. Our vecor field a does no necessarily saisfy e assumpions in [15]. Indeed one can ceck a i violaes drasically e sric monooniciy condiion of [15], for large Qu. For is reason, we only sudy uniqueness of soluions wi bounded iniial condiions even if, for is class of soluion, a is sill no sricly monoone in e sense of [2] or [15]. Te sraegy consiss firs in sowing a ere exiss a Hamilonian H Hx, ϱ, m cfr. equaion 5.3 suc a for eac x, ax, ϱ, m is conained in e subdifferenial of Hx,, a ϱ, m. Ten, assuming Hx,, convex and Q Lipsciz, we esablis a conracion propery for bounded soluions of 1.1. As a by produc we conclude uniqueness of bounded soluions. Te paper is srucured as follows: in secion 2 we sar wi some preliminaries and se up e general framework for our sudy. Te proof of e exisence of soluions is en spli ino wo cases. Secion 3 is concerned wi e case were is bounded, and we prove exisence of soluions of equaions 1.1 by applying e discree algorim described before. In secion 4 we relax e assumpion a is bounded: under e ypoeses a ϱ P1 ac and Uϱ dx is finie, we consruc by approximaion a soluion of equaion 1.1 as described above. Secion 5 is concerned wi uniqueness and sabiliy in L 1 of bounded soluions of equaion 1.1 wen Q is Lipsciz. To acieve a goal, we impose e sronger condiion 5.5 on e Hamilonian H. We avoid repeaing known facs as muc as possible, wile rying o provide all e necessary deails for a complee proof. 2. Preliminaries, Noaion and Definiions 2.1. Main assumpions. We fix a convex superlinear funcion θ : [, + [, + suc a θ =. Te main examples we ave in mind are funcions θ wic are posiive combinaions of funcions like α wi α > 1 for funcions like ln + or e, cfr. remark We consider a funcion L : R d R d R wic we call Lagrangian. We assume a: L1 L C 2 R d R d, and Lx, = for all x R d. L2 Te marix vv Lx, v is sricly posiive definie for all x, v R d. L3 Tere exis consans A, A, C > suc a C θ v + A Lx, v θ v A x, v R d. Le us remark a e condiion Lx, = is no resricive, as we can always replace L by L Lx,, and is would no affec e sudy of e problem we are going o consider. We also noe a L1, L2 and L3 ensure a L is a so-called Tonelli Lagrangian cfr. for insance

5 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES 5 [8, Appendix B]. To prove a maximum principle for e soluions of 1.1, we will also need e assumpion: L4 Lx, v Lx, for all x, v R d. Te global Legendre ransform L : R d R d R d R d of L is defined by Lx, v := x, v Lx, v. We denoe by Φ L : [, + R d R d R d R d e Lagrangian flow defined by { d [ 2.1 d v L Φ L, x, v ] = x L Φ L, x, v, Φ L, x, v = x, v. Furermore, we denoe by Φ L 1 : [, + Rd R d R d e firs componen of e flow: Φ L 1 := π 1 Φ L, π 1 x, v := x. Te Legendre ransform of L, called e Hamilonian of L, is defined by Hx, p := sup v R d { v, p Lx, v }. Moreover we define e Legendre ransform of θ as θ { } s := sup s θ, s R. I is well-known a L saisfies L1, L2 and L3 if and only if H saisfies e following condiions: H1 H C 2 R d R d, and Hx, p for all x, p R d. H2 Te marix pp Hx, p is sricly posiive definie for all x, p R d. H3 θ : R [, + is convex, superlinear a +, and we ave A + C θ p C Moreover L4 is equivalen o: H4 p Hx, = for all x R d. Hx, p θ p + A x, v R d. We also inroduce some weaker condiions on L, wic combined wi L3 make i a weak Tonelli Lagrangian: L1 w L C 1 R d R d, and Lx, = for all x R d. L2 w For eac x R d, Lx, is sricly convex. Under L1 w, L2 w and L3, e global Legendre ransform is an omeomorpism, and e Hamilonian associaed o L saisfies H3 and H1 w H C 1 R d R d, and Hx, p for all x, p R d. H2 w For eac x R d, Hx, is sricly convex. Cfr. for insance [8, Appendix B]. In is paper we will mainly work assuming L1, L2 and L3, excep in secion 4 were we relax e assumpions on L and correspondingly a on H o L1 w, L2 w and L3. Le U : [, + R be a given funcion suc a 2.2 U C 2, + C[, +, U >,

6 6 A. FIGALLI, W. GANGBO, AND T. YOLCU and 2.3 U =, U lim = +. + We se U = + for,, so a U remains convex and lower-semiconinuous on e wole R. We denoe by U e Legendre ransform of U : 2.4 U { } { } s := sup s U = sup s U. R Wen ϱ is a Borel probabiliy densiy of R d suc U ϱ L 1 R d we define e inernal energy Uϱ := Uϱ dx. R d If ϱ represens e densiy of a fluid, one inerpres P ϱ as a pressure, were 2.5 P s := U ss Us. Noe a P s = su s, so a P is increasing on [, Noaion and definiions. If ϱ is a probabiliy densiy and α >, we wrie M α ϱ := x α ϱx dx R d for is momen of order α. If R d is a Borel se, we denoe by P ac e se of all Borel probabiliy densiies on. If ϱ P ac, we acily idenify i wi is exension defined o be ouside. We denoe by P e se of Borel probabiliy measures µ on R d a are concenraed on : µ = 1. Finally, we denoe by Pα ac P ac e se of ϱ probabiliy densiy on suc a M α ϱ is finie. Wen α 1, is is a meric space wen endowed wi e Wassersein disance W α cfr. equaion 2.1 below. We denoe by L d e d dimensional Lebesgue measure. Le u, v : R d R {± }. We denoe by u v e funcion x, y ux + vy were i is well-defined. Te se of poins x suc a ux R is called e domain of u and denoed by domu. We denoe by ux e subdifferenial of u a x. Similarly, we denoe by + ux e superdifferenial of u a x. Te se of poin were u is differeniable is called e domain of u and is denoed by dom u. Le u : R d R {+ }. Is Legendre ransform is u : R d R {+ } defined by u { } y = sup x, y ux. x In case u : R d R {+ }, is Legendre ransform is defined by idenifying u wi is exension wic akes e value + ouside. Finally, for f : a, b R, we se d + f d =c := lim sup + fc + fc, d f d =c := lim inf fc + fc.

