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ESTIMATES IN THE GENERALIZED CAMPANATO-JOHN-NIRENBERG SPACES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS LUIS A. CAFFARELLI and QINGBO HUANG

Abstract We establish estimates in BMO and Campanato-John-Nirenberg spaces BMOψ for the second derivatives of solutions to the fully nonlinear elliptic equation F(D 2 u, x) = f (x). 1. Introduction In this paper we consider estimates in the generalized Campanato-John-Nirenberg spaces BMOψ for solutions to the fully nonlinear elliptic equation F(D 2 u, x) = f (x),

(1.1)

where x ∈ and is a bounded domain in Rn . We assume that F(M, x) is Lipschitz continuous in M, bounded measurable in x, and uniformly elliptic; that is, there exist constants 0 < λ ≤ 3 such that λkN k ≤ F(M + N , x) − F(M, x) ≤ 3kN k

(1.2)

for M, N ∈ S , N ≥ 0, a.e. x ∈ , where S denotes the space of real n × n symmetric metrices and kN k denotes the norm of N . Throughout the paper we assume that F(0, x) = 0 and f ∈ L n (). Let us recall the generalized Campanato-John-Nirenberg spaces BMOψ (). Let ψ be a nondecreasing continuous function on [0, ∞) such that ψ(t) > 0 for t > 0 and t/ψ(t) is almost increasing, which means that t/ψ(t) ≤ K s/ψ(s) for 0 < t < s. A function f (x) ∈ L 1 () belongs to BMOψ () if Z 1 [ f ]BMOψ () = sup | f (x) − f x0 ,r | d x < ∞, x0 ∈ ψ(r ) Br (x0 )∩ 0 n) for the equation (1.1) when f ∈ L p , the mean oscillation of F with respect to x is small, and F(D 2 w, x0 ) = 0 has C 1,1 -estimates. These W 2, p -estimates were extended by L. Escauriaza [E] to the case p > n − ε for some ε > 0. If f is Dini continuous, J. Kovats [K] showed that if u is a solution of the concave equation F(D 2 u) = f (x), then D 2 u ∈ C 0 . If f is only continuous, by Caffarelli’s W 2, p -estimates, D 2 u ∈ L p for any p < ∞, but in general D 2 u ∈ / C 0 even in the case of Poisson’s equation. It is our main goal in this paper to consider the delicate case between L p - and α C -spaces. First, we obtain the estimate in BMO. If f ∈ BMO, F(M, x) is a small multiplier of BMO in x, and Evans-Krylov estimates hold, then D 2 u ∈ BMO. Second, we prove that if f ∈ BMOψ , where ψ does not necessarily satisfy the Dini R1 condition 0 ψ(t)/t dt < ∞, F(M, x) is a small multiplier of BMOψ in x (smallness can be removed if a Dini condition for ψ holds), and Evans-Krylov estimates hold, then D 2 u ∈ BMOψ (for a more precise description, see Theorem A). Even when ψ satisfies a Dini condition, our estimates in BMOψ sharpen the estimates on the modulus of continuity in [K] under the framework in Dini spaces, since assumptions on the modulus of continuity of given data are replaced by those on the mean oscillation and the modulus of continuity can be derived from estimates in BMOψ R |x−y| and the fact that | f (x) − f (y)| ≤ C[ f ]BMOψ 0 ψ(t)/t dt. Our method of obtaining BMOψ -estimates is based on a perturbation argument. We first prove that the Evans-Krylov estimates imply Campanato inequalities. In contrast to the case of linear equations for which the Caccipolli inequality holds, in fully nonlinear setting, when one compares an estimated solution u to a solution w of a good equation, D 2 (u − w) can have only estimates in L δ (0 < δ < 1). Thus one usually obtains estimates of mean oscillations in L δ -norms. With the aid of results of John-Strömberg type in BMOψ , one can still get BMOψ -estimates.

ESTIMATES IN CAMPANATO-JOHN-NIRENBERG SPACES

3

Before stating the main result, let us recall the notion of L n -viscosity solutions. A function u(x) ∈ C() is an L n -viscosity subsolution (resp., supersolution) of (1.1) 2,n in if, for x0 ∈ and φ(x) ∈ Wloc (), u − φ attains the local maximum (resp., minimum) at x0 ; then ess lim sup F(D 2 φ(x), x) − f (x) ≥ 0, x→x0

respectively, ess lim inf F(D 2 φ(x), x) − f (x) ≤ 0. x→x0

If u is both an L n -viscosity subsolution and an L n -viscosity supersolution, then we say that u is an L n -viscosity solution. It is well known, by [CCKS], that if u ∈ W 2,n (), then u is a W 2,n strong solution to (1.1) if and only if u is an L n -viscosity solution of (1.1). Moreover, W 2,n strong solutions are unique in the class of L n viscosity solutions. Let Z Fx0 ,R (M) = F(M, x) d x, B R (x0 )

and let the oscillation of F(M, x) in x be measured by β(x, x0 , R) = β F (x, x0 , R) =

|F(M, x) − Fx0 ,R (M)| . kMk M∈S \{0} sup

(1.3)

