SCHAUDER ESTIMATES FOR ELLIPTIC AND PARABOLIC EQUATIONS. Xu-Jia Wang The Australian National University

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1 SCHAUDER ESTIMATES FOR ELLIPTIC AND PARABOLIC EQUATIONS Xu-Jia Wang The Austalian National Univesity Intouction The Schaue estimate fo the Laplace equation was taitionally built upon the Newton potential theoy. Diffeent poofs wee foun late by Campanato [Ca], in which he intouce the Campanato space; Peete [P], who use the convolution of functions; Tuinge [T], who use the mollification of functions; an Simon [Si], who use a blowup agument. Also a petubation agument was foun by Safonov [S1,S2] an Caffaelli [C1, CC] fo fully nonlinea unifomly elliptic equations, which also applies to the Laplace equation. In this note we give an elementay an simple poof fo the Schaue estimates fo elliptic an paabolic equations. Ou poof allows the ight han sie to be Dini continuous an also give a shap estimate fo the moulus of continuity of the secon eivatives. It also yiels the log-lipschitz continuity of the gaient fo equations with boune ight han sie. Moeove, it also applies to nonlinea equations. Consie the Laplace equation 1. The Laplace equation u = f in (), (1.1) whee () is the unit ball in the Eucliean space R n. Suppose f is Dini continuous, namely <, whee = sup x y < f(x) f(y). Then we have the following estimate fo the moulus of continuity of D 2 u. Theoem 1. Let u C 2 be a solution of (1.1). Then x, y /2 (), D 2 u(x) D 2 u(y) C n [ sup u + + ], (1.2) 2 whee = x y, C n > epens only on n. It follows that if f C α ( ), then [ u C 2,α (/2 ) C n sup u + f Cα() ] α(1 α) if α (, 1), (1.3) D 2 u(x) D 2 u(y) C n (sup u + f C,1 log ) if α = 1. (1.4) * Wok suppote by the Austalian Reseach Council. 1

2 Poof. We will use the following elementay estimates fo hamonic functions, D k w() C n,k k sup B w, (1.5) whee C n,k epens only on n an k. Simple poofs of (1.5) can be foun in [E2]. Denote B k = B ρ k() (ρ = 1 2 ). Fo k =, 1,, let u k be the solution of u k = f() in B k, u k = u on B k. Then (u k u) = f() f. By the maximum pinciple, Hence Since u k+1 u k is hamonic, by (1.5) we have u k u L (B k ) Cρ 2k ω(ρ k ). (1.6) u k u k+1 L (B k+1 ) Cρ 2k ω(ρ k ). (1.7) D(u k u k+1 ) L (B k+2 ) Cρ k ω(ρ k ), D 2 (u k u k+1 ) L (B k+2 ) Cω(ρ k ). (1.8) Since u C 2, by (1.6), u k minus the quaatic pat of u is hamonic an is equal to o(ρ 2k ) in B k. Hence by (1.5), Du() = lim k Du k (), Fo any given point z nea the oigin, we have D 2 u() = lim k D 2 u k (). (1.9) D 2 u(z) D 2 u() I 1 + I 2 + I 3 =: (1.1) D 2 u k (z) D 2 u k () + D 2 u k () D 2 u() + D 2 u(z) D 2 u k (z). Let k 1 such that ρ k+4 z ρ k+3. Then by (1.8), we have z I 2 C ω(ρ k ) C. (1.11) j=k Similaly we can estimate I 3, though the solutions of v = f(z) in B j (z) an v = u on B j (z) fo j = k, k + 1,. To estimate I 1, enote h j = u j u j 1. By (1.5) an (1.7) we have D 2 h j (z) D 2 h j () Cρ j ω(ρ j ) z. (1.12) Hence I 1 D 2 u k 1 (z) D 2 u k 1 () + D 2 h k (z) D 2 h k () D 2 u (z) D 2 u () + k j=1 D2 h j (z) D 2 h j () C z ( u L + C ρ j ω(ρ j ) ) C z ( u L + C z ). (1.13) 2 Combining (1.1), (1.11), an (1.13) we obtain (1.2). This completes the poof. 2

