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1 Georgian Mahemaical Journal Volume ), Number 3, ESTIMATES OF THE LOCATION OF A FREE BOUNDARY FOR THE OBSTACLE AND STEFAN PROBLEMS OBTAINED BY MEANS OF SOME ENERGY METHODS JESÚS ILDEFONSO DÍAZ Dedicaed o he memory of Jacques-Louis Lions Absrac. In his aer we use some energy mehods o sudy he locaion and formaion) of a free boundary arising in some unilaeral roblems, for insance, in he obsacle roblem and he Sefan roblem Mahemaics Subjec Classificaion: 35K35, 35R35, 35R70. Key words and hrases: Formaion and locaion of he free boundary, obsacle roblem, Sefan roblem, energy mehod 1. Inroducion Unilaeral roblems or variaional inequaliies arising in mechanics, hysics and economics were formulaed and successfully solved during he las half of he as cenury. Afer he ioneer works by J. L. Lions and G. Samacchia [16], [17], his ye of roblems araced he aenion of Jacques-Louis Lions for many years. He resened he resuls of his research in his shere in he domain in a series of works: [14], [10], [3], [4] and [13]. This ye of roblems may give rise o a free boundary searaing he domains of sace where he soluion behaves in differen ways. This free boundary has an imoran hysical meaning and, besides, i conains rich informaion on he soluion of he roblem since i usually includes he oins where he soluion loses is regulariy). The main goal of his aer is o exend he, so-called, energy mehod develoed since he beginning of he 80s for he sudy of free boundaries aearing in he soluions of nonlinear PDEs see, e.g., he monograh [1] by S. N. Anonsev, J. I. Díaz and S. I. Shmarev) in he case of some unilaeral roblems, for insance, he obsacle roblem and he Sefan roblem. The qualiaive behavior of he coincidence se for he obsacle roblem was iniiaed in 1976 by Brezis and Friedman [5] and Bensoussan and Lions [2] see also Tarar [20] and Evans and Knerr [11]) by using he maximum rincile. Some oher general references o he Sefan roblem are Friedman [12] and Meirmanov [18]. The energy mehods are of secial ineres in siuaions where he radiional mehods based on comarison rinciles fail. A yical examle of such a siuaion is eiher a higher-order equaion or a sysem of PDEs. Moreover, even when he comarison rincile holds, i may be exremely difficul o consruc suiable sub- or suer-soluions if, for insance, he equaion under ISSN X / $8.00 / c Heldermann Verlag

2 476 J. I. DÍAZ sudy conains ransor erms and has eiher variable or unbounded coefficiens or he righ-hand side. Here we shall deal wih he formaion of a free boundary for local soluions of he obsacle roblem ψu) div Ax,, u, D u) + Bx,, u, D u) + Cx,, u) + βu) fx, ), 1) where βu) is a maximal monoone grah given by βu) = {0} if u 0 and βu) = he emy se) if u < 0. We make he following general srucural assumions: Ax,, r, q) C 1 q 1, C 2 q Ax,, r, q) q, Bx,, r, q) C 3 r α q β, 0 Cx,, r) r, C 6 r γ+1 Gr) C 5 r γ+1, where r Gr) = ψr) r ψτ) dτ. 2a) 2b) 2c) Here C 1 C 6,, α, β, σ, γ, k are osiive consans which will be secified laer. We also consider he Sefan roblem ψu) 0 div Ax,, u, D u) + Bx,, u, D u) + Cx,, u) fx, ), 3) where ψu) is a maximal monoone grah ψu) = k + u + L if u > 0, ψu) = k u if u < 0, ψ0) = [0, L] wih osiive consans k +, k and L. In boh cases, we deal wih weak soluions saisfying he iniial condiion ux, 0) = u 0 x), x Ω. 4) We oin ou ha his energy mehod can also be alied o some roblems arising in Climaology. For insance, in [7] he exisence of mushy region is roved by giving a arial answer o a quesion raised by Jacques-Louis Lions see, e.g., [15]). 2. Saemen of he Main Resuls Le us sar by considering he obsacle roblem. Definiion. A funcion ux, ), wih ψu) C [0, T ] : L 1 loc Ω)), is called weak soluion of roblem 1), 4) if u L 0, T ; L γ+1 Ω )) L 0, T ; W 1, Ω )), Ω Ω, A,, u, Du), B,, u, Du), C,, u) L 1 Q); lim inf 0 Gu, )) = Gu 0 ) in L 1 Ω); ux, ) 0 and cx, ) βu, x)) a.e., x) 0, T ) Ω for some c L 1 0, T ) Ω), and for every es funcion ϕ L 0, T ; W 1, 0 Ω) )

