The Harnack inequality for a class of degenerate elliptic operators

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1 The Harnack inequaliy for a class of degenerae ellipic operaors arxiv: v1 [mah.ap] 14 Dec 2011 François Hamel a and Andrej Zlaoš b a Aix-Marseille Universié & Insiu Universiaire de France LATP, Faculé des Sciences e Techniques, F Marseille Cedex 20, France b Deparmen of Mahemaics, Universiy of Wisconsin, Madison, WI 53706, USA Absrac We prove a Harnack inequaliy for disribuional soluions o a ype of degenerae ellipic PDEs in N dimensions. The differenial operaors in quesion are relaed o he Kolmogorov operaor, made up of he Laplacian in he las N 1 variables, a firs-order erm corresponding o a shear flow in he direcion of he firs variable, and a bounded measurable poenial erm. The firs-order coefficien is a smooh funcion of he las N 1 variables and is derivaives up o cerain order do no vanish simulaneously a any poin, making he operaors in quesion hypoellipic. 1 Inroducion We prove a Harnack inequaliy for disribuional soluions o he degenerae ellipic PDE y u+β(y)u x +γ(x,y)u = 0 (1.1) in cylindrical domains in R N wih axes in he direcion of he firs variable x. Here γ is bounded measurable and β is a smooh funcion such ha he operaor L = N 1 n=1 N 1 Xn 2 +X 0 := ( yn ) 2 +β(y) x = y +β(y) x (1.2) n=1 saisfies Hörmander s hypoellipiciy condiion. Tha is, vecor fields {X n } N 1 n=0 and heir commuaors up o cerain order span he whole angen space R N a each (x,y). Moreover, β changes sign so ha L is no parabolic, since hen he ellipic Harnack inequaliy (1.4) below would no hold in general. These condiions on β are equivalen o hypohesis (1.3) below and our resul is hen as follows: 1

2 Theorem 1.1 Le D R N 1 be open conneced and u : (a,b) D [0, ) a bounded disribuional soluion of (1.1) wih γ bounded measurable and β saisfying for some r N, β C (D), inf β < 0 < β, and D D 0 ζ r D ζ β(y) > 0 for all y D. 1 (1.3) Then for each [a,b ] (a,b) and bounded open D wih D D, here is C > 0, depending only on D, D, β and an upper bound on (a a) 1,(b b ) 1,b a, and γ, such ha u C inf (a,b ) D (a,b ) D u. (1.4) Remark. We noe ha y could be replaced by any x-independen, uniformly ellipic in y operaor on D, bu for he sake of simpliciy we sae he heorem wih y insead. This resul is moivaed by is applicaion in our work [6] on large ampliude A asympoics of raveling frons in he x-direcion, and heir speeds, for he reacion-adveciondiffusion equaion v +Aα(y)v x = x,y v+f(v) (1.5) on R N+1, wih he firs order erm represening a shear flow in he x-direcion and f a nonnegaive reacion funcion vanishing a 0 and 1. The fron speeds in quesion are proved o saisfy lim A c (Aα,f)/A = κ(α,f) for some consan κ(α,f) 0, so afer subsiuing he fron ansaz v(,x,y) = u(x c (Aα,f),y) ino (1.5) and scaling by A in he x variable, one formally recovers (1.1) in he limi A, wih β(y) := κ(α,f) α(y) and γ(x,y) := f(u(x,y))/u(x,y). The sudy of hypoellipic operaors of he form L = M Xn 2 +X 0 n=1 (where X n are firs order differenial operaors wih smooh coefficiens), possibly wih an addiional poenial erm, has been sysemaically pursued since Hörmander s fundamenal paper [7]. Alhough various regulariy and maximum principle resuls have been obained soon hereafer (see, e.g., [2, 3, 4, 14, 19, 20]), Harnack inequaliies and relaed hea kernel esimaes for such operaors have iniially been proved only in he case when he angen space a each poin is spanned by he fields {X n } M n=1 and heir commuaors, someimes wih X 0 eiher zero or a linear combinaion of {X n } M n=1 [2, 9, 10, 12, 13]. More recenly, Harnack inequaliies have been obained wihou his assumpion for cerain special classes of operaors, no including (1.1) wih general β, γ. Specifically, some operaors wih consan and linear coefficiens, such as he Kolmogorov operaor L = yy 2 +y x, were considered in [5, 16], and cases of more general coefficiens saisfying somewha rigid srucural assumpions (see hypohesis [H.1] in [17]) were sudied in [11, 17] and wih a poenial erm in [18]. The domains involved in he obained inequaliies have o 1 For ζ = (ζ 1,...,ζ N 1 ) N N 1, we le ζ = ζ 1 + +ζ N 1 and D ζ β(y) = ζ β y ζ 1 1 yζ N 1 N 1 (y). 2

