ON THE REGULARITY OF SOLUTIONS TO FULLY NONLINEAR ...

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 7, Pages 1955–1959 S 0002-9939(02)06503-6 Article electronically published on February 8, 2002

ON THE REGULARITY OF SOLUTIONS TO FULLY NONLINEAR ELLIPTIC EQUATIONS VIA THE LIOUVILLE PROPERTY QINGBO HUANG (Communicated by David S. Tartakoff) Abstract. We show that any C 1,1 solution to the uniformly elliptic equation F (D 2 u) = 0 must belong to C 2,α , if the equation has the Liouville property.

§1. Introduction In this paper, we consider the interior regularity of solutions to the following fully nonlinear elliptic equation: (1)

F (D2 u) = 0.

We assume that F is uniformly elliptic, i.e., there exist constants 0 < λ ≤ Λ such that (2)

λkN k ≤ F (M + N ) − F (M ) ≤ ΛkN k,

for M, N ∈ S, N ≥ 0,

where S denotes the space of real n × n symmetric matrices and kN k denotes the norm of N . For simplicity, we also assume that F (0) = 0. There have been a number of works concerning equation (1). For instance, see [CC], [GT], [K] and the references cited there. When F is a concave or convex functional, it is well known that the Evans-Krylov estimate [D2 u]C α (B1/2 ) ≤ CkukC 1,1 (B1 ) holds, and C 1,1 viscosity solutions of (1) are C 2,α for some α > 0. On the contrary, in the case when F is not concave nor convex, C 1,1 viscosity solutions of (1) may not be in the C 2 class. This has recently been shown by Nadirashvili in [N] in which he found a C 1,1 viscosity solution u to the equation F (D2 u) = 0 where F is smooth, uniformly elliptic and u is not C 2 . Therefore, it would be interesting to know under what condition a C 1,1 solution of (1) is actually in the C 2 class. It is our purpose in this paper to show that any C 1,1 viscosity solution of (1) must be C 2,α if the elliptic operator F has the Liouville property. A continuous function u(x) is said to be a viscosity subsolution (resp., supersolution) of (1) in a domain Ω if for x0 ∈ Ω and φ(x) ∈ C 2 , u − φ attains the local Received by the editors September 20, 1999. 2000 Mathematics Subject Classification. Primary 35J60; Secondary 35B65. Key words and phrases. Fully nonlinear elliptic equation, regularity, Liouville property, VMO. c

2002 American Mathematical Society

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QINGBO HUANG

maximum (resp., minimum) at x0 , then F (D2 φ(x0 )) ≥ 0 (resp., ≤ 0). If u is both a subsolution and a supersolution, then we say u is a viscosity solution. We mention that if u ∈ C 1,1 , then u is a viscosity solution of (1) if and only if u is a strong solution to (1). 1,1 (Rn ) is an Equation (1) or F is said to satisfy the Liouville property if u ∈ Cloc entire viscosity solution of (1) with bounded D2 u in Rn , |D2 u| ≤ C, then u must be a polynomial of degree at most 2. Let Br (x0 ) = {x ∈ Rn : |x − x0 | < r}. Now we state the main theorem. Theorem. Suppose that F ∈ C 1 satisfies (2) and F (0) = 0. Let u ∈ C 1,1 (B1 (0)) be a viscosity solution of (1) in B1 (0). If equation (1) satisfies the Liouville property, then for any 0 < α < 1, u ∈ C 2,α (B1/2 (0)) and [D2 u]C α (B1/2 (0)) ≤ C, where C depends only on n, λ, Λ, α, kukC 1,1 (B1 (0)) , F , and the modulus of continuity of DF . §2. The proof of the Theorem We will use the blow-up technique to prove the Theorem. The tool needed to 2,2 (Rn ) is the W 2,δ obtain a subsequence of blow-up solutions converging in Wloc estimate for nondivergent uniform elliptic equations. For the convenience of our readers, let us give a little more preliminary information. Recall that u ∈ BMO(Ω) is in VMO(Ω) if Z |u(x) − ux0 ,r | dx −→ 0, as R −→ 0, ηu (R, Ω) = sup x0 ∈Ω 0 0 we have Z |D2 u − (D2 u)0,ρ | dx Bρ (0)

Z =

B ρ (0)

|D2 vk − (D2 vk )0, kρ | dy

k

ρ ρ as k −→ ∞. ≤ ηD2 vk ( ) ≤ η( ) −→ 0, k k This implies that D2 u = const in Rn and hence u is a polynomial of degree at most 2. Suppose that F satisfies the Liouville property. We want to show (i). Let XM = {u ∈ C 1,1 (B1 (0)) : F (D2 u) = 0 and |D2 u| ≤ M in B1 (0)}. To prove that (i) holds, it suffices to show the following claim: Z (3) |D2 u − (D2 u)x0 ,r |2 dx −→ 0, as R −→ 0. sup u∈XM x0 ∈B1/2 (0) r≤R

Br (x0 )

We will show (3) by contradiction. If (3) is false, then there exist ε0 > 0, rk −→ 0, xk ∈ B1/2 (0), uk ∈ XM such that for k ≥ 1 Z |D2 uk − (D2 uk )xk ,rk |2 dx ≥ ε0 . Brk (xk )

Let Tk y = xk + rk y, vk (y) =

Ωk = Tk−1 B1 (0);

uk (xk + rk y) − uk (xk ) − Duk (xk )rk y . rk2

It is easy to check that

(4)

in Ωk . F (D2 vk ) = 0, Z |D2 vk − (D2 vk )0,1 |2 dy ≥ ε0 .

(5)

kvk kC 1,1 (B2A (0)) ≤ Cn,A M,

B1 (0)

if B2Ark (xk ) ⊂ B1 (0).