7 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES 7 Definiion 2.1 c-ransform. Le c : R d R d R {+ }, le R d and le u, v : R { }. Te firs c-ransform of u, u c : R { }, and e second c-ransform of v, v c : R { }, are respecively defined by 2.6 u c { } { y := inf cx, y ux, vc x := inf cx, y vy}. x y Definiion 2.2 c-concaviy. We say a u : R { } is firs c-concave if ere exiss v : R { } suc a u = v c. Similarly, v : R { } is second c-concave if ere exiss u : R { } suc a v = u c. For simpliciy we will omi e words firs and second wen referring o c-ransform and c-concaviy. For >, we define e acion A σ of an absoluely coninuous curve σ : [, ] R d as and e cos funcion A σ := Lστ, στ dτ { } 2.7 c x, y := inf σ A σ : σ W 1,1, ; R d, σ = x, σ = y. For µ, µ 1 PR d, le Γµ, µ 1 be e se of probabiliy measures on R d R d wic ave µ and µ 1 as marginals. Se { } 2.8 C µ, µ 1 := inf γ c x, ydγx, y : γ Γµ, µ 1 R d R d and 2.9 W θ, µ, µ 1 := inf γ { R d R d θ y x } dγx, y : γ Γµ, µ 1. Remark 2.3. By remark 2.11 c is coninuous. In paricular, ere always exiss a minimizer for 2.8 rivial if C is idenically + on Γϱ, ϱ 1. We denoe e se of minimizers by Γ ϱ, ϱ 1. Similarly, ere is a minimizer for 2.9, and we denoe e se of is minimizers by Γ θ ϱ, ϱ 1. We also recall e definiion of e α-wassersein disance, α 1: { } 1/α 2.1 W α µ, µ 1 := inf y x α dγx, y : γ Γµ, µ 1. γ R d R d I is well-known cfr. for insance [3] a W α merizes e weak opology of measures on bounded subses of R d. Aloug we define W α ere for all α 1, only W 1 will be used excep afer secion 3.5. Te following fac can be cecked easily: 2.11 C µ, µ 2 C µ, µ 1 + C µ 1, µ 2 for all [, ] and µ, µ 1, µ 2 PR d.

8 8 A. FIGALLI, W. GANGBO, AND T. YOLCU 2.3. Properies of enalpy and pressure funcionals. In is subsecion, we assume a 2.2 and 2.3 old. Lemma 2.4. Te following properies old: i U : [, + R is sricly increasing, and so inverible. Is inverse is of class C 1 and lim + U = +. ii U C 1 R is nonnegaive, and U s for all s R. iii lim s + U s = +. U iv lim s s + s = +. v P : [, + [, + is sricly increasing, bijecive, lim + P = +, and is inverse Q : [, + [, + saisfies lim s + Qs = +. Proof: i Since U is convex and U =, we ave U U/. Tis ogeer wi U > and e superlineariy of U easily imply e resul. ii U follows from U =. Te remaining par is a consequence of U U = for >, ogeer wi U s = and so U s = for s U +. iii Follows from i and e ideniy U U = for >. iv Since U is convex and nonnegaive we ave U s s 2 U s 2, so a e resul follows from iii. v Observe a P = U U by ii. Since U is monoone nondecreasing, for < 1 we ave P U 1 U. We conclude a lim + P =. Te remaining saemens follow. Remark 2.5. Le R d be a bounded se, and le ϱ P ac be a probabiliy densiy. Recall a we exend ϱ ouside by seing is value o be idenically. If R > is suc a B R, we ave R d θ x ϱx dx θr. Moreover, since by convexiy U U1 + U 1 1 a + b for, R d U ϱ dx is bounded on P ac by a + b L d. Hence Uϱ is always well-defined on P ac, and is finie if and only if U + ϱ L 1. Te following lemma is a sandard resul of e calculus of variaions, cfr. for insance [5] for a more general resul on unbounded domains, cfr. secion 4: Lemma 2.6. Le R d and suppose {ϱ n } n N P ac converges weakly o ϱ in L 1. Assume a eier is bounded, or is unbounded and U. Ten lim inf n Uϱn Uϱ Properies of H and e cos funcions. Lemma 2.7. Te following properies old for < < and x, y R d : i c x, x. ii c x, y c x, y. iii x y C θ + A c x, y θ x y A A. Proof: i Se σ x for [, ] and recall a Lx, = o ge c x, x A σ =. ii Given σ W 1,1, ; R d saisfying σ = x and σ = y, we can associae an exension o

9 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES 9, ], wic we sill denoe σ, suc a σ = y for, ]. We ave σ W 1,1, ; R d, σ = x and σ = y. Hence, c x, y A σ = A σ + Ly, d = A σ. Since σ W 1,1, ; R d is arbirary, is concludes e proof of ii. iii Te firs inequaliy is obained using L3 and c x, y A T σ wi σ = 1 /x+/y, wile e second one follows from Jensen s inequaliy. Te nex proposiion can readily be derived from e sandard eory of Hamilonian sysems cfr. e.g. [8, Appendix B]: Proposiion 2.8. Under e assumpions L1, L2 and L3, 2.7 admis a minimizer σ x,y for any x, y R d. We ave a σ x,y C 2 [, ] and saisfies e Euler-Lagrange equaion 2.12 σ x,y τ, σ x,y τ = Φ L τ, x, σ x,y τ [, ], were Φ L is e Lagrangian flow defined in equaion 2.1. Moreover, for any r > and S, + a compac se, ere exiss a consan k S r, depending on S and r only, suc a σ x,y C 2 [,] k S r if x, y r and S. Remark 2.9. Le σ be a minimizer of e problem 2.7, and se pτ := v L στ, στ. a Te Euler-Lagrange equaion 2.12 implies a σ and p are of class C 1 and saisfy e sysem of ordinary differenial equaions 2.13 στ = p Hστ, pτ, ṗτ = x Hστ, pτ b Te Hamilonian is consan along e inegral curve στ, pτ, i.e. Hστ, pτ = Hσ, p for τ [, ]. Te following lemma is sandard cfr. for insance [8, Appendix B]: Lemma 2.1. Under e assumpions in proposiion 2.8, le σ be a minimizer of 2.7, and define p i := v Lσi, σi for i =,. For r, m > ere exiss a consan l r, m, depending on, r, m only, suc a if x, y B r and w B m, en: a c x + w, y c x, y p, w l r, m w 2 ; b c x, y + w c x, y + p, w l r, m w 2. Remark Tis lemma says a p + c, yx, and for y B r e resricion of c, y o B r is l r, m-concave. Similarly, p + c x, y, and for x B r e resricion of cx, o B r is l r, m-concave. Lemma Suppose L1, L2 and L3 old. Le a, b, r, + be suc a a < b and se S = [a, b]. Ten ere exiss a consan k S r, depending on S and r only, suc a c x, y c x, y k S r for all, S and all x, y R d saisfying x, y r.

10 1 A. FIGALLI, W. GANGBO, AND T. YOLCU Proof: Le k S r be e consan appearing in proposiion 2.8 and le { E 1 := sup{ Lx, v : x, v k S r}, E 2 := sup v Lx, v : x k S r, v k S r b }. x,v x,v a Fix, S suc a <. For x, y R d suc a x, y r we denoe by σ a minimizer of 2.7. Define σ = σ / for [, ]. Ten σ C 2 [, ], σ = x and σ = y. Ten c x, y L σ, σ d = L σ, σ ds = c x, y + L σ, σ Lσ, σ ds. Tis implies and so c x, y c x, y + E 2 1 k S r = c x, y + E 2 k S r, 2.14 c x, y c x, y c x, y + E 2 k S r E 1 + E 2 k S r, were we used e rivial bound c x, y E 1. Since by lemma 2.7ii c x, y c x, y, 2.14 proves e lemma Toal works and eir properies. In is subsecion we assume a 2.2 and 2.3 old. Lemma Te following properies old: i For any µ PR d we ave C µ, µ. In paricular, for any µ, µ PR d, C µ, µ C µ, µ if <. ii For any >, µ, µ PR d, A A + W θ, µ, µ C µ, µ C W θ, µ, µ + A. iii For any K > ere exiss a consan CK > suc a 2.15 W 1 µ, µ 1 K W θ,µ, µ + CK K >, µ, µ PRd. Proof: i Te firs par follows from c x, x, wile e second saemen is a consequence of e firs one and C µ, µ C µ, µ + C µ, µ. ii I follows direcly from Lemma 2.7iii. iii Tanks o e superlineariy of, for any K > ere exiss a consan CK > suc a 2.16 θs Ks CK s. Fix now γ Γ θ µ, µ 1. Ten W 1 µ, µ x y dγx, y R d R d [ ] x y K CK K R d R d 1 x y K R d R d θ dγx, y + CK K dγx, y + CK K = 1 K W θ,µ, µ + CK K.