3 ESTIMATES OF THE LOCATION OF A FREE BOUNDARY 477 W 1,2 0, T ; L Ω)), {ψu)ϕ A Dϕ B ϕ C ϕ cϕ} dxd Q Ω ψu)ϕ dx = =T =0 f ϕ dx d. 5) In conras o he consideraion of a finie roagaion rae or he uniform localizaion of a suor, we use some energy funcions defined in he domains of secial form. Le us inroduce he following noaion: given x 0 Ω and he nonnegaive arameers ϑ and υ, we define he energy se P ) P ; ϑ, υ) = {x, s) Q : x x 0 < ρs) ϑs ) υ, s, T )}. The shae of he local energy se P ) is deermined by he choice of arameers ϑ and υ. Here we ake ϑ > 0, 0 < υ < 1 and so P ) becomes a araboloid oher roeries should be sudied by oher choices, see [1]). We define he local energy funcions EP ) := Dux, τ) dxdτ, CP ) := ux, τ) dxdτ, P ) bt ) := ess su s,t ) x x 0 <ϑs ) υ P ) ux, s) γ+1 dx. Alhough our resuls are of local naure for insance, hey are indeenden of he boundary condiions), we need some global informaion on he global energy funcion Du, )) := ess su ux, s) γ+1 dx + Du + u ) dxd. 6) s 0,T ) Ω For he sake of simliciy, we addiionally assume ha 1 γ 1. Our main assumion concerns fx, ): we assume ha here exiss Θ > 0 and ρ > 0 such ha fx, ) < Θ on x 0 ) Ω, a.e. 0, T ). 7) In he resence of he firs order erm, B,, u, Du), we need he exra condiions α = γ 1 + γ)β/, ) β)/ ) β/ C 3 < Θ C 1 2 β if 0 < β <, 8) C 3 < Θ if β = 0 resecively Θ < C 2 if β = ). The nex resul shows how he mulivalued erm causes he formaion of he null-se of he soluion, even for he osiive iniial daa. Q Q

4 478 J. I. DÍAZ Theorem 1. There exis some osiive consans M,, and υ 0, 1) such ha any weak soluion of roblem 1), 4) wih Du) M saisfies ux, ) 0 in P : 1, υ). In he case of he Sefan roblem a definiion of weak soluion can be given in similar erms bu he inegral ideniy reads now as follows: =T {ψ u ϕ A Dϕ B ϕ C ϕ} dxd ψu)ϕ dx = f ϕ dx d, 9) Q for some ψ u L 1 0, T ) Ω), ψ u x, ) ψu, x)) a.e., x) 0, T ) Ω. To simlify he exosiion we assume now ha A and B are indeenden of u. Theorem 2. Assume ha fx, )< Θ resec. fx, )>Θ) on x 0 ) Ω, a.e. 0, T ). Then here exis some osiive consans M,, and υ 0, 1) such ha any weak soluion of roblem 3), 4) wih Du) M saisfies ux, ) 0 resec. ux, ) 0) in P : 1, υ). Ω =0 Q 3. Proofs and Remarks The roof of Theorem 1 consiss of several ars: Se 1. The inegraion-by-ars formula: i 1 + i 2 + i 3 + i 4 = Gux, ))dx + P {=T } A Du dxdθ + B udxdθ + C udxdθ uf dxdθ P P n x A u dγdθ + P P n τ Gux, ))dγdθ + P {=0} Gux, ))dx+ := j 1 + j 2 + j 3, where denoes he laeral boundary of P, i.e. = {x, s) : x x 0 = ϑs ) υ, s, T )}, dγ is he differenial form on he hyersurface { = cons}, n x and n τ are he comonens of he uni normal vecor o. This inequaliy can be roved by aking he cuing funcion ζx, θ) := ψ ε x x 0, θ) ξ k θ) 1 h θ+h T m ux, s)) ds, h > 0, θ