3 depend on he merics associaed o he operaors raher han he Euclidian meric, as shows a couner-example o a Harnack inequaliy in [5]. This is relaed o he need for he signchanging assumpion on β here. We also noe ha he operaors considered in hese papers involve he erm and appropriae parabolic-ype Harnack inequaliies are obained, bu corresponding ellipic inequaliies follow from hese. I was a mild surprise o us ha we were no able o find in he lieraure a sufficienly general resul which would include our case (1.1). I appears ha Harnack inequaliies and hea kernel esimaes become much more involved when he field X 0 is required for Hörmander s condiion o be saisfied. One hin in his direcion is he fac ha he signchanging hypohesis on β is necessary for (1.4) o hold, so hypoellipiciy of L is in iself no a sufficien condiion. We herefore believe ha our mehod of proof of Theorem 1.1 in he nex secion is iself also a valuable conribuion o he problem of quaniaive esimaes for hypoellipic operaors. The proof is based on he Feynman-Kac formula for he sochasic process associaed wih he operaor L, and uses he independence of L, and hus also of he sochasic process, on x. I is no immediaely obvious wheher his requiremen can be lifed and replaced, for insance, by some assumpion on he relaion of he sochasic processes associaed o L and saring from wo differen poins which can be conneced by a pah wih angen vecor X 0 a each poin. We leave his as an open problem. We hank Luis Caffarelli, Nicola Garofalo, Nicolai Nadirashvili, Brian Sree, and Daniel Sroock for useful discussions and poiners o references. FH is indebed o he Alexander von Humbold Foundaion for is por. His work was also pored by he French Agence Naionale de la Recherche hrough he projec PREFERED. AZ was pored in par by NSF grans DMS and DMS , and by an Alfred P. Sloan Research Fellowship. Par of his work was carried ou during visis by FH o he Deparmens of Mahemaics of he Universiies of Chicago and Wisconsin and by AZ o he Faculé des Sciences e Techniques, Aix-Marseille Universié, he hospialiy of which is graefuly acknowledged. 2 Proof of Theorem 1.1 Wihoulosswecanassumeinf D β < 0 < D β andd conneced, aferpossiblyenlarging D. Wewillalsoassumea = 5,a = 0,b = 1,b = 6,D = B 3 (0),D = B 1 (0),and γ 1, wih C hen only depending on β, because he general case is analogous. We also noe ha [19, Theorem 18(c)] and boundedness of u show ha u is acually coninuous. We firs claim ha for each d > 0 here is C d,β 1 such ha u C d,β inf u, (2.6) [0,1] A d [0,1] B 1 (0) wih A d := A + d A d and A± d := {y B 1(0) ±β(y) > d}. Clearly i suffices o show his for all small enough d such ha A ± d, which we shall assume. To his end, noe ha parabolic regulariy heory wih x as he ime variable, applied on [ 1,5] {y B 2 (0) β(y) > d/2}, yields [0,1] A d u C d,β 3 inf u, (2.7) [2,5] A d