11 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES 11 Lemma Le >. Suppose a {ϱ n } n N converges weakly o ϱ in L 1 R d and a {M 1 ϱ n } n N is bounded. Ten M 1 ϱ is finie, and we ave lim inf n C ϱ, ϱ n C ϱ, ϱ ϱ P ac 1. Proof: Te fac a M 1 ϱ is finie follows from e weak lower-semiconinuiy in L 1 R d of M 1. Le now γ n Γ ϱ, ϱ n. Since {M 1 ϱ n } n N is bounded we ave 2.17 sup n N x + y γ n dx, dy < +. R d As x + y is coercive, equaion 2.17 implies a {γ n } n N admis a cluser poin γ for e opology of e narrow convergence. Furermore i is easy o see a γ Γ ϱ, ϱ and so, since c is coninuous and bounded below, we ge lim inf C ϱ, ϱ n = lim inf c x, y dγ n x, y c x, y dγx, y C ϱ, ϱ. n n R d R d R d R d 3. Exisence of soluions in a bounded domain Trougou is secion we assume a 2.2 and 2.3 old. We recall a L saisfies L1, L2 and L3. We also assume a R d is an open bounded se wose boundary is of zero Lebesgue measure, and we denoe by is closure. Te goal is o prove exisence of disribuional soluions o equaion 1.1 by using an approximaion by discreizaion in ime. More precisely, in subsecion 3.1 we consruc approximae soluions a discree imes {, 2, 3,...} by an implici Euler sceme, wic involves e minimizaion of a suiable funcional. Ten in subsecion 3.2 we explicily caracerize e minimizer inroducing a dual problem. We en sudy e properies of an augmened acion funcional wic allows o prove a priori bounds on e De Giorgi s variaional and geodesic inerpolaions cfr. subsecion 3.4. Finally, using ese bounds we can ake e limi as and prove exisence of disribuional soluions o equaion 1.1 wen θ beaves a infiniy like α, α > Te discree variaional problem. We fix a ime sep > and for simpliciy of noaion we se c = c. We fix ϱ P ac, and we consider e variaional problem 3.1 inf ϱ P ac C ϱ, ϱ + Uϱ. Lemma 3.1. Tere exiss a unique minimizer ϱ of problem 3.1. Suppose in addiion a L4 olds. If M, + and ϱ M, en ϱ M. In oer words, e maximum principle olds. Proof: Exisence of a minimizer ϱ follows by classical meods in e calculus of variaion, anks o e lower-semiconinuiy of e funcional ϱ C ϱ, ϱ + Uϱ in e weak opology of measures and o e superlineariy of U wic implies a any limi poin of a minimizing sequence sill belongs o P ac. To prove uniqueness, le ϱ 1 and ϱ 2 be wo minimizers, and ake γ 1 Γ ϱ, ϱ 1, γ 2 Γ ϱ, ϱ 2 cfr. remark 2.3. Ten γ 1+γ 2, so a C ϱ, ϱ 1 + ϱ Γ ϱ, ϱ 1+ϱ 2 2 γ1 + γ 2 cx, y d 2 = C ϱ, ϱ 1 + C ϱ, ϱ 2. 2

12 12 A. FIGALLI, W. GANGBO, AND T. YOLCU Moreover by sric convexiy of U ϱ1 + ϱ 2 U 2 Uϱ 1 + Uϱ 2 2 wi equaliy if and only if ϱ 1 = ϱ 2. Tis implies uniqueness. Tanks o L1 and L4 one easily ges a c x, x < c x, y for all x, y, x y. Tanks o is fac e proof of e maximum principle is a folklore wic can be found in [18] Caracerizaion of minimizers via a dual problem. Te aim of is paragrap is o compleely caracerize e minimizer ϱ provided by lemma 3.1. We are going o idenify a problem, dual o problem 3.1, and o use i o acieve a goal. We define E E c o be e se of pairs u, v C C suc a ux + vy cx, y for all x, y, and we wrie u v c. We consider e funcional Ju, v := uϱ dx U v dx. To alleviae e noaion, we ave omied o display e ϱ dependence in J. We recall some well-known resuls: Lemma 3.2. Le u C b. Ten u c c u, u c c u, u c c c = u c, and u c c c = u c. Moreover: i If u = v c for some v C, en: a Tere exiss a consan A = Ac,, independen of u, suc a u is A-Lipsciz and A-semiconcave. b If x is a poin of differeniabiliy of u, ȳ, and u x + vȳ = c x, ȳ, en x is a poin of differeniabiliy of c, ȳ and u x = x c x, ȳ. Furermore ȳ = Φ L 1, x, p H x, u x, and in paricular ȳ is uniquely deermined. ii If v = u c for some u C, en: a Tere exiss a consan A = Ac,, independen of v, suc a v is A-Lipsciz and A-semiconcave. b If x, ȳ is a poin of differeniabiliy of v, and u x + vȳ = c x, ȳ, en ȳ is a poin of differeniabiliy of c x, and vȳ = y c x, ȳ. Furermore, x = Φ L 1, ȳ, p H ȳ, vȳ, and in paricular ȳ is uniquely deermined. In paricular, if K R is bounded, e se {v c : v C, v c K } is compac in C, and weak compac in W 1,. Proof: Despie e fac a e asserions made in e lemma are now par of e folklore of e Monge-Kanorovic eory, we skec e main seps of e proof. Te firs par is classical, and can be found in [12, 16, 17]. Regarding i-a, we observe a by remark 2.11 e funcions c, y are uniformly semiconcave for y, so a u is semiconcave as e infimum of uniformly semiconcave funcions cfr. for insance [8, Appendix A]. In paricular u is Lipsciz, wi a Lipsciz consan bounded by x c L. To prove i-b, we noe a u x c, ȳ x. Since by remark c, ȳ x is nonempy, we conclude a c, ȳ is differeniable a x if u is. Hence u x = x c x, ȳ = v Lσ, σ