5 ESTIMATES OF THE LOCATION OF A FREE BOUNDARY 479 as a es funcion,where T m is he runcaion a he level m, 1 if θ [, T k] 1, ξ k θ) := kt θ) for θ [ T 1, T ], 0 oherwise, k N, 1 if d > ε, 1 ψ ε x x 0, θ) := d if d < ε, ε 0 oherwise, wih d = disx, θ), )) and ε > 0. So su ζx, θ) P ), ζ, ζ L 0, T ) Ω) and ζ x i L 0, T ) Ω). Using he monooniciy of β and assing o he limi we ge he inequaliy. Se 2. A differenial inequaliy for some energy funcion. We assume a choice P such ha i does no ouch he iniial lane { = 0} and P x 0 ) [0, T ]. Then i 1 + i 2 + i 3 j 1 + j 2. In order o esimae j 1, le us menion 1 ha n = n x, n τ ) = θ ) 1 υ e ϑ 2 υ 2 +θ ) 21 υ) ) 1/2 x υe τ ) wih e x, e τ being he orhogonal uni vecors o he hyerlane = 0 and he axis, resecively. If we denoes by ρ, ω), ρ 0 and ω B 1 he sherical coordinae sysem in R N, if Φρ, ω, θ) is he sherical reresenaion of a general funcion F x, ), hen we have T ρθ,) I) := F x, θ)dxdθ dθ ρ N 1 dρ Φρ, ω, θ) J dω, P 0 B 1 where J is he Jacobi marix and ρθ, ) = ϑθ ) υ. So, ρθ,) di) = ρ N 1 dρ Φρ, ω, θ) J dω d + 0 T B 1 θ= ρ ρ N 1 dθ Φρ, ω, ) J dω = B 1 Then, by Hölder s inequaliy, we ge n x A u dγdθ M 2 n x u 1 u dγdθ 1)/ M 2 ρ u dγdθ n x ρ 1 u dγdθ = M 2 de d ) 1)/ T n x ρ 1 θ,) ρ F x, θ)dγdθ. 10) u dγ dθ 1/ 1/. 11)

6 480 J. I. DÍAZ To esimae he righ-hand side of 11) we use he inerolaion inequaliy [9]) : if 0 σ 1, hen here exiss L 0 > 0 such ha v W 1, ) v,sρ L 0 v,bρ + ρ δ ) θ ) 1 θ v σ+1,bρ v r,bρ, 12) ) r [1, 1 + γ], θ = N rn 1), δ = σ N. In our case, we aly N+1) Nr 1+σ) 12) o he limi case σ = 0. By Hölder s inequaliy ) 1/r ) 1/qr ) q 1)/qr u r dx u dx u γ+1 dx, γ wih q =. Then γ r+1 u dγ L 0 u + ρ δ u ) θ L 0 ρ δ θ u + u ) /2 ) θ u Kρ δ θ E + C ) θ C 1 θ)/qr b q 1)1 θ)/qr u r ) 1 θ)/qr ) 1 θ)/r u γ+1 ) q 1)1 θ)/qr Kρ δ θ E + C ) θ+1 θ)/qr b q 1)1 θ)/qr, 13) where E, ρ) := u dx, C, ρ) := u dx and K is a suiable osiive consan. Taking r γ+1), γ θ [ ] we ge ha µ = θ+ < 1. Alying +γ qr once again Hölder s inequaliy wih he exonen µ, we have from 13) j 1 L de ) 1)/ T 1/ n x Kρδ θ E d ρ 1 + C ) µ b q 1)1 θ)/qr dτ L de ) 1)/ b q 1)1 θ)/qr d T µ T E + C ) dτ nx ρδ θ τ) ρ 1 Lσ) d E + C) d ) 1 1 µ dτ 1 µ ) 1)/ b q 1)1 θ)/qr E + C) θ + 1 θ qr, 14) for a suiable osiive consan L. To obain 14) we have assumed ha T 1 µ ) 1 σ) := 1 ρδ θ 1 µ τ) dτ < ρ 1