4 where C d,β > 0 depends only on d and β. Similarly, we obain [3,4] A + d u C d,β inf u, (2.8) [ 1,2] A + d Nex, consider he sochasic process (X x,y ) saring a (x,y) R B 2 (0) and saisfying he sochasic differenial equaion (dx x,y,dy x,y ) = (β(y x,y )d, 2dB ), (X x,y 0,Y x,y 0 ) = (x,y). Here is a new ime variable and B is a normalized Brownian moion on R N 1 wih B 0 = 0 (defined on a probabiliy space (Ω,B,P)). We hen have for any (x,y) R B 2 (0), in paricular, Y x,y also define he sopping ime (X x,y ) = (X 0,y +x, 2B +y). (2.9) τ = τ y := inf { > 0 Y x,y / B 2 (0) }. is independen from x. For any y B 2 (0) we If τ := min{,τ}, hen by he Feynman-Kac formula, γ 1, and he parabolic comparison principle, we have for each 0 and (x,y) R B 2 (0), e E(u(X x,y τ,y x,y τ)) u(x,y) e E(u(X x,y τ,y x,y τ)). (2.10) (The Feynman-Kac formula is usually saed for C 2 funcions so we provide a proof of (2.10) in Lemma 2.1 below.) Here E(u(X τ,y x,y τ)) x,y = u(x τ(ω),y x,y τ(ω))dp(ω) x,y = u(x,y )dµ x,y (x,y ), (2.11) Ω R B 2 (0) wih he probabiliy measure µ x,y on R B 2 (0) such ha µ x,y (A) = P((X τ,y x,y τ) x,y A) for Borel ses A R B 2 (0). Noice ha µ x,y is pored on [x β,x+ β ] B 2 (0) and µ x,y (R B 2 (0)) = P(τ y ). By (2.9), ranslaion in x equally ranslaes he µ x,y, and he (x-independen) measure on B 2 (0) given by ν(a) y = µ x,y (R A) is jus he law of 2B τy +y, he Brownian moion on B 2 (0) saring a y, wih sopping ime τ y, and wih ime scaled by a facor of wo. In paricular for each > 0 here is h > 0 such ha for any y 1,y 2 B 1 (0) and any Borel ses A 1 B 1 (0) and A 2 B 2 (0), h ν y 1 (A 2 ) ν y 2 (A 2 ) h 1 ν y 1 (A 2 ) and h A 1 ν y 1 (A 1 ) h 1 A 1. (2.12) From his i follows for := β 1 ha wih C d,β := e h min{ A + d, A d } and, similarly, we obain inf u [0,1] B 1 (0) C d,β inf u (2.13) [ 1,2] A + d inf u C d,β inf u. (2.14) [3,4] B 1 (0) [2,5] A d 4

5 Using (2.7), (2.14), (2.8), and (2.13) (in ha order) yields [0,1] A d u C d,β inf u, [0,1] B 1 (0) wih C d,β > 0 depending only on d and β. An analogous argumen gives [0,1] A + d u C d,β inf u, [0,1] B 1 (0) and (2.6) follows. Nex we le v(x,y) := z u(x+s,y)ds for some z (0,1/3] z v C z,β inf u (2.15) [0,1] B 1 (0) [0,1] B 1 (0) holds for some C z,β 1. Indeed, i follows from (2.9), (2.10), (2.11) ha for each (x,y) R B 2 (0), e u(x,y )dµ x,y;z (x,y ) v(x,y) e u(x,y )dµ x,y;z (x,y ), R B 2 (0) R B 2 (0) where µ x,y;z (x,y ) = µ x,y (x,y ) (χ [ z,z] (x )dx δ 0 (y )). The above claims abou µ x,y and he definiion of ν y imply ha µ x,y;z (x,y ) κ x;z (x ) ν(y y ) M m= M µ x+2mz,y;z (x,y ), whereκ x;z ishemeasureonrwihκ x;z (B) = B [x z β,x+z+ β ] foranyborel seb R,andM issuchha(2m+1)z 2(z+ β ),forinsance, M := 1/2+ β /z. This and he firs claim in (2.12) means ha v(x,y 1 ) e 2 h 2 M m= M v(x+2mz,y 2 ) (2.16) for any x R, y 1,y 2 B 1 (0) and > 0. Now we ake any x [0,1], y 1 B 1 (0), and y 2 A d for some fixed d > 0 such ha A ± d. Pick := (2 β ) 1 z and M = 1 o obain using (2.16), v(x,y 1 ) e 2 h 2 3z 3z u(x+s,y 2 )ds e 2 h 2 Since (2.6) and is shifs in x give for c = 1,0,1, 2 u C d,β inf u C d,β u, [c 1,c] A d {c} A d [c,c+1] A d [c 1,c] A d u C d,β inf {c 1} A d u C d,β 5 1 u, [c 2,c 1] A d u(x,y 2 )dx.