13 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES 13 were σ : [, ] is e unique curve suc a c x, ȳ = Lσ, σ d cfr. [8, Secion 4 and Appendix B]. Tis ogeer wi equaion 2.12 implies 3.2 ȳ = Φ L 1, x, p H x, u x. Te proof of ii is analogous. Remark 3.3. By lemma 3.2, if u = v c for some v C b we can uniquely define L d -a.e. a map T : dom u suc a ux + vt x = cx, T x. Tis map is coninuous on dom u, and since u can be exended o a Borel map on we conclude a T can be exended o a Borel map on, oo. Moreover we ave ux = x cx, T x L d -a.e., and T is e unique opimal map pusing any densiy ϱ P ac forward o µ := T # ϱl d P cfr. for insance [12, 16, 17]. Lemma 3.4. If u, v E and ϱ P ac, en Ju, v C ϱ, ϱ + Uϱ. Proof: Le γ Γϱ, ϱ. Since Uϱy + U vy ϱyvy and u, v E, inegraing e inequaliy we ge 3.3 Uϱy+U vy dy ϱyvy dy cx, y dγx, y+ ϱ xux dx. Rearranging e expressions in equaion 3.3 and opimizing over Γϱ, ϱ we obain e resul. Lemma 3.5. Tere exiss u, v E maximizing Ju, v over E and saisfying u c = v and v c = u. Furermore: i u and v are Lipsciz wi a Lipsciz consan bounded by c L. ii ϱ v := U v is a probabiliy densiy on, and e opimal map T associaed o u cfr. remark 3.3 puses ϱ L d forward o ϱ v L d. Proof: Noe a if u c = v and v c = u, en i is a direc consequence of lemma 3.2. Before proving e firs saemen of e lemma, le us sow a i implies ii. Le ϕ C and se v ε := v + εϕ, u ε := v ε c. Remark 3.3 says a for L d -a.e. x e equaion u x + v y = cx, y admis a unique soluion T x. As done in [1] cfr. also [11] we ave a u ε x u x u ε u ε ϕ, lim = ϕt x ε ε for L d -a.e. x. Hence by e Lebesgue dominaed convergence eorem u ε x u x 3.4 lim ϱ x dx = ϕt xϱ x dx. ε ε Since u, v maximizes J over E, by equaion 3.4 we obain Terefore 3.5 Ju ε, v ε Ju, v = lim = ε ε ϕt xϱ x dx + ϕt xϱ x dx = U v xϕx dx. U v xϕx dx.

14 14 A. FIGALLI, W. GANGBO, AND T. YOLCU Coosing ϕ 1 in equaion 3.5 and recalling a U cfr. lemma 2.4ii we discover a ϱ v := U v is a probabiliy densiy on. Moreover equaion 3.5 means a T puses ϱ L d forward o ϱ v L d. Tis proves ii. We evenually proceed wi e proof of e firs saemen. Observe a e funcional J is coninuous on E, wic is a closed subse of C C. Tus i suffices o sow e exisence of a compac se E E suc a E {u, v : u c = v, v c = u} and sup E J = sup E J. If u, v E en u v c, and so Ju, v Jv c, v. Bu as poined ou in lemma 3.2 v v c c, and since by lemma 2.4ii U C 1 R is monoone nondecreasing we ave Ju, v Jv c, v Jv c, v c c. Se ū = v c and v = v c c. Observe a by lemma 3.2 ū = v c and v = ū c. As U C 1 R and U, e funcional λ eλ := U vx + λ dx is differeniable and e λ = U vx + λ dx. Since by lemma 2.4iv U grows superlinearly a infiniy, so does eλ. Hence 3.6 lim Jū + λ, v λ = lim ūϱ dx + λ eλ =. λ + λ + Moreover, as U cfr. lemma 2.4ii, 3.7 lim Jū + λ, v λ lim ūϱ dx + λ =. λ λ Since λ Jū + λ, v λ is differeniable, 3.6 and 3.7 imply a Jū + λ, v λ acieves is maximum a a cerain value λ wic saisfies 1 = e λ. Terefore we ave ũ, ṽ := ū + λ, v λ E, Jū, v Jũ, ṽ, and U ṽ dx = 1. Tis las inequaliy and e fac a U ṽ is coninuous on e compac se ensure e exisence of a poin x suc a ṽx = U 1/L d. In lig of lemma 3.2 and e above reasoning we ave esablised a e se E := {u, v : u, v E, u c = v, v c = u, vx = U 1/L d for some x } saisfies e required condiions. Se φϱ := C ϱ, ϱ + Uϱ. Lemma 3.6. Le ϱ be e unique minimizer of φ provided by lemma 3.1, and le u, v be a maximizer of J obained in lemma 3.5. Ten ϱ = U v, and max J = Ju, v = φϱ = min φ. E P ac Proof: Le T be as in lemma 3.5ii, and define ϱ v := U v. Noe a since T puses ϱ L d forward o ϱ v L d, we ave a id T # ϱ L d Γϱ, ϱ v. Terefore, as cx, T x = u x+v T x for ϱ L d -a.e. x, C ϱ, ϱ v cx, T xϱ x dx = u x + v T x ϱ x dx 3.8 = u xϱ x dx + v xϱ v x dx.

15 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES 15 Since C ϱ, ϱ v u xϱ x dx + v xϱ v x dx rivially olds as u v c, inequaliy 3.8 is in fac an equaliy, and so 3.9 C ϱ, ϱ v + Uϱ v = u xϱ x dx + v xϱ v x + Uϱ v dx. Combining 3.9 wi e equaliy v ϱ v = Uϱ v +U v wic follows from ϱ v = U v we ge C ϱ, ϱ + Uϱ = Ju, v, wic ogeer wi lemma 3.4 gives a ϱ v minimizes φ over P ac and sup E J = φϱ v. Since e minimizer of φ over P ac is unique cfr. lemma 3.1, is concludes e proof. Remark 3.7. Tanks o lemma 3.2, on dom v we can uniquely define a map S by u Sy+v y = csy, y, and we ave v y = y csy, y. Tis map is e inverse of T up o a se of zero measure, i puses ϱ L d forward o ϱ L d, and Sy = Φ L 1, y, p H y, v y. Moreover, anks o lemma 3.6, U ϱ = v is Lipsciz, and v y = [U ϱ ]y. In paricular Sy = Φ L 1, y, p H y, [U ϱ ]y. We observe a e dualiy meod allows o deduce in an easy way e Euler-Lagrange equaion associaed o e funcional φ, by-passing many ecnical problems due o regulariy issues. Moreover i also gives y csy, y = [U ϱ ]y L d -a.e. in and no only ϱ L d -a.e Augmened acions. We now inroduce e funcional and we define Φτ, ϱ, ϱ := C τ ϱ, ϱ + Uϱ φ τ ϱ := inf Φτ, ϱ, ϱ. ϱ P ac ϱ, ϱ P ac, Te goal of is subsecion is o sudy e properies of Φ and φ τ, in e same spiri as in [3, Caper 3]. In e sequel, we fix ϱ P ac. Lemma 3.6 provides exisence of a unique minimizer of Φτ, ϱ, ϱ over P ac, wic we call ϱ τ. Remark 3.8. i Noe a φ τ ϱ Φτ, ϱ, ϱ Uϱ since C τ ϱ, ϱ, cfr. lemma 2.7. ii Tanks lemma 3.9 below, τ φ τ is monoone nonincreasing on, +. Terefore seing φ ϱ = Uϱ ensures a τ φ τ remains monoone nonincreasing on [, +, and we ave Uϱ Uϱ = φ ϱ φ ϱ + C ϱ, ϱ. Lemma 3.9. Te funcion τ φ τ ϱ is nonincreasing, and saisfies 3.1 C τ1 ϱ, ϱ τ1 C τ ϱ, ϱ τ1 τ 1 τ φ τ 1 ϱ φ τ ϱ τ 1 τ C τ 1 ϱ, ϱ τ C τ ϱ, ϱ τ τ 1 τ < τ τ 1.

