Remarks on he fracional Laplacian wih Dirichle boundary condiions and applicaions Peer Consanin and Mihaela Ignaova ABSTRACT. We prove nonlinear lower bounds and commuaor esimaes for he Dirichle fracional Laplacian in bounded domains. The applicaions include bounds for linear drif-diffusion equaions wih nonlocal dissipaion and global exisence of weak soluions of criical surface quasi-geosrophic equaions.. Inroducion Drif-diffusion equaions wih nonlocal dissipaion naurally occur in hydrodynamics and in models of elecroconvecion. The sudy of hese equaions in bounded domains is hindered by a lack of explici informaion on he kernels of he nonlocal operaors appearing in hem. In his paper we develop ools adaped for he Dirichle boundary case: he Córdoba-Córdoba inequaliy (3] and a nonlinear lower bound in he spiri of (2], and commuaor esimaes. Lower bounds for he fracional Laplacian are insrumenal in proofs of regulariy of soluions o nonlinear nonlocal drif-diffusion equaions. The presence of boundaries requires naural modificaions of he bounds. The nonlinear bounds are proved using a represenaion based on he hea kernel and fine informaion regarding i (4], 7], 8]. Nonlocal diffusion operaors in bounded domains do no commue in general wih differeniaion. The commuaor esimaes are proved using he mehod of harmonic exension and resuls of (]. We apply hese ools o linear drif-diffusion equaions wih nonlocal dissipaion, where we obain srong global bounds, and o global exisence of weak soluions of he surface quasi-geosrophic equaion (SQG in bounded domains. We consider a bounded open domain R d wih smooh (a leas C 2,α boundary. We denoe by he Laplacian operaor wih homogeneous Dirichle boundary condiions. Is L 2 ( - normalized eigenfuncions are denoed w j, and is eigenvalues couned wih heir mulipliciies are denoed λ j : w j = λ j w j. ( I is well known ha < λ... λ j and ha is a posiive selfadjoin operaor in L 2 ( wih domain D ( = H 2 ( H (. The ground sae w is posiive and holds for all x, where c d(x w (x C d(x (2 d(x = dis(x, (3 and c, C are posiive consans depending on. Funcional calculus can be defined using he eigenfuncion expansion. In paricular ( α f = λ α j f j w j (4 wih f j = f(yw j (ydy
2 PETER CONSTANTIN AND MIHAELA IGNATOVA for f D (( α = {f (λ α j f j l 2 (N}. We will denoe by Λ s D = ( α, s = 2α (5 he fracional powers of he Dirichle Laplacian, wih α and wih f s,d he norm in D (Λ s D : I is well-known and easy o show ha Indeed, for f D ( we have f 2 L 2 ( = f 2 s,d = λ s jfj 2. (6 D (Λ D = H (. f ( fdx = Λ D f 2 L 2 ( = f 2,D. We recall ha he Poincaré inequaliy implies ha he Dirichle inegral on he lef-hand side above is equivalen o he norm in H ( and herefore he ideniy map from he dense subse D ( of H ( o D (Λ D is an isomery, and hus H ( D (Λ D. Bu D ( is dense in D (Λ D as well, because finie linear combinaions of eigenfuncions are dense in D (Λ D. Thus he opposie inclusion