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QINGBO HUANG

Now we want to show {D2 vk } is precompact in L2 . By [CC] (see Prop. 5.5) 4τ e vk (y) =

vk (y + τ e) − vk (y) ∈ S, τ

in B3A/2 (0), |e| = 1.

By the W 2,δ estimate (δ > 0) (see Prop. 7.4, [CC]) for functions in S Z |D2 4τ e vk |δ (y) dy ≤ CA k4τ evk kδL∞ (B3A/2 (0)) ≤ CA,M M δ , BA (0)

where δ and CA are independent of k. Therefore, by (5) we have Z |D2 vk (y + τ e) − D2 vk (y)|2 dy BA (0) Z 2−δ 2 |D2 vk (y + τ e) − D2 vk (y)|δ dy ≤CkD vk kL∞ (B2A (0)) BA (0)

≤Cτ δ . By Proposition 1, this fact together with kD2 vk kL2 (B2A (0)) ≤ CA implies that {D2 vk } is precompact in L2 (BA (0)). Since vk (0) = 0, Dvk (0) = 0, and kvk kC 1,1 (B2A (0)) ≤ CA , we may assume that by the diagonalizing process vk −→ v,

2,2 1 in Wloc (Rn ) ∩ Cloc (Rn ),

D2 vk −→ D2 v,

a.e. in Rn .

Therefore, F (D2 v) = 0 in Rn . Since kD2 vk kL∞ (BA (0)) ≤ kD2 uk kL∞ (B1 (0)) ≤ M , |D2 v| ≤ M in Rn . By the Liouville property, v must be a polynomial of degree at most 2, and hence D2 v = const. This contradicts the following: Z Z 2 2 2 |D v − (D v)0,1 | dy = lim |D2 vk − (D2 vk )0,1 |2 dy ≥ ε0 . k→∞

B1 (0)

B1 (0)

Thus, Lemma 1 follows. Lemma 2. Let F ∈ C 1 satisfy (2) and F (0) = 0. If u is a viscosity solution of (1) in B1 (0) and D2 u ∈ VMO(B1 (0)), then for any 0 < α < 1, u ∈ C 2,α (B1/2 (0)) and [D2 u]C 2,α (B1/2 (0)) ≤ C, where C depends on n, α, λ, Λ, the modulus of continuity of DF , kukC 1 (B1 (0)) and the VMO modulus of D2 u. Proof. By differentiating (1), we obtain (6)

aij (x)Dij 4he u(x) = 0,

in B3/4 (0),

where 4he u(x) = [u(x + he) − u(x)]/h, h < 14 , |e| = 1, and Z 1 ∂F ((1 − θ)D2 u(x) + θD2 u(x + he)) dθ. aij (x) = ∂M ij 0 Let uh (x) = u(x + he) and Z 1 ∂F ((1 − θ)(D2 u)x0 ,r + θ(D2 uh )x0 ,r ) dθ. cij = ∂M ij 0


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1959

Without loss of generality, we can assume that the continuity modulus of DF denoted by ω(R) is concave. Obviously by Jensen’s inequality Z |aij (x) − cij | dx Br (x0 )

Z

Z

1

ω[|(1 − θ)(D2 u − (D2 u)x0 ,r ) + θ(D2 uh − (D2 uh )x0 ,r )|] dθdx

≤ Br (x0 )

0

≤ ω(ηD2 u (r)) −→ 0,

as r −→ 0.

Therefore aij ∈ VMO(B3/4 (0)). Since (2) holds, λ0 I ≤ (aij ) ≤ Λ0 I and (6) is uniformly elliptic. By the Lp estimate in [CFL], we obtain k4he ukW 2,p (B1/2 (0)) ≤ Ck4he ukL∞ (B3/4 (0)) . Thus, we finish the proof of Lemma 2. The Theorem follows immediately from Lemma 1 and Lemma 2. Acknowledgement The author expresses his gratitude to Professor L. Caffarelli for discussions on several occasions. The author also thanks the referee for some suggestions. References T. Aubin, Nonlinear Analysis on Manifolds. Monge-Amp` ere Equations, Springer-Verlag, 1982. MR 85j:58002 [CC] L. Caffarelli & X. Cabr´e, Fully nonlinear elliptic equations, AMS Colloquium Publications V. 43, AMS, Rhode Island, 1993. MR 96h:35046 [CFL] F. Chiarenza, M. Frasca & P. Longo, W 2,p –solvability of the Dirichlet Problem for Nondivergence Elliptic Equations with V M O Coefficients, Trans. Amer. Math. Soc. 336 (1993), 841-853. MR 93f:35232 [GT] D. Gilbarg & N. S. Trudinger, Elliptic partial differential equations of second order, 2nd edition, Springer-Verlag, 1983. MR 86c:35035 [Gi] M. Giaquinta & J. Neˇcas, On the Regularity of Weak Solutions to Nonlinear Ellptic Systems of Partial Differential Equations, J. Reine Angew. Math. 316 (1980), 140-159. MR 81m:35056 [H] Q. Huang, Estimates on the Generalized Morrey Spaces L2,λ φ and BMOψ for Linear Elliptic Systems, Indiana Univ. Math. J. 45 (1996), 397-439. MR 97i:35033 [K] N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order, Mathematics and its Applications, Reidel, 1987. MR 88d:35005 [N] N. Nadirashvili, Nonclassical Solutions to Fully Nonlinear Elliptic Equations, Preprint.

[A]

Department of Mathematics, University of Texas at Austin, Austin, Texas 78712 E-mail address: [email protected] Current address: Department of Mathematics & Statistics, Wright State University, Dayton, Ohio 45435 E-mail address: [email protected]