6 weobain(2.6)wih[0,1]andc d,β replacedby[ 1,2]andCd,β 3. Thisproves(2.15). Similarly, (2.15) wih [ 1,0] and [1,2] in place of [0,1], ogeher wih (2.6), yield In a similar way one can also obain v C z,β C d,β inf u. [ 1,2] B 1 (0) [0,1] B 1 (0) v C z,d,β inf u. (2.17) [ 1,2] B 2 (0) [0,1] B 2 (0) for some C z,d,β > 0 (recall ha B 2 (0) D = B 3 (0)). We will now need o use (1.3) o finish he proof. This assumpion makes he differenial operaor on he lef-hand side of (1.1) hypoellipic in he sense of Hörmander. I follows is absoluely coninuous when resriced o R B 2 (0) and also o R B 2 (0) (as an (N 1)-dimensional measure in he laer case), wih densiies p (x,y,, ),q (x,y,, ) 0 such ha ha for > 0, he measure µ x,y p (x,y,x,y ) = p (0,y,x x,y ), q (x,y,x,y ) = q (0,y,x x,y ), and p,q are bounded funcions when resriced o y B 1 (0) (wih y B 2 (0) for p and y B 2 (0) for q ). For p his follows from he same claim for he corresponding measure µ x,y on R N given by (2.11) wih in place of τ and β smoohly exended o a periodic funciononr N 1 (whosedensiy issmoohinallargumens, [8, Theorem 3]). This isbecause µ x,y (A) µ x,y (A) for any Borel se A R B 2 (0). For q his would follow from he same claim for he corresponding escape measure µ x,y τ on R B 2 (0) given by (2.11) wih τ = τ y in place of τ. We know of such a resul for bounded domains only [1, Corollary 2.11] bu since µ x,y is pored on a bounded cylinder, i applies in our case as well. Specifically, ake any a < β and a + > β. There is a convex open domain G wih a smooh boundary whose inersecion wih [a,a + ] R N 1 is [a,a + ] B 2 (0), and he inersecion wih [(,a ) (a +, )] R N 1 are wo smooh slaned conical caps G ± R B 2 (0) over he (N 1)-dimensional balls {a ± } B 2 (0) wih he wo (rounded) ips a poins wih y coordinaes y ± such ha ±β(y ± ) > 0 and sufficienly long so ha for any (x,y ) G ± G, he uni ouer normal vecor n(x,y ) o G ± a (x,y ) saisfies n(x,y ) (1,0,,0) 1 2 ( β 1 +1) whenever ±β(y ) 0. Then G saisfies he hypoheses of [1, Corollary 2.11] (i saisfies he escape condiion and all poins of G are τ -regular). I follows ha he escape measure µ x,y τ has a densiy q τ (x,y,, ) whichisaconinuousfuncionof(x,y,x,y ) G G,where Gisheseof good poins of G, ha is, all (x,y ) G excep of he wo cone ips, where n(x,y ) = (±1,0,,0). Thus q τ is bounded on S := {0} B 1 (0) (a,a + ) B 2 (0). Since {Xs 0,y } s τ almos surely says in (a,a + ), we obain q q τ on S and q = 0 on [{0} B 1 (0) R B 2 (0)] \ S. Finally, q (x,y,x,y ) = q (0,y,x x,y ) shows ha q is bounded on R B 1 (0) R B 2 (0). 6