16 16 A. FIGALLI, W. GANGBO, AND T. YOLCU Te funcion τ φ τ ϱ is Lipsciz on an inerval of e form [τ, τ 1 ], +, [, +, and L 1 loc 3.11 φ τ1 ϱ φ τ ϱ = τ1 dφ τ ϱ τ dτ τ dτ τ τ 1. Proof: I is an immediae consequence of e definiion of φ τ and ϱ τ a, for all τ, τ 1 >, C τ1 ϱ, ϱ τ C τ ϱ, ϱ τ φ τ1 ϱ φ τ ϱ. dφ τ ϱ dτ Tis gives e second inequaliy in 3.1, wic ogeer wi lemma 2.13i implies a τ φ τ ϱ is nonincreasing. Te firs inequaliy in 3.1 can be esablised in a similar manner. Se S := [τ, τ 1 ], +, and le r > be suc a is conained in e closed ball of radius r cenered a e origin. Le k S r be e consan appearing in lemma 2.12, and fix γ Γ τ ϱ, ϱ τ cfr. remark 2.3. Since ϱ and ϱ τ are suppored inside we ave a γ is suppored on, wic implies a c τ x, y c τ1 x, y k S r for γ-a.e. x, y. We conclude a C τ ϱ, ϱ τ = c τ dγ c τ1 + k S r τ τ 1 dγ C τ1 ϱ, ϱ τ + k S r τ τ 1. R d R d R d R d Tis ogeer wi lemma 2.13i implies 3.12 C τ ϱ, ϱ τ C τ1 ϱ, ϱ τ = C τ ϱ, ϱ τ C τ1 ϱ, ϱ τ k S r τ τ 1. Similarly 3.13 C τ1 ϱ, ϱ τ1 C τ ϱ, ϱ τ1 k S r τ τ 1. We combine 3.1, 3.12 and 3.13 o conclude a τ φ τ ϱ is Lipsciz on [τ, τ 1 ], +. Now, since τ φ τ ϱ is nonincreasing, recalling remark 3.8 we ave τ1 dφ τ ϱ τ dτ = φ τ ϱ φ τ1 ϱ Uϱ φ τ1 ϱ. dτ τ Since τ 1 > τ > are arbirary, we conclude a dφ τ ϱ dτ L 1 loc [, + and 3.11 olds. For >, we denoe by T e opimal map a puses ϱ L d forward o ϱ L d as provided by e previous subsecion 3.2. We ave T x = Φ L 1, x, p H x, u x, wi u, v a maximizer of u, v ϱ udx U v dx over e se of u, v C C suc a u v c. We recall a u and v are Lipsciz cfr. lemma 3.5 and U v = ϱ cfr. lemma 3.6. Moreover, if we define e inerpolaion map beween ϱ and ϱ by 3.14 T s x := ΦL 1 s, x, p H x, u x, s [, ], we ave 3.15 c x, T x = L σ x s, σ x s ds, wi σ x s := T s x. Finally, since v = U ϱ, denoing by S e inverse of T we also ave 3.16 y c S y, y = [U ϱ ]y = v L σ S y, σ S y for L d -a.e. y.

17 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES 17 Lemma 3.1. For L 1 -a.e. > we ave dφ ϱ 3.17 = = d H y, [U ϱ ]y ϱ y dy. Proof: For ε /2, s [, + ε] and L d -a.e. x we define s σε x s := Φ L 1 + ε, x, ph x, u x. Because u is a Lipsciz funcion we ave 3.18 sup{ σɛ C 1 [,+ɛ] : ε /2, x } < + x,ε Since σε x = x and σ ε + ε = T x, by e definiion of C +ε we ge +ε 3.19 C +ε ϱ, ϱ ϱ x L σε x, σ ε x + ε ds dx = ϱ x Moreover, since Lx, is convex, 3.2 L σ x, σ x L σ x, Recall a 3.21 v L σ x, + ε σx, + ε σx We combine equaions o obain 3.22 C +ε ϱ, ϱ C ϱ, ϱ ε + ε + ε σx + ε v L σ x, + ε = L σ x, ϱ x + ε σx, σ x. + ε σx + H σ x, v L σ x, L σ x, + ε σx ds dx + ε σx, H σ x, v L σ x, + ε σx ds dx. Tanks o 3.18 we can apply e Lebesgue dominaed convergence eorem and en use o e conservaion of e Hamilonian H cfr. remark 2.9ii o obain ϱ x H σ x, v L σ x, + ε σx ds dx = ϱ x H σ x, v L σ x, σ x ds dx = ϱ xh σ x, v L σ x, σ x dx. lim ε Recalling a σ x = T x puses ϱ L d forward o ϱ L d and exploiing 3.16 in e previous equaliy we conclude a H σ x, v L σ x, 3.23 lim ϱ x ε + ε σx ds dx = H y, [U ϱ ]y ϱ y dy. We now disinguis wo cases, depending on e sign of ε. 1. If < ε < /2 we se τ 1 = + ε and τ = in e second inequaliy in 3.1 o obain φ +ε ϱ φ ϱ ε Tis ogeer wi 3.22 yields φ +ε ϱ φ ϱ 1 ϱ x ε + ε C +εϱ, ϱ C ϱ, ϱ. ε H σ x, v L σ x, + ε σx ds dx.

18 18 A. FIGALLI, W. GANGBO, AND T. YOLCU Leing ε end o and exploiing 3.23 we conclude a d + φ ϱ 3.24 = H y, [U ϱ ]y ϱ y dy. d 2. If /2 < ε < we se τ 1 = and τ = + ε in e firs inequaliy in 3.1, and by rearranging e erms we obain Tis, ogeer wi 3.22 yields ε C +ε ϱ, ϱ C ϱ, ϱ ε H σ x, v L σ x, + ε σx φ +εϱ φ ϱ. ε ds dx φ +εϱ φ ϱ. ε We combine 3.23 and 3.25 o ge 3.26 H y, [U ϱ ]y ϱ y dy d φ ϱ =. d Since by lemma 3.9 τ φ τ ϱ is locally Lipsciz on, +, i is differeniable L 1 -a.e. Hence 3.24 and 3.26 yield 3.17 a any differeniabiliy poin De Giorgi s variaional and geodesic inerpolaions. We fix ϱ P ac, a ime sep >, and se ϱ := ϱ. We consider ϱ P ac e unique minimizer of Φτ, ϱ, provided by lemma 3.1, and we inerpolae beween ϱ and ϱ along pas minimizing e acion A : anks o [8, Teorem 5.1] ere exiss a e unique soluion ϱ s P ac of C s ϱ, ϱ s + C s ϱ s, ϱ = C ϱ, ϱ, wic is also given by cfr for e definiion of T s ϱ s L d := T s #ϱ L d, s. Moreover [8, Teorem 5.1] ensures a T s is inverible ϱ s-a.e., so a in paricular ere exiss a unique vecor field Vs defined ϱ s -a.e. suc a V s T s = st s ϱ s -a.e. Recall a by lemma 3.5i u L c L. Exploiing equaion 3.14 and e fac a s Φ L maps bounded subses of R d R d ono bounded subses of R d R d, we obain a sup s s T s L ϱ s < +. Terefore sup s Vs L ϱ s < +. Finally a direc compuaion gives a 3.27 s ϱ s + div ϱ s V s = in e sense of disribuion on, R d. Observe a ϱ = ϱ and ϱ = ϱ. Remark Noe a aloug e range of T is conained in, a of T s may fail o be in a se. Indeed even if x and T x are bo in, e Lagrangian flow provided by L and connecing x o T x may no lie enirely in.