is also rue, by he same isomery argumen. Noe ha in view of he ideniy wih valid for α <, we have he represenaion λ α = c α ( e λ α d, (7 = c α ( e s s α ds, (( α f (x = c α f(x e f(x ] α d (8 for f D (( α. We use precise upper and lower bounds for he kernel H D (, x, y of he hea operaor, (e f(x = H D (, x, yf(ydy. (9 These are as follows (4],7],8]. There exiss a ime T > depending on he domain and consans c, C, k, K, depending on T and such ha ( ( c min w (x x y, min w (y x y, d 2 e x y 2 k ( ( ( H D (, x, y C min w (x x y, min w (y x y, d 2 e x y 2 K holds for all T. Moreover x H D (, x, y H D (, x, y C holds for all T. Noe ha, in view of H D (, x, y = { d(x, ( + x y if d(x,, if d(x ( e λ j w j (xw j (y, (2
ellipic regulariy esimaes and Sobolev embedding which imply uniform absolue convergence of he series (if is smooh enough, we have ha β H D(, y, x = β 2 H D(, x, y = e λ j y β w j (yw j (x (3 for posiive, where we denoed by β and β 2 derivaives wih respec o he firs spaial variables and he second spaial variables, respecively. Therefore, he gradien bounds ( resul in { y H D (, x, y d(y C, if d(y, H D (, x, y ( + x y, if (4 d(y. 2. The Córdoba - Córdoba inequaliy PROPOSITION. Le Φ be a C 2 convex funcion saisfying Φ( =. Le f C ( and le s 2. Then Φ (fλ s Df Λ s D(Φ(f (5 holds poinwise almos everywhere in. Proof. In view of he fac ha boh f H ( H2 ( and Φ(f H ( H2 (, he erms in he inequaliy (5 are well defined. We define ( α f] ɛ (x = c α f(x e f(x ] α d (6 and approximae he represenaion (8: ɛ (( α f (x = lim ɛ ( α f] ɛ (x. (7 The limi is srong in L 2 (. We sar he calculaion wih his approximaion and hen we rearrange erms: Φ (f(x Λ 2α D f] ɛ (x Λ 2α D (Φ(f] ɛ (x = c α ɛ α d { ] } Φ (f(x f(x H D(, x, yf(y Φ(f(x + H D(, x, yφ(f(y dy = c α ɛ α d H D(, x, y Φ(f(y Φ(f(x Φ (f(x(f(y f(x] dy +c α ɛ α d f(xφ (f(x Φ(f(x] ( H D(, x, ydy = c α ɛ α d H D(, x, y Φ(f(y Φ(f(x Φ (f(x(f(y f(x] dy + f(xφ (f(x Φ(f(x] c α ɛ α ( e d Because of he convexiy of Φ we have and because Φ( = we have Φ(b Φ(a Φ (a(b a, a, b R, aφ (a Φ(a, a R. Consequenly f(xφ (f(x Φ(f(x holds everywhere. The funcion θ = e solves he hea equaion θ θ = in, wih homogeneous Dirichle boundary condiions, and wih iniial daa equal everywhere o. Alhough is no in he domain of, e has a unique exension o L 2 ( where does belong, and on he oher hand, by he maximum principle θ(x, holds for, x. I is only because / D( ha we had o use he ɛ approximaion. Now we discard he nonnegaive erm f(xφ (f(x Φ(f(x ] c α ( θ(x, α d ɛ 3