3 Similaly we have the estimate at the bounay. Theoem 1. Let u C 2 ( {x n }) be a solution of u = f an u = on {x n = }. Suppose f is Dini continuous. Then x, y /2 {x n }, the estimate (1.2) hols. The poof is the same as that of Theoem 1, povie we eplace B k by B k {x n } an note that if w is a hamonic function in {x n } an w = on T, then w is hamonic in afte o extension in x n. Replacing the secon eivatives in the poof of (1.4) by the fist eivatives, an letting u k be the solution of u k = in B k, u k = u on B k, we also obtain the following log-lipschitz continuity fo Du, which was use in [Y] to establish the global existence of smooth solutions to the 2- Eule equation. Coollay 1. Let u C 1 be a solution of (1.1). Then x, y /2 (), Du(x) Du(y) C n (sup u + f L log ). (1.14) 2. Linea paabolic equations The above poof also applies to equations with vaiable coefficients. Let us consie the linea paabolic equation aij (x, t)u xi x j u t = f(x, t) in Q 1. (2.1) We enote Q = {(x, t) R n R 1 : x <, 2 < t }. Theoem 2. Let u C 2,1 x,t be a solution of (2.1). Suppose f an a ij ae Dini continuous. Then fo any points p 1 = (x 1, t 1 ), p 2 = (x 2, t 2 ) Q 1/2, [ xu(p 2 1 ) xu(p 2 2 ) C n sup u + Q 1 [ + C n sup xu 2 Q 1 ω f () ω a () + + ] ω f () 2 ω a () 2 ], (2.2) whee = p 1 p 2 (paabolic istance), ω f () = sup p1 p 2 < f(p 1 ) f(p 2 ), an ω a () = sup i,j ω aij (). Note that the moulus of continuity of t u follows fom (2.2) an equation (2.1). If a ij an f ae Höle continuous, by the intepolation inequality [L], we obtain the Schaue estimate fo paabolic equations. That is if a ij, f C α (Q 1 ) fo some α (, 1), then u 2+α,1+α/2 C x,t (Q 1/2 ) C[ ] sup u + f C α (Q 1 ). (2.3) Q 1 If α = 1, we have an estimate simila to (1.4). 3

4 Poof. Denote Q k = Q ρ k() (ρ = 1 2 ). Let u k be the solution of aij ()u xi x j u t = f() in Q k, u k = u on p Q k, whee p enotes the paabolic bounay. Then v = u u k satisfies aij ()v xi x j v t = f f() + (a ij () a ij (x))u xi x j. (2.4) By the maximum pinciple, u k u L (Q k ) Cρ 2k [ω f (ρ k ) + ω a (ρ k )η], whee η = sup xu. 2 Hence u k u k+1 L (Q k+1 ) Cρ 2k [ω f (ρ k ) + ω a (ρ k )η]. (2.5) Theefoe similaly as (1.8), sup Qk+2 { x(u 2 k u k+1 ), t (u k u k+1 ) } C[ω f (ρ k ) + ω a (ρ k )η]. (2.6) The est of the poof is the same as that of Theoem 1 an is omitte hee. 3. Fully nonlinea equations 3.1. Fully nonlinea, unifomly elliptic equations. The agument in 1 also applies to fully nonlinea unifomly elliptic equations. simplicity we consie the equation Fo F (D 2 u) = f(x) in (), (3.1) whee F is C 1,1. The estimates can be extene to opeatos of the fom F (D 2 u, x) by the feezing coefficient metho as in 2. We nee an a pioi estimate as (1.4). Assumption (a). Fo any solution to F (D 2 u + M) = c in B, whee c is a constant an M is a symmetic constant matix such that F (M) = c, we have the estimate whee ᾱ (, 1], C is inepenent of M, c an. u C 2,ᾱ (B /2 ) C 2 ᾱ u L ( ), (3.2) If F is concave o convex, the inteio C 2,ᾱ estimate fo some ᾱ (, 1] was establishe inepenently by Evans [E1] an Kylov [K]. Similaly to Theoem 1 we then have 4

5 Theoem 3. Let u C 2 be a solution of (3.1). Then x, y /2 (), If f C α ( ), we have D 2 u(x) D 2 u(y) C [ 1 ᾱ sup u + + ᾱ u C 2,α (/2 ) C [ ] sup u + f C α ( ) 1+ᾱ ]. (3.3) if < α < ᾱ, (3.4) D 2 u(x) D 2 u(y) Cᾱ[ sup u + f C α log ] if α = ᾱ, (3.5) u C 2,ᾱ (/2 ) C [ ] sup u + f C α ( ) if ᾱ < α 1. (3.6) The constant C epens on n, ᾱ, C in (3.2), an the ellipticity constants (least an lagest eigenvalues of { ij F ()}). Poof. The poof is vey simila to that of Theoem 1. Let u k be the solution of F (D 2 u k ) = f() in B k, u k = u on B k. (3.7) By Assumption (a), u k u k+1 satisfies a lineaize equation of F with coefficients in Cᾱ. Hence by the Schaue estimate fo linea elliptic equations, D 2 (u k u k+1 ) Bk+2 Cρ 2k u k u k+1 L Cω(ρ k ). (3.8) It follows that D 2 u k () is convegent if f is Dini continuous. By Assumption (a) an since u C 2, we have D 2 u k () D 2 u(). The only iffeence in the est pat of the poof is that (1.12) shoul be eplace by whee h k = u k u k+1. D 2 h k (z) D 2 h k () Cρ kᾱ ω(ρ k ) z ᾱ, (3.9) 3.2. The Monge-Ampèe equations. Estimate (1.2) (o (3.3) with ᾱ = 1) hols fo stictly convex solutions to the Monge- Ampèe equation etd 2 u = f(x) in, (3.1) whee C f C fo positive constants C, C, an = sup x y < f(x) f(y) (equivalent to sup x y < f 1/n (x) f 1/n (y) ). The constant C epens on n, C, C, an the moulus of convexity of u. The poof is simila to that of Theoem 1, except that we fist nee to nomalize the solution as follows. By subtacting a linea function, we assume u() = an Du() =. Denote S h = {x, u(x) < h}. Fo a sufficiently small h >, fist make unimoula linea tansfom T such that B R T (S h ) B nr. Then make a ilation x x/r an u u/r 2 such that S 1 B n an is invaiant une the changes. 5 is small. The Monge-Ampèe opeato