19 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES 19 We se { } ϱ s := argmin C s ϱ, ϱ + Uϱ : ϱ P ac, s. In a meric space S, dis wi sc s = dis 2, e inerpolaion s ϱ s is due o De Giorgi [6] cfr. also [3, 7]. Teorem Te following energy inequaliy olds: Uϱ Uϱ ds Ly, V s ϱ s dy + R d ds H x, U ϱ s ϱ s dx. Proof: By lemma 3.1 s φ s ϱ is locally Lipsciz on [, ], and 3.17 olds. Hence, by exploiing remark 3.8 we conclude a d H x, [U ϱ s ] ϱ s dx φ ϱ φ ϱ = Uϱ Uϱ C ϱ, ϱ = Uϱ Uϱ = Uϱ Uϱ d LT s x, st s xϱ x dx d Ly, V s y ϱ s y dy. R d We remark a e las inegral as o be aken on e wole R d, as we do no know in general a e measures ϱ s are concenraed on, cfr. remark We now ierae e argumen above: lemma 3.6 ensures e exisence of a sequence {ϱ k } k= P ac suc a } ϱ k+1 {C := argmin ϱ k, ϱ + Uϱ : ϱ Pac. As above, we define { } 3.28 ϱ k+s := argmin C s ϱ k, ϱ + Uϱ : ϱ Pac 1, s. Te argumens used before can be applied o ϱ k, ϱ k+1 in place of ϱ, ϱ o obain a unique map T k : suc a id T k # ϱ k L d Γ ϱ k, ϱ k+1. Moreover, for s, we define ϱ k+s o be e inerpolaion along pas minimizing e acion A, a is ϱ k+s is e unique soluion of C s ϱ k, ϱ k+s + C s ϱ k+s, ϱ k+1 = C ϱ k, ϱ k+1. We denoe by u k+s, v k+s e soluion o e dual problem o 3.28 provided by lemma 3.5. Replacing u by u k in 3.14 we obain e inerpolaion maps T k s. As before, we consider e inerpolaion measures ϱ k+s Ld := Tk s #ϱ k Ld, and we define ϱ k+s-a.e. e velociies V k+s by Vk+s T k s = stk s. As in 3.27, one can easily see a e curve of densiies s ϱ s saisfies e coninuiy equaion 3.29 s ϱ s + div ϱ s V s = in e sense of disribuion on, + R d.

20 2 A. FIGALLI, W. GANGBO, AND T. YOLCU Corollary For >, for any j k N, we ave k Uϱ j Uϱ k ds Ly, Vs ϱ s dy + j R d k Proof: Te resul is a direc consequence of eorem j ds H x, [U ϱ s ] ϱ s dx Sabiliy propery and exisence of soluions. As before, we fix ϱ P ac and make e addiional assumpion a Uϱ is finie. We fix T > and we wan o prove exisence of soluions o equaion 1.1 on [, T ]. Recall a by lemma 2.13i C s ϱ, ϱ for any s, ϱ P ac 1. Tis ogeer wi e definiion of ϱ k+s yields C ϱ k, ϱ k+s + U ϱ k+s U ϱ k, s. By adding over k N e above inequaliy, anks o remark 2.5 we ge 3.3 C ϱ k, ϱ k+1 U ϱ lim inf U ϱ n n U ϱ + a + b L d. k= Similarly, using corollary 3.13, e fac a H and L3, for any N > ineger we ave N Uϱ Uϱ N + ds Ly, Vs ϱ s dyϱ s dx R d N 3.31 ds θ Vs ϱ s dyϱ s dx A N a b L d. R d We also recall a v : R is a Lipsciz funcion cfr. lemma 3.5i wic saisfies v = U ϱ, so a seing β := U v = P ϱ we ave 3.32 ϱ [U ϱ ] = U v v = [U v ] = [P ϱ ] = β L d -a.e. We sar wi e following: Lemma Le A be e consan provided in assumpion L3. We ave 3.33 Uϱ Uϱ + A. Moreover, for any K > ere exiss a consan CK > suc a, for any, 1], 3.34 W 1 ϱ, ϱ C K + 2A + CK [, T ], K 3.35 W 1 ϱ, ϱ k C K + A + CK [k, k + 1], k N, K 3.36 W 1 ϱ, ϱ C s K + 2A + CK [ s + ] s, K 3.37 W 1 ϱ, ϱ C s K + A + CK [ s + ] s. K Here C is a posiive consan independen of, K, and, 1].

21 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES 21 Proof: Le [k, k + 1] for some k N. Ten by lemma 2.13ii we ave 3.38 Uϱ A k Uϱ + C k ϱ, ϱ k Uϱ k. In paricular Uϱ k+1 Uϱ k + A for all k N, so a adding over k we ge Uϱ Uϱ k + A k Uϱ k 1 + A [ + k]... Uϱ + A [k + k] = Uϱ + A. Tis proves Now, since C C k cfr. lemma 2.13i, we ave C ϱ k, ϱ Uϱ k Uϱ wic combined wi equaion 3.38 and remark 2.5 gives [k, k + 1], 3.39 C ϱ k, ϱ Uϱ + A + a + b L d [k, k + 1]. Moreover, as ϱ k = ϱ k for any k N, using again lemma 2.13ii we ge C ϱ k, ϱ k+1 = C kϱ k, ϱ + C k+1 ϱ, ϱ k+1 C k ϱ k, ϱ A C ϱ k, ϱ A. Tanks o lemma 2.13ii-iii, for any K > ere exiss a consan CK > suc a W 1 ϱ k, ϱ 1 K C ϱ k, ϱ + A + CK, K W 1 ϱ k, ϱ 1 K C ϱ k, ϱ + A + CK. K Using e riangle inequaliy for W 1 and combining ogeer e esimaes above, 3.34 and 3.35 follow. Finally, o prove equaions 3.36 and 3.37, we observe a 3.3 combined wi lemma 2.13iii gives W 1 ϱ N, ϱ M N 1 j=m 1 K W 1 ϱ j+1, ϱ j N 1 j=m C ϱ j+1, ϱ j + A + CK N M K 1 [ U ϱ K + a + b L d ] + A + CK N M. K Combining is esimae wi 3.34 and 3.35 we obain e desired resul. Lemma i Te curve ϱ P ac 1 Rd is coninuous on [, T ]. ii Te curve ϱ P ac is coninuous on [, T ] wi respec o W 1. Proof: i Tanks o 3.29 and 3.31, i is no difficul o sow ϱ coninuous on [, T ] cfr. [3, Caper 8]. P ac 1 Rd is uniformly