For simplicity, from now on, let Br = Br (0). We say that Fx0 ,R (D 2 w) = C has the Evans-Krylov estimate with constants A and 0 < α0 ≤ 1 if for h ∈ C(∂ B1 ) there exists a solution w ∈ C 2 (B1 ) ∩ C(B 1 ) of ( Fx0 ,R (D 2 w) = C in B1 , (1.4) w=h on ∂ B1 , such that for 0 < ρ < 1/2, osc Bρ D 2 w ≤ Aρ α0 osc B1/2 D 2 w.

(1.5)

Note that if F(M, x) is concave in M, then Fx0 ,R (D 2 w) = C has the Evans-Krylov estimate (see [GT] or [CC]). Now we state the main theorem. THEOREM A

Let u be a W 2,n strong solution of (1.1) in B1 , and let D 2 u ∈ BMOψ (B1 ). Assume that F satisfies (1.2) and F(0, x) = 0 in B1 and that for C0 ∈ R, x0 ∈ B1 , and

4

CAFFARELLI and HUANG

R < 1 − |x0 |, Fx0 ,R (D 2 w) = C0 has the Evans-Krylov estimate with constants A and α0 . Suppose that f ∈ BMOψ (B1 ) and ρ α /ψ(ρ) is almost increasing for some 0 < α < α0 . Then we have the following. R1 (i) If 0 ψ(t)/t dt < ∞ and there exists A1 such that for any B R (x0 ) ⊂ B1 , Z 1 β n (x, x0 , R) d x ≤ A1 , (1.6) ψ(R)n B R (x0 ) then there exist C and 0 < δ < 1 such that 1/δ i h Z |D 2 u|δ d x + [ f ]BMOψ (B1 ) , [D 2 u]BMOψ (B1/2 ) ≤ C

(1.7)

B1

(ii)

where C depends only on structure constants and δ depends on n, λ, and 3. R1 If 0 ψ(t)/t dt = ∞ and if for σ > 0 there exists ε0 > 0 depending on structure constants such that for any B R (x0 ) ⊂ B1 with R ≤ R ∗ (0 < R ∗ < 1), Z 1 β n+σ (x, x0 , R) d x ≤ ε0n+σ , (1.8) ω(R)n+σ B R (x0 ) R1 where ω(R) = ψ(R)/ R ψ(t)/t dt, then BMOψ -estimate (1.7) holds.

Remark B In Theorem A, by letting ψ = 1, one obtains the estimate in BMO. The organization of the paper is as follows. In §2, we discuss results of JohnStrömberg type for BMOψ . In §3, the Campanato inequality is derived from the Evans-Krylov estimate. In §4, the estimates in BMOψ -spaces are established. 2. BMOψ -spaces The main purpose of this section is to discuss the John-Strömberg results for BMOψ , which are needed to obtain the estimates in BMOψ . Now let us recall the median values of a function discussed in F. John [J]. Let g(x) be a measurable function and a.e. finite in a bounded domain . A real number m g () is said to be a median value of g over if it satisfies x ∈ : g(x) > m g () ≤ 1 || 2 and

x ∈ : g(x) < m g () ≤ 1 ||, 2 where |A| denotes the Lebesgue measure of A.


5

One can easily check that a = inf{N : |{x ∈ : g > N }| ≤ (1/2)||} and b = sup{K : |{x ∈ : g < K }| ≤ (1/2)||} are two median values of g. Indeed, t is a median value of g if and only if a ≤ t ≤ b. Although m g () is not unique in general, any one of these median values fits our purpose here. Unlike the average value of a function, median values are well defined for functions in L p (0 < p < 1) and measurable functions. It is easy to see by the definition of m g () that for g ∈ L p () ( p > 0), Z Z Cp |g − m g ()| p d x ≤ inf |g − C| p d x, (2.1)

C∈R

where R is the set of all real numbers. For 0 < s < 1/2 and C ∈ R, one has n o |{x ∈ : |g − C| > t}| inf t : ≤s || n o |{x ∈ : |g − m g ()| > t}| 1 ≥ inf t : ≤ s . (2.2) 2 || To prove this, assume m g () = 0, and let t satisfy |{x ∈ : |g − C| > t}| ≤ s||. Since s < 1/2, by the definition of m g (), one obtains C − t ≤ 0 ≤ C + t. Thus |C| ≤ t