7 ESTIMATES OF THE LOCATION OF A FREE BOUNDARY 481 which is fulfilled if we choose ν 0, 1) sufficienly small because he condiion of convergence ) of he inegral σ) has he form 1 ν) 1) + νδ θ > 1 θ) 1. So, we have obained an esimae of he following ye qr ) 1)/ j 1 L 1 Λ)Du) q 1)1 θ)/qr λ E + C + b) 1 ω+λ de + C), 15) d where L 1 is a universal osiive consan, Du) is he oal energy of he soluion under invesigaion, λ [0, q 1)1 θ)/qr] ) and ω := 1 θ 1 θ 1 1, 1. qr ω λ) 1 This allows us o choose λ so ha 0, 1). Le us esimae j 2. Using he exression for n τ, we have j 2 C 5 u 1+γ dγdθ. We aly hen he inerolaion inequaliy for he limi case σ = 0) v γ+1, Bρ L 0 v,bρ + ρ δ v σ+1,bρ ) s v 1 s r, v W 1, ) 16) wih a universal osiive consan L 0 > 0 and exonens s = γ+1)n rn 1) N+r) Nr r [1 + σ, 1 + γ]. Again sγ+1)/ u γ+1 dx L 1+γ K sγ+1)/ θ u σ+1 dx Here K is he same as before. Le η = sγ+1) η + π 1. Then j 2 = T dτ τ) L bt )) π u dx + 1/qr q 1)/qr u σ+1 dx u γ+1 dx u γ+1 dγ T + 1 s)γ+1) qr K sγ+1)/ θ E + C ) η n τ dτ L E + C + bt, Ω)) bt, Ω)) κ T 1 s)γ+1), γ+1. 17) < 1, π = q 1)1 s)γ+1) qr, ) ε K sγ+1)/ θ dτ 1/ε, 18) for some L = LC 5, L 0 ) and exonens κ := η +π 1, ε = 1/1 η). Therefore we have u 1+γ dx + E + CΘ i 1 + i 2 + i 3, 19) C 5 P {=T }

8 482 J. I. DÍAZ β i 4 εc 3 Cρ, ) + βc 3 ε β)/β Eρ, ), 20) C 2 K u 1+γ dx + E + C i 1 + i 2 + i 3 + i 4, 21) P {=T } for differen osiive consans K. Now, assuming T and Du) so small ha T 1/ε ) ε L bt, Ω)) κ K sγ+1)/ θ dτ < K 2, we arrive a he inequaliy E + C + bt, Ω) L 1 Λ)Du) q 1)1 θ)/qr λ ) 1)/ E + C + bt, Ω)) 1 ω+λ de + C), 22) d whence follows he desired differenial inequaliy for he energy funcion Y ) := E + C Y ω λ)/ 1) ) c) Y )), 23) where / 1) c) = L 1 Du)) σ)) q 1)1 θ)/qr λ, L1 = cons > 0. Noice ha c) 0 as T. Moreover, he exonen ω λ) belongs o 1 he inerval 0, 1), which leads o he resul see [1]). In order o rove Theorem 2, le us sar wih he case fx, ) > Θ. I is easy o see ha a similar inegraion-by-ars formula can be obained for v = u = minu, 0) i suffices o relace T m ux, s)) by T m u x, s))). In ha case Gu) = k 2 u2. So, γ = 1 and he argumens of Theorem 1 aly. For he sudy of he case fx, ) < Θ i is convenien o inroduce v = ψu) L) + and hen mulily he equaion by a regular localized aroximaion of he funcion T m vx, s)). Then, aking ino accoun ha ψu) sign + 0 v = v sign + 0 v in a weak sense) and ha Cx, u) sign + 0 v 0 here sign + 0 v = 1 if v > 0, and 0 if v 0) we ge he inegraion-by-ars formula for v wih Gv) = k + 2 v2. The argumens of he roof of Theorem 1 lead o he esimae v = 0 in P : 1, υ). Finally, he conclusion follows from he fac ha v = 0 if and only if u + = 0, u + = maxu, 0). Remark 1. The resen echniques can be alied o he sudy of oher roeries such as he roagaion rae, he shrinking of a suor, delay ime or he sudy of locally vanishing soluions of he associaed saionary roblems. For an illusraion of hese roeries, global consequences, alicaions o fluid mechanics and he deailed bibliograhy we refer he reader o he monograh [1]. Remark 2. The assumions on Bx, u, Du) can be imroved. For insance, he case of Bx, u, Du) = wx, ) Du is considered in [6].