7 Le d > 0 be such ha A ± d C := max{, le z := 1/3, := β 1, and R B 1 (0) R B 2 (0) p, R B 1 (0) R B 2 (0) q } <. Then p (x,y,x,y ),q (x,y,x,y ) C χ [x 1,x+1] (x ) because he measure µ x,y is pored on [x 1,x+1] B 2 (0), so we obain from (2.10) and (2.11) u C e u(x,y )dx dy +C e u(x,y )dx dy [0,1] B 1 (0) [ 1,2] B 2 (0) [ 1,2] B 2 (0) 10C e 10C C z,d,β e v [ 1,2] B 2 (0) inf u [0,1] B 1 (0) by using [ 1,2] = [ 1, 1/3] [ 1/3,1/3] [1/3,1] [1,5/3] [4/3,2] and (2.17). This is (1.4), so he heorem will be proved once we esablish (2.10). Lemma 2.1 If u,β,γ,x x,y,τ y are as in he proof of Theorem 1.1 (in paricular, γ 1), hen (2.10) holds for (x,y) R B 2 (0). Proof. Le Z x,y process (X x,y,z x,y = so ha dz x,y he sense of disribuions, ha is, = d and K := y +β(y) x + z is he generaor of he ). If we le v(x,y,z) := e z u(x,y), hen Kv 0 on R B 3 (0) R in R B 3 (0) R vk φdxdydz 0 for any φ C0 (R B 3 (0) R), wih K := y β(y) x z he adjoin of K. For any ε > 0 le δ ε (0,1/2 N 1) be such ha β(y) β(y ) ε 2 whenever y,y B 5/2 (0) and y y N 1δ ε. Le η : R [0,1] be a smooh non-negaive funcion pored in [ 1,1], wih 1 1 η(x )dx = 1 and η 2. For ε > 0 define he mollifier ( x η ε (x,y,z) := ε 2 δε 1 N η η ε) ( N 1 z η ε) and le v ε := v η ε and φ ε;x,y,z (x,y,z ) := η ε (x x,y y,z z ). For ε (0,1) he smooh funcion v ε hen saisfies on R B 2 (0) R (Kv ε )(x,y,z) = vk φ ε;x,y,z dx dy dz R B 3 (0) R + v(x,y,z )[β(y ) β(y)]φ ε;x,y,z x (x,y,z )dx dy dz. R B 3 (0) R The firs inegral is non-negaive. The inegrand in he second vanishes when x x > ε or y n y n > δ ε for some n or z z > ε, and φ ε;x,y,z x (x,y,z ) 2ε 3 δ 1 N ε, so we have n=1 (Kv ε )(x,y,z) 2 N+2 εe z+ε u. 7 ( yn δ ε ),

8 We nex apply Dynkin s formula [15, Theorem 7.4.1] o he smooh funcion v ε, he process (X x,y,z x,y ), and sopping ime τ (wih τ = τ y ), o obain [ τ ] E[v ε (X τ,y x,y τ,z x,y τ)] x,y = v ε (x,y,0)+e (Kv ε )(Xs x,y,ys x,y,zs x,y )ds v ε (x,y,0) 2 N+2 εe +ε u. Since v ε v uniformly on [x β,x+ β ] B 2 (0) [0,] as ε 0 (by coninuiy of v) and Z x,y τ, i follows ha e E[u(X x,y τ,y x,y τ)] E[v(X x,y τ,y x,y τ,z x,y τ)] u(x,y). This is he second inequaliy in (2.10). The firs inequaliy is obained in he same way, his ime wih v(x,y,z) := e z u(x,y), so ha Kv 0 on R B 3 (0) R and References (Kv ε )(x,y,z) 2 N+2 εe z+ε u. [1] G. Ben Arous, S. Kusuoka, and D.W. Sroock, The Poisson kernel for cerain degenerae ellipic operaors, J. Func. Anal 56 (1984), [2] J.-M. Bony, Principe du maximum, inégalie de Harnack e unicié du problème de Cauchy pour les opéraeurs ellipiques dégénérés, Ann. Ins. Fourier (Grenoble) 19 (1969), [3] C. Fefferman and D.H. Phong, Subellipic eigenvalue problems, Conference on harmonic analysis in honor of Anoni Zygmund, Vol. I, II (Chicago, Ill., 1981), , Wadsworh Mah. Ser., Wadsworh, Belmon, CA, [4] G.B. Folland, Subellipic esimaes and funcion spaces on nilpoen Lie groups, Ark. Ma. 13 (1975), [5] N. Garofalo and E. Lanconelli, Level ses of he fundamenal soluion and Harnack inequaliy for degenerae equaions of Kolmogorov ype, Trans. Amer. Mah. Soc. 321 (1990), [6] F. Hamel and A. Zlaoš, Speed-up of combusion frons in shear flows, preprin, [7] L. Hörmander, Hypoellipic second order differenial equaions, Aca Mah. 119 (1967), [8] K. Ichihara and H. Kunia, A classificaion of he second order degenerae ellipic operaors and is probabilisic characerizaion, Z. Wahrscheinlichkeisheorie und Verw. Gebiee 30 (1974), [9] D. Jerison, The Poincaré inequaliy for vecor fields saisfying Hörmander s condiion, Duke Mah. J. 53 (1986),

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