22 22 A. FIGALLI, W. GANGBO, AND T. YOLCU ii Wiou loss of generaliy we can resric e sudy of ii o e se [, ]. Before saring our argumen, le us recall a since is bounded e weak convergence coincides wi e W 1 convergence on P ac. Fix s, ], and le S, ] be a closed inerval. Lemma 2.12 yields a C ϱ o, ϱ is Lipsciz on S, wi a Lipsciz consan k S independen of ϱ. Since is bounded and P ac is precompac for e weak convergence, i is also precompac for W 1. Now, given ϱ P ac, by definiion of ϱ we ave Φ, ϱ, ϱ Φ, ϱ, ϱ, and so 3.4 Φs, ϱ, ϱ Φs, ϱ, ϱ + 2 s k S Le {ϱ n } n N be an arbirary subsequence of {ϱ } S converging o ϱ as n s. Ten, anks o 3.4 and e fac a Φs, ϱ, is lower semiconinuous for e weak opology we obain Φs, ϱ, ϱ Φs, ϱ, ϱ, a is ϱ minimizes Φs, ϱ, over P ac. Hence, anks o lemma 3.1 ϱ = ϱ s. Since e limi ϱ is independen of e subsequence { n } n N and we are on a meric space, we conclude a {ϱ } S converges o ϱ s as s. Tis prove a ϱ is coninuous a s. I remains o sow a ϱ is coninuous a. Te fac a c x, x cfr. lemma 2.7 sows a C ϱ, ϱ. Hence, using e definiion of ϱ we obain C ϱ, ϱ + Uϱ C ϱ, ϱ + Uϱ Uϱ. Combining e above esimae wi 2.13iii we conclude a for eac K > ere exiss a consan CK independen of suc a 3.41 A + CK + KW 1 ϱ, ϱ + Uϱ Uϱ. Le {ϱ n } n N is be subsequence of {ϱ } S converging o ϱ as n. Ten e lower semiconinuiy of U wi respec o e weak opology ogeer wi 3.41 give KW 1 ϱ, ϱ + Uϱ Uϱ. Leing K + we obain W 1 ϱ, ϱ, a is ϱ = ϱ. By e arbirariness of e sequence { n } n N we conclude as before a ϱ is coninuous a. We can now prove e compacness of our discree soluions. Proposiion Tere exiss a sequence n, a densiy ϱ P ac [, T ], and a Borel funcion V : [, T ] R d suc a: i Te curves ϱ n P ac and ϱ n P ac R d converge o ϱ := ϱ, wi respec o e narrow opology. Moreover e curve ϱ := ϱ, is uniformly coninuous and lim + ϱ = ϱ in P ac, W 1. ii Te vecor-valued measures ϱ n xv n xdxd converge narrowly o ϱ xv xdxd, were V := V,. iii ϱ + divϱ V = olds on, T in e sense of disribuion. Proof: Firs of all, le us recall a e narrow opology is merizable cfr. [3, Caper 5]. Tanks o lemma 3.15 and e esimaes 3.36 and 3.37, as K > is arbirary i is easy o see a e curves ϱ and ϱ are equiconinuous wi respec o e 1-Wassersein disance. Since bounded ses wi respec o W 1 are precompac wi respec o e narrow opology on R d cfr. for insance [3, Caper 7], by Ascoli-Arzelà Teorem we can find a sequence n suc a ϱ n P ac and ϱ n narrow-coninuous curve µ P wic is e same for bo ϱ n P ac R d converge uniformly locally in ime o a and ϱ n anks o 3.34.

23 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES 23 Moreover µ is suppored in as so is ϱ n, and e iniial condiion w -lim + ϱ n = ϱ olds in e limi. Concerning e vecor-valued measure V ϱ, recalling a H, anks o corollary 3.13 and remark 2.5 we ave T d Lx, V ϱ dx Uϱ + a + b L d. R d By L3 is gives T d θ V ϱ dx Uϱ + a + b L d + A T =: C 1. R d Te above inequaliy, ogeer wi e superlineariy of θ and e uniform convergence of ϱ o µ, implies easily a e vecor-valued measure V ϱ ave a limi poin λ, wic is concenraed on [, T ]. Moreover, e superlineariy and e convexiy of θ ensure a λ µ, and ere exiss a µ-measurable vecor field V : [, T ] R d suc a λ = V µ, and T d θ V dµ C 1. To conclude e proof of i and ii we ave o sow a µ L d+1. We observe a anks o equaion 3.33 T T T 2 d Uϱ n dx = Uϱ n d T Uϱ + A 2, so a by e superlineariy of U any limi poin of ϱ n is absoluely coninuous. Hence µ = ϱl d, and i and ii are proved. Finally from equaion 3.29 we deduce a 3.42 ϱ n + div ϱ n V n = on, T in e sense of disribuion, so a iii follows aking e limi as n. We are now ready o prove e following exisence resul. To simplify e noaion, given wo nonnegaive funcions f and g, we wrie f g if ere exiss wo nonnegaive consans c, c 1 suc a c f + c 1 g. If bo and old, we wrie f g. Teorem Le R d be an open bounded se wose boundary is of zero Lebesgue measure, and assume a H saisfies H1, H2 and H3. Assume a U saisfies 2.2 and 2.3, and le ϱ P1 ac be suc a Uϱ is finie. Le ϱ and V be as in proposiion Ten we ave P ϱ L 1, T ; W 1,1, [P ϱ ] is absoluely coninuous wi respec o ϱ, and T [ 3.43 Uϱ Uϱ T d L x, V + H x, ϱ 1 [P ϱ ] ] ϱ dx. Furermore, if θ α for some α > 1 and U saisfies e doubling condiion 3.44 U + s CU + Us + 1, s, en ϱ AC α, T ; P ac α, 3.45 Uϱ T1 Uϱ T2 = T2 T 1 d [P ϱ ], V dx,

24 24 A. FIGALLI, W. GANGBO, AND T. YOLCU for T 1, T 2 [, T ]. In paricular V x = p H x, ϱ 1 [P ϱ ] ϱ -a.e., and ϱ is a disribuional soluion of equaion 1.1 saring from ϱ. Suppose in addiion a H4 olds. If ϱ M for some M en ϱ M for every, T maximum principle. Proof: Te maximum principle is a direc consequence of lemma 3.1. We firs remark a e las par of e saemen is a simple consequence of equaions 3.43 and 3.45 combined wi proposiion 3.16i-iii. So i suffices o prove equaions 3.43 and We firs prove Corollary 3.13 implies a, if T [k n, k + 1 n ] for some k N, since L A and H we ave Uϱ n Uϱ n k+1 n T [ d R d T Lx, V n ϱ n + H x, [U ϱ n ] ] ϱ n dx A n. We now consider wo coninuous funcions w, w : [, T ] R d R d wi compac suppor. Ten T [ d Lx, V n ϱ n + H x, [U ϱ n ] ] ϱ n dx R d T [ ] d V n, w, x ϱ n Hx, w, x ϱ n dx + d R d [ [U ϱ n Tanks o proposiion 3.16i-ii we immediaely ge T [ lim d n R d V n, w, x ϱ n ], w, x ϱ n ] Lx, w, xϱ n dx. ] Hx, w, x ϱ n dx T ] = d [ V, w, x ϱ Hx, w, xϱ dx, R d so a aking e supremum among all coninuous funcions w : [, T ] R d R d wi compac suppor we obain T T lim inf d Lx, V n n ϱ n dx d Lx, V ϱ dx. R d R d Concerning e oer erm, we observe a, anks o remark 2.5, as L A we ave a T d H x, [U ϱ n ] ϱ n dx R d is uniformly bounded wi respec o n. In paricular, since by H3 Hx, p p C 1 for some consan C 1, anks o equaion 3.32 we ge a T T d [P ϱ n ] dx = d [U ϱ n ] ϱ n dx R d R d