9 ESTIMATES OF THE LOCATION OF A FREE BOUNDARY 483 Remark 3. The sudy of he case of erurbaions Cx, u) growing wih negaive exonens is he main goal of he work [8]. In ha case, he alied inerolaion inequaliy is obained from [19]. Acknowledgmens Research is arially suored by he rojecs MTM of he DG- ISGPI Sain) and CCG07-UCM/ESP-2787 of he DGUIC of he CAM and he UCM. References 1. S. N. Anonsev, J. I. Diaz, and S. I. Shmarev, Energy mehods for free boundary roblems. Alicaions o nonlinear PDEs and fluid mechanics. Progress in Nonlinear Differenial Equaions and heir Alicaions, 48. Birkhäuser Boson, Inc., Boson, MA, A. Bensoussan and J.-L. Lions, On he suor of he soluion of some variaional inequaliies of evoluion. J. Mah. Soc. Jaan ), No. 1, A. Bensoussan and J.-L. Lions, Alicaions des inéquaions variaionnelles en conrole sochasique. Méhodes Mahémaiques de l Informaique, No. 6. Dunod, Paris, A. Bensoussan and J.-L. Lions, Conrôle imulsionnel e inéquaions quasi variaionnelles. French) [Imulse conrol and quasivariaional inequaliies] Méhodes Mahémaiques de l Informaique [Mahemaical Mehods of Informaion Science], 11. Gauhier- Villars, Paris, H. Brézis and A. Friedman, Esimaes on he suor of soluions of arabolic variaional inequaliies. Illinois J. Mah ), No. 1, N. Calvo, J. I. Diaz, J. Durany, E. Schiavi, and C. Vázquez, On a doubly nonlinear arabolic obsacle roblem modelling ice shee dynamics. SIAM J. Al. Mah ), No. 2, elecronic). 7. J. I. Diaz, Mahemaical analysis of some diffusive energy balance models in climaology. Mahemaics, climae and environmen Madrid, 1991), 28 56, RMA Res. Noes Al. Mah., 27, Masson, Paris, J. I. Diaz, work in rearaion. 9. J. I. Diaz and L. Véron, Local vanishing roeries of soluions of elliic and arabolic quasilinear equaions. Trans. Amer. Mah. Soc ), No. 2, G. Duvau and J.-L. Lions, Les inéquaions en mécanique e en hysique. Travaux e Recherches Mahémaiques, No. 21. Dunod, Paris, L. C. Evans and B. Knerr, Insananeous shrinking of he suor of nonnegaive soluions o cerain nonlinear arabolic equaions and variaional inequaliies. Illinois J. Mah ), No. 1, A. Friedman, Variaional rinciles and free-boundary roblems. A Wiley-Inerscience Publicaion. Pure and Alied Mahemaics. John Wiley & Sons, Inc., New York, R. Glowinski, J.-L. Lions, and R. Tremolieres, Analyse numérique des inéquaions variaionnelles. Tome 2. Alicaions aux hénomènes saionnaires e d évoluion. Méhodes Mahémaiques de l Informaique, 5. Dunod, Paris, J.-L. Lions, Quelques méhodes de résoluion des roblèmes aux limies non linéaires. Dunod; Gauhier-Villars, Paris, J.-L. Lions, El lanea Tierra. El ael de las Maemáicas y de los suerordenadores. Serie del Insiuo de Esaña 8, Esasa-Cale, Madrid, 1990.

10 484 J. I. DÍAZ 16. J.-L. Lions and G. Samacchia, Inéquaions variaionnelles non coercives. C. R. Acad. Sci. Paris ), J.-L. Lions and G. Samacchia, Variaional inequaliies. Comm. Pure Al. Mah ), A. M. Meirmanov, The Sefan roblem. Translaed from he Russian) de Gruyer Exosiions in Mahemaics, 3. Waler de Gruyer & Co., Berlin, L. Nirenberg, An exended inerolaion inequaliy. Ann. Scuola Norm. Su. Pisa 3) ), L. Tarar, Personal communicaion, Auhor s address: Received ) Dearameno Maemáica Alicada Universidad Comluense de Madrid Madrid Sain diaz.racefyn@insde.es