4 PETER CONSTANTIN AND MIHAELA IGNATOVA in he calculaion above, and deduce ha Φ (f(x Λ 2α D f ] ɛ (x Λ 2α D (Φ(f ] (x (8 ɛ as an elemen of L 2 (. (This simply means ha is inegral agains any nonnegaive L 2 ( funcion is nonnegaive. Passing o he limi ɛ we obain he inequaliy (5. If Φ and he boundary of he domain are smooh enough hen we can prove ha he erms in he inequaliy are coninuous, and herefore he inequaliy holds everywhere. 3. The Nonlinear Bound We prove a bound in he spiri of (2]. The nonlinear lower bound was used as an essenial ingredien in proofs of global regulariy for drif-diffusion equaions wih nonlocal dissipaion. THEOREM. Le f L ( D(Λ 2α D, α <. Assume ha f = q wih q L ( and a firs order derivaive. Then here exis consans c, C depending on and α such ha fλ 2α D f 2 Λ2α D f 2 c q 2α L f d 2+2α (9 holds poinwise in, wih ( f(x, if f(x C q L f d (x = ( max (, if f(x C q L ( max Proof. We sar he calculaion using he inequaliy fλ 2α D f 2 Λ2α D f 2 ( 2 c α ψ α d τ diam(, d(x diam(, d(x,. (2 H D (, x, y(f(x f(y 2 dy (2 where τ > is arbirary and ψ(s is a smooh funcion, vanishing idenically for s and equal idenically o for s 2. This follows repeaing he calculaion of he proof of he Córdoba-Córdoba inequaliy wih Φ(f = 2 f 2 : f(x Λ 2α D f] ɛ (x 2 Λ 2α D f 2] ɛ (x = c α ɛ α { ] } f(x2 f(xh D (, x, yf(y 2 f 2 (x + 2 H D(, x, yf 2 (y dy = c α ɛ α d { 2 HD (, x, y(f(x f(y 2] ]} + 2 f 2 (x H D(, x, y dy = c α ɛ α d { 2 HD (, x, y(f(x f(y 2] dy + 2 f 2 (x e ] (x } c α ɛ α d 2 H D(, x, y (f(x f(y 2 dy where in he las inequaliy we used he maximum principle again. Then, we choose τ > and le ɛ < τ. I follows ha f(x Λ 2α D f ] ɛ (x Λ 2α 2 D f 2] ɛ (x ( 2 c α ψ α d H D (, x, y (f(x f(y 2 dy. τ We obain (2 by leing ɛ. We resric o T, fλ 2α D f ] 2 Λ2α D f 2 (x T ( 2 c α ψ α d H D (, x, y (f(x f(y 2 dy (22 τ and open brackes in (22: fλ 2α D f 2 Λ2α D f 2] (x 2 f 2 T (xc α ψ ( τ α d H T D(, x, ydy f(xc α ψ ( τ α d H D(, x, yf(ydy f(x 2 f(x I(x J(x] (23
wih T I(x = c α ψ ( α d H D (, x, ydy, (24 τ and T J(x = c α ψ ( τ α d H D(, x, yf(ydy T = c α ψ ( τ α d (25 yh D (, x, yq(ydy. We proceed wih a lower bound on I and an upper bound on J. For he lower bound on I we noe ha θ(x, = H D (, x, ydy H D (, x, ydy x y d(x 2 because H D is posiive. Using he lower bound in (2 we have ha x y d(x 2 implies w (x x y 2c, and hen, using he lower bound in ( we obain Inegraing i follows ha w (y x y c, H D (, x, y 2cc 2 d 2 e x y 2 k. θ(x, 2cc 2 ω d k d 2 d(x 2 k ρ d e ρ2 dρ If d(x 2 hen he inegral is bounded below by k ρd e ρ2 dρ. If d(x 2 hen ρ implies ha he k exponenial is bounded below by e and so { ( } d(x d θ(x, c min, (26 for all T where c is a posiive consan, depending on. Because T ( I(x = ψ α θ(x, d τ we have min(t,d I(x c 2 (x ψ ( τ α d = c τ α τ (min(t,d 2 (x ψ(ss α ds Therefore we have ha I(x c 2 τ α (27 2 wih c 2 = c ψ(ss α ds, a posiive consan depending only on and α, provided τ is small enough, τ 2 min(t, d2 (x. (28 In order o bound J from above we use he upper bound (4 which yields y H D (, x, y dy C 2 (29 wih C depending only on. Indeed, d(y yh D (, x, y dy C 2 2 R ( + x y d d 2 e x y 2 k dy = C 3 2 5