6 Now efine u k as in (3.7) (fo the Monge-Ampee equation it is moe convenient to use level sets than balls in (3.7)). We nee to veify Assumption (a) (with ᾱ = 1) fo all k. It suffices to show that the set E k = {x R n u k (x) < inf u k + ρ 2(k+1) } has a goo shape, namely B Rk E k B 2nRk fo concentate balls B Rk an B 2nRk. But this is guaantee at k = an also at k > by inuction, as long as ρ is chosen small an is sufficiently small. We wish to iscuss the egulaity of the Monge-Ampèe equation with moe etails in a sepaate wok Remaks. (i) By Aleksanov s maximum pinciple [GT], we can eplace = osc B f by = n f f() L n (B ) in the above poofs. (ii) By the existence an uniqueness of weak o viscosity solutions to the Diichlet poblem, the above theoems also hol fo weak o viscosity solutions. (iii) The shap estimate (1.2) fo the Laplace equation was establishe in [B] by elicate singula integal estimates. (iv) Theoem 3 with f C α (α < ᾱ) was pove by Safonov [S1, S2] an Caffaelli [C1, CC] by a petubation agument, using appoximation by quaatic polynomials. See also [K] fo the case when f is Dini continuous. Ou poof allow the case α ᾱ in (3.5) an (3.6) above. (v) The C 2,α estimate fo stictly convex solutions to the Monge-Ampèe equation (3.1) was pove in [C2]. When the Höle continuity of f is elaxe to Dini continuity, the continuity of D 2 u was pove in [W1]. (vi) Simila estimate to (1.14) also hols fo the paabolic equation (2.1) an fully nonlinea equation (3.1), but it is not tue fo the Monge-Ampèe equation (3.1), by an example in [W2]. Refeences [B] C. Buch, The Dini conition an egulaity of weak solutions of elliptic equations, J. Diff. Equa. 3 (1978), [C1] L.A. Caffaelli, Inteio a pioi estimates fo solutions of fully nonlinea equations, Ann. of Math. (2) 13 (1989), [C2] L.A. Caffaelli, Inteio W 2,p estimates fo solutions of Monge-Ampèe equations, Ann. Math., 131(199), [CC] L.A. Caffaelli an X. Cabe, Fully nonlinea elliptic equations, Ame. Math. Soc., [Ca] S. Campanato, Popiet i una famiglia i spazi funzionali, Ann. Scuola Nom. Sup. Pisa (3) 18(1964), [E1] L.C. Evans, Classical solutions of fully nonlinea, convex, secon oe elliptic equations, Comm. Pue Appl. Math., 25(1982), [E2] L.C. Evans, Patial Diffeential equations, Ame. Math. Soc., [GT] D. Gilbag an N.S. Tuinge, Elliptic patial iffeential equations of secon oe, Spinge, [K] J. Kovats, Fully nonlinea elliptic equations an the Dini conition, Comm. PDE 22 (1997), no , [K] N.V. Kylov, Bounely inhomogeneous elliptic an paabolic equations, Iz. Aka. Nauk. SSSR 46(1982), ; English tanslation in Math. USSR (1983). [L] G.M. Liebeman, Secon oe paabolic iffeential equations, Wol Scientific, [P] J. Peete, On convolution opeatos leaving L p,λ spaces invaiant, Ann. Mat. Pua Appl. 72 (1966),

7 [S1] M.V. Safonov, The classical solution of the elliptic Bellman equation, Dokl. Aka. Nauk SSSR 278(1984), ; English tanslation in Soviet Math Doklay 3(1984), [S2] M.V. Safonov, Classical solution of secon-oe nonlinea elliptic equations, Izv. Aka. Nauk SSSR Se. Mat. 52(1988), ; English tanslation in Math. USSR-Izv. 33(1989), [Si] L. Simon, Schaue estimates by scaling, Calc. Va. PDE, 5(1997), [T] N.S. Tuinge, A new appoach to the Schaue estimates fo linea elliptic equations, Poc. Cente Math. Anal., Austal. Nat. Univ., 14(1986), [Y] V. Yuovich, Non-stationay flows of an ieal incompessible flui (in Russian), Z. Vyčisl. Mat. i Mat. Fiz. 3(1963), [W1] X.-J. Wang, Remaks on the Regulaity of Monge-Ampèe equations, Poc. Inte. Conf. Nonlinea PDE, es: G.C. Dong an F.H. Lin, 1992, pp [W2] X.-J. Wang, Some counteexamples to the egulaity of Monge-Ampèe equations. Poc. Ame. Math. Soc. 123 (1995), Cente fo Mathematics an its Applications, Austalian National Univesity, Canbea ACT 2, Austalia aess: wang@maths.anu.eu.au 7

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