25 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES 25 is uniformly bounded. Tis implies a, up o a subsequence, e vecor-valued measures [P ϱ n ]dxd converges weakly o a measure ν of finie oal mass. Terefore we obain T + > lim inf d H x, [U ϱ n n ] ϱ n dx R d T [ = lim n T d d [U ϱ n w, x, νd, dx ], w, x ϱ n T d ] Lx, w, xϱ n dx Lx, w, xϱ dx. By e arbirariness of w we easily ge a e measure νd, dx is absoluely coninuous wi respec o ϱ dxd, so a νd, dx = e xϱ xdxd for some Borel funcion e : [, T ] R d. We now observe a by Faou Lemma we also ave wic gives 3.46 lim inf n T lim inf n [P ϱ n R d ] dx d < +, R d [P ϱ n ] dx < + for [, T ] \ N, wi L 1 N =. Hence, for any [, T ] \ N ere exiss a subsequence ϱ n k suc a lim inf ] dx = lim ] dx, n k [P ϱ n R d [P ϱ nk R d and P ϱ n k converges weakly in BV and L d -a.e. o a funcion β. As a consequence Dβ = e ϱ L d, so a β W 1,1. Since Q is coninuous, we deduce a ϱ n k = Q P ϱ n k converges L d -a.e. o Qβ. Recalling a ϱ n k also converges weakly o ϱ, we obain Qβ = ϱ, a is β = P ϱ. Moreover, from e equaliy β = e ϱ, we ge [P ϱ ] = e ϱ. We ave proved a P ϱ L 1, T ; W 1,1 and [P ϱ ] is absoluely coninuous wi respec o ϱ. Finally ϱ n k+1 n converges weakly o ϱ T, and e erm Uϱ n T is lower-semiconinuous under weak convergence, and is concludes e proof of equaion We now prove equaion Le us observe a e assumpion θ α implies a Lx, v v α and Hx, p p α, were α = α/α 1. Hence, anks o equaion 3.43 we ave T T > d Lx, V ϱ dx V α L α ϱ d and > Since T T d Hx, e ϱ dx e α L α ϱ d. ϱ + div ϱ V =, 3.47 implies a e curve ϱ is absoluely coninuous wi values in e α-wassersein space P α, and we denoe by V is velociy field of minimal norm cfr. [3, Caper 8]. Moreover, anks o equaion 3.48, e L α ϱ for a.e., T.

26 26 A. FIGALLI, W. GANGBO, AND T. YOLCU Denoing by ϱ e meric derivaive of e curve ϱ wi respec o e α-wassersein disance, cfr. equaion 2.1, by 3.47 and [3, eorem 8.3.1] we ave 3.49 ϱ V L α ϱ V L α ϱ < +. Since e ϱ = P ϱ wi P ϱ W 1,1 for a.e., we can apply [3, Teorem 1.4.6] o conclude a, for L 1 -a.e., U as a finie slope a ϱl d, U ϱ = e L α ϱ, and e = o Uϱ. Te las saemen means a e is e elemen of minimal norm of e convex se Uϱ, and so i belongs o e closure of { ϕ : ϕ Cc } in L α ϱ. Le Λ, T be e se of suc a a Uϱ ; b U is approximaely differeniable a ; c of [3] olds. We use equaions 3.48, 3.49, and e fac a U ϱ = e L α ϱ for L1 -a.e., T, o conclude a 3.5 T U ϱ ϱ d 1 α T e α L α ϱ d + 1 α T V α L α ϱ d < +. By [3, Proposiion 9.3.9] U is convex along α-wassersein geodesics, and so exploiing equaion 3.5 and invoking [3, Proposiion ] we obain a L 1, T \ Λ = and Uϱ is absoluely coninuous. Tus is poinwise, disribuional, and approximae derivaives coincide almos everywere, and by [3, Proposiion ] and e fac a e Uϱ we ge 3.51 d d Uϱ = e, V ϱ dx. Because V and V are bo velociy fields for ϱ we ave φ, V V ϱ dx = for all φ Cc for L 1 -a.e., T, and since e belongs o e closure of { ϕ : ϕ Cc } in L α ϱ we conclude by a densiy argumen a e, V V ϱ dx = for L 1 -a.e., T. Tis ogeer wi equaion 3.51 finally yields T Uϱ T1 Uϱ T2 = d e, V T2 ϱ dx = d as desired. T 1 T 1 e, V ϱ dx, Remark If ϱ is a general nonnegaive inegrable funcion on wic does no necessarily ave uni mass, we can sill prove exisence of soluions o equaion 1.1. Indeed, defining c := ϱ dx, we consider ϱ c P ac a soluion of equaion 1.1 for e Hamilonian H c x, p := chx, p/c and e inernal energy U c := Uc, saring from ϱ c := ϱ /c. Ten ϱ := cϱ c solves equaion 1.1. Moreover, using is scaling argumen also a a discree level, we can also consruc discree soluions saring from ϱ.

27 A VARIATIONAL METHOD FOR A CLASS OF PARABOLIC PDES 27 Remark We believe a e above exisence resul could be exend o more general funcions θ by inroducing some Orlicz-ype spaces as follows: for θ : [, + [, + convex, superlinear, and suc a θ =, we define e Orlicz-Wassersein disance W θ µ, µ 1 := inf We also define e Orlicz-ype norm f θ,µ := inf { λ > : { λ > : inf θ γ Γµ,µ 1 θ x y λ } f dµx 1. λ } dγ 1. I is no difficul o prove a e following dynamical formulaion of e Orlicz-Wassersein disance olds: { 1 } 3.53 W θ µ, µ 1 = inf V θ,µ d : µ + divµ V =. Now, in order o prove e ideniy 3.45 of e previous eorem in e case were θ does no necessarily beave as a power funcion, one sould exend e resuls of [3] o is more general seing. We believe suc an exension o be reacable aloug no sraigforward. However is kind of effor goes beyond e scope of is paper. 4. Exisence of soluions in unbounded domains for weak Tonelli Lagrangians Te aim of is secion is o exend e exisence resul proved in e previous secion o unbounded domains, using an approximaion argumen were we consruc our soluions in B m for smooed Lagrangians L m, and en we le m +. In order o be able o pass o e limi in e esimaes and find a soluion, we need o assume e exisence of wo consans c > and a d d+1, 1 suc a 4.1 U := max{ U, } c a. Te above assumpion, ogeer wi 2.2 and 2.3, are saisfied by posiive muliples of e following funcions: ln, or α wi α > 1. Under is addiional assumpion we now prove some lemmas and proposiion wic easily allow o consruc our soluion as a limi of soluions in bounded domains cfr. subsecion 4.2. Tanks o assumpion 4.1 we can prove a if M 1 ϱ is finie en U ϱ := R U ϱ dx is d finie, and so Uϱ = R Uϱ dx is well-defined. d Lemma 4.1. Tere exiss C = Cd, a suc a U ϱ CM 1 ϱ a + 1. Consequenly Uϱ is well defined wenever M 1 ϱ is finie. Furermore C can be cosen so a B R c U ϱ dx CM 1 ϱ a R d1 a a R.