6 PETER CONSTANTIN AND MIHAELA IGNATOVA and, in view of he upper bound in (2, d(y w (y C and he upper bound in (, d(y yh D (, x, y dy Now and herefore, in view of (29 and herefore wih J q L ( C 4 R d T x y d 2 e x y 2 ψ J C q L ( K dy = C 5 2 ( α d y H D (, x, y dy τ T ψ ( 3 2 α d τ J C 6 q L (τ 2 α (3 C 6 = C ψ(ss 3 2 α ds a consan depending only on and α. Now, because of he lower bound (23, if we can choose τ so ha J(x 4 f(x I(x hen i follows ha fλ 2α D f 2 Λ2α D f 2 ] (x 4 f 2 (xi(x. (3 Because of he bounds (27, (3 he choice τ(x = c 3 q 2 L f(x 2 (32 wih c 3 = 6C 2 6 c 2 2 achieves he desired bound. The requiremen (28 limis he possibiliy of making his choice o he siuaion q 2 L c 3 f(x 2 2 min(t, d2 (x (33 which leads o he saemen of he heorem. Indeed, if (32 is allowed hen he lower bound in (3 becomes fλ 2α D f ] 2 Λ2α D f 2 (x c q 2α L f d 2+2α (34 wih c = 4 c 2c α 3. 4. Commuaor esimaes We sar by considering he commuaor, Λ D ] in = R d +. The hea kernel wih Dirichle boundary condiions is H(x, y, = c d 2 (e x y 2 4 e x ey 2 4 where ỹ = (y,..., y d, y d. We claim ha ( x + y H(x, y, dy C 2 e x 2 d 4. (35 Indeed, he only nonzero componen occurs when he differeniaion is wih respec o he normal direcion, and hen ( ( xd + yd H(x, y, = c d 2 e x y 2 xd + y d 4 e (x d +y d 2 4
where we denoed x = (x,..., x d and y = (y,..., y d. Therefore Consequenly ( x + y H(x, y, dy C 2 = C xd 2 ξe ξ 2 4 dξ = C 2 e x 2 d 4. K(x, y = ( xd +y d 3 2 ( x + y H(x, y, d e (x d +y d 2 4 dy d obeys K(x, ydy C 2 e x 2 d C 4 d = x 2. d The commuaor, Λ D ] is compued as follows, Λ D ]f(x = 3 2 xh D (x, y, f(y H D (x, y, y f(y] dyd = 3 2 ( x + y H D (x, y, f(ydyd = K(x, yf(ydy. We have proved hus ha he kernel K(x, y of he commuaor obeys K(x, ydy Cd(x 2 (36 and herefore we obain, for insance, for any p, q, ] wih p + q = ( ( g, Λ D ]fdx C d(x 2 f(x p p dx d(x 2 g(x q q dx. In general domains, he absence of explici expressions for he hea kernel wih Dirichle boundary condiions requires a less direc approach o commuaor esimaes. We ake hus an open bounded domain R d wih smooh boundary and describe he square roo of he Dirichle Laplacian using he harmonic exension. We denoe and consider he races of funcions in H,L (Q, Q = R + = {(x, z x, z > } H,L(Q = {v H (Q v(x, z =, x, z > } V ( = {f v H,L(Q, f(x = v(x,, x } (37 where we slighly abused noaion by referring o he images of v under resricion operaors as v(x, z for x, and as v(x, for x. We recall from (] ha, on one hand, V ( = {f H f 2 (x 2 ( dx < } (38 d(x wih norm and on he oher hand V ( = D(Λ 2 D, i.e. f 2 V = f 2 H 2 ( + f 2 (x d(x dx, 7 V ( = {f L 2 ( f = j f j w j, j λ 2 j f 2 j < } (39
8 PETER CONSTANTIN AND MIHAELA IGNATOVA wih equivalen norm f 2 2,D = λ 2 j fj 2 = Λ 2 D f 2 L 2 (. The harmonic exension of f will be denoed v f. I is given by v f (x, z = f j e z λ j w j (x (4 and he operaor Λ D is hen idenified wih Λ D f = ( z v f z= (4 Noe ha if f V ( hen v f H (Q. Noe also, ha v f decays exponenially in he sense ha v f e zl H (Q = e zl v f L 2 (Q + e zl v f L 2 (Q C f V (42 holds wih l = λ 4. We use a lemma in Q: LEMMA. Le F H (Q (he dual of H (Q. Then he problem { u = F, in Q, u =, on Q has a unique weak soluion u H (Q. If F L2 (Q and if here exiss l > so ha e zl F 2 L 2 (Q = e 2zl F (x, z 2 dxdz < (43 hen u H (Q H2 (Q and i saisfies wih C a consan depending only on and l. u H 2 (Q C e zl F L 2 (Q Proof. We consider he domain U = R and ake he odd exension of F o U, F (x, z = F (x, z. The exisence of a weak soluion in H (U follows by variaional mehods, by minimizing ( I(v = 2 v 2 + vf dxdz U among all odd funcions v H (U. The domain U has finie widh, so he Poincaré inequaliy v 2 L 2 (U c v 2 L 2 (U is valid for funcions in H (U. This allows o show exisence and uniqueness of weak soluions. If F L 2 (U we obain locally uniform ellipic esimaes u H 2 (U j C F L 2 (V j where U j = {(x, z x, z (j, j + }, V j = {(x, z x, z (j 2, j + 2}, and j = ± 2, ±, ± 3 2,..., i.e. j 2Z. The consan C does no depend on j. Because of he decay assumpion on F, he esimaes can be summed. THEOREM 2. Le a B( where B( = W 2,d ( W, (, if d 3, and B( = W 2,p ( wih p > 2, if d = 2. There exiss a consan C, depending only on, such ha holds for any f V (, wih a, Λ D ]f 2,D C a B( f 2,D (44 a B( = a W 2,d ( + a W, (
9 if d 3 and wih p > 2, if d = 2. a B( = a W 2,p ( Proof. In order o compue v af, le us noe ha av f H,L (Q, and (av f = v f x a + 2 x a v f and, because v f e zl H (Q and a B( we have ha Solving (av f L 2 (e zl dzdx C a B( v f e zl H (Q. { u = (avf in Q, u = on Q, we obain u H (Q H2 (Q. This follows from Lemma above. Noe ha z u H,L (Q. Then and v af = av f u aλ D f Λ D (af = a( z v f z= + z (av f u z= = z u z=. The esimae follows from ellipic esimaes and resricion esimaes z u z= V C z u H (Q C a B( v f e zl H (Q C a B( f V THEOREM 3. Le a vecor field a have componens in B( defined above, a (B( d. Assume ha he normal componen of he race of a on he boundary vanishes, a n = (i.e he vecor field is angen o he boundary. There exiss a consan C such ha a, Λ D ]f 2,D C a B( f 3 2,D (45 holds for any f such ha f D (Λ 2D 3. Proof. In order o compue v a f we noe ha (a v f = a v f + a v f, and because v f e zl H 2 (Q and a B( we have ha Then solving (a v f L 2 (e zl dzdx C a B( v f e zl H 2 (Q. { u = (a vf in Q, u = on Q, we obain u H 2 (Q (by Lemma and herefore z u H,L (Q. Consequenly zu z= V (. Because v f vanishes on he boundary and a is angen o he boundary, i follows ha a v f H,L (Q (vanishes on he laeral boundary of Q and is in H (Q and herefore Consequenly v a f = a v f u. a, Λ D ]f = z u z=. The esimae (45 follows from he ellipic esimaes and resricion esimaes on u, as above: z u z= V C z u H (Q C a B( v f e zl H 2 (Q C a B( f 3 2,D
PETER CONSTANTIN AND MIHAELA IGNATOVA We sudy he equaion wih iniial daa 5. Linear ranspor and nonlocal diffusion θ + u θ + Λ D θ = (46 θ(x, = θ (47 in he bounded open domain R d wih smooh boundary. We assume ha u = u(x, is divergence-free u =, (48 ha u is smooh u L 2 (, T ; B( d, (49 and ha u is parallel o he boundary u n =. (5 We consider zero boundary condiions for θ. Sricly speaking, because his is a firs order equaion, i is beer o hink of hese as a consrain on he evoluion equaion. We sar wih iniial daa θ which vanish on he boundary, and mainain his propery in ime. The ranspor evoluion θ + u θ = and, separaely, he nonlocal diffusion θ + Λ D θ = keep he consrain of θ =. Because he operaors u and Λ D have he same differenial order, neiher dominaes he oher, and he linear evoluion needs o be reaed carefully. We sar by considering Galerkin approximaions. Le m P m f = f j w j, for f = f j w j, (5 and le obey wih iniial daa θ m (x, = m θ (m j (w j (x (52 θ m + P m (u θ m + Λ D θ m = (53 θ m (x, = (P m θ (x. (54 These are ODEs for he coefficiens θ (m j (, wrien convenienly. We prove bounds ha are independen of m and pass o he limi. Noe ha by consrucion θ m D (Λ r D, r. We sar wih d 2 d θ m 2 L 2 ( + θ m 2 V = (55 which implies T sup T 2 θ m(, 2 L 2 ( + θ m 2 V d 2 θ 2 L 2 (. (56 This follows because of he divergence-free condiion and he fac ha u is parallel o he boundary. Nex, we apply Λ D o (53. For convenience, we denoe Λ D, u ]f = Γf (57 because u is fixed hroughou his secion. Because P m and Λ D commue, we have hus Λ D θ m + P m (u Λ D θ m + Γθ m + Λ 2 Dθ m =. (58
Now, we muliply (58 by Λ 3 D θ m and inegrae. Noe ha P m (u Λ D θ m + Γθ m Λ 3 Dθ m dx = (u Λ D θ m + Γθ m Λ 3 Dθ m dx because P m θ m = θ m and P m is selfadjoin. We bound he erm Γθ m Λ 3 Dθ m dx Γθ m V Λ 2.5 D θ m L 2 ( and use Theorem 3 (45 o deduce Γθ m Λ 3 Dθ m dx C u B( Λ D θ m V Λ 2.5 D θ m L 2 (. We compue (u Λ Dθ m Λ 3 D θ mdx = Λ2 D (u Λ Dθ m Λ D θ m = ( u Λ Dθ m 2 u Λ D θ m ] Λ D θ m dx + (u Λ3 D θ mλ D θ m dx = ( u Λ Dθ m 2 u Λ D θ m ] Λ D θ m dx Λ3 D θ m(u Λ D θ m dx = (( u Λ Dθ m Λ D θ m + 2 u Λ D θ m Λ D θ m ] dx (u Λ Dθ m Λ 3 D θ mdx. In he firs inegraion by pars we used he fac ha Λ 3 D θ m is a finie linear combinaion of eigenfuncions which vanish a he boundary. Then we use he fac ha Λ 2 D = is local. In he las equaliy we inegraed by pars using he fac ha Λ D θ m is a finie linear combinaion of eigenfuncions which vanish a he boundary and he fac ha u is divergence-free. I follows ha Λ D θ m Λ (u 3 Dθ m dx = (( u Λ D θ m Λ D θ m + 2 u Λ D θ m Λ D θ m ] dx 2 and consequenly We obain hus (u Λ D θ m Λ 3 Dθ m dx C u B( Λ 2 Dθ m 2 L 2 ( T sup Λ 2 Dθ m (, 2 L 2 ( + Λ 2 Dθ m 2 V d C Λ 2 Dθ 2 R T L 2 ( ec u 2 B( d. (59 T Passing o he limi m is done using he Aubin-Lions Lemma (6]. We obain THEOREM 4. Le u L 2 (, T ; B( d be a vecor field parallel o he boundary. Then he equaion (46 wih iniial daa θ H ( H2 ( has unique soluions belonging o If he iniial daa θ L p (, p, hen θ L (, T ; H 2 ( H ( L 2 (, T ; H 2.5 (. holds. sup θ(, L p ( θ L p ( (6 T The esimae (6 holds because, by use of Proposiion for he diffusive par and inegraion by pars for he ranspor par, we have for soluions of (46 d d θ p L p (, p <. The L bound follows by aking he limi p in (6.
2 PETER CONSTANTIN AND MIHAELA IGNATOVA wih and We consider now he equaion 6. SQG θ + u θ + Λ D θ = (6 u = R Dθ (62 R D = Λ D (63 in a bounded open domain in R 2 wih smooh boundary. Local exisence of smooh soluions is possible o prove using mehods similar o hose developed above for linear drif-diffusion equaions. We will consider weak soluions (soluions which saisfy he equaions in he sense of disribuions. THEOREM 5. Le θ L 2 ( and le T >. There exiss a weak soluion of (6 saisfying lim θ( = θ weakly in L 2 (. Proof. We consider Galerkin approximaions, θ m θ L (, T ; L 2 ( L 2 (, T ; V ( θ m (x, = m θ j (w j (x obeying he ODEs (wrien convenienly as PDEs: θ m + P m R D(θ m θ m ] + Λ D θ m = wih iniial daum θ m ( = P m (θ. We observe ha, muliplying by θ m and inegraing we have which implies ha he sequence θ m is bounded in d 2 d θ m 2 + θ m 2 2,D = θ m L (, T ; L 2 ( L 2 (, T ; V ( I is known (] ha V ( L 4 ( wih coninuous inclusion. I is also known (5] ha R D : L 4 ( L 4 ( are bounded linear operaors. I is hen easy o see ha θ m are bounded in L 2 (, T ; H (. Applying he Aubin-Lions lemma, we obain a subsequence, renamed θ m converging srongly in L 2 (, T ; L 2 ( and weakly in L 2 (, T ; V ( and in L 2 (, T ; L 4 (. The limi solves he equaion (6 weakly. Indeed, his follows afer inegraion by pars because he produc (RD θ mθ m is weakly convergen in L 2 (, T ; L 2 ( by weak-imes-srong weak coninuiy. The weak coninuiy in ime a = follows by inegraing (θ m (, φ (θ m (, φ = and use of he equaion and uniform bounds. We omi furher deails. d ds θ m(sds Acknowledgmen. The work of PC was parially suppored by NSF gran DMS-29394
3 References ] X. Cabre, J. Tan, Posiive soluions of nonlinear problems involving he square roo of he Laplacian, Adv. Mah. 224 (2, no. 5, 252-293. 2] P. Consanin, V. Vicol, Nonlinear maximum principles for dissipaive linear nonlocal operaors and applicaions, GAFA 22 (22 289-32. 3] A. Córdoba, D. Córdoba, A maximum principle applied o quasi-geosrophic equaions. Comm. Mah. Phys. 249 (24, 5 528. 4] E.B. Davies, Explici consans for Gaussian upper bounds on hea kernels, Am. J. Mah 9 (987 39-333. 5] D. Jerison, C. Kenig, The inhomogeneous Dirichle problem in Lipschiz domains, J. Func. Analysis 3 (995, 6-22. 6] J.L. Lions, Quelque mehodes de résoluion des problèmes aux limies non linéaires, Paris, Dunod (969. 7] Q. S. Zhang, The boundary behavior of hea kernels of Dirichle Laplacians, J. Diff. Eqn 82 (22, 46-43 8] Q. S. Zhang, Some gradien esimaes for he hea equaion on domains and for an equaion by Perelman, IMRN (26, aricle ID9234, -39. DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, PRINCETON, NJ 8544 E-mail address: cons@mah.princeon.edu DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, PRINCETON, NJ 8544 E-mail address: ignaova@mah.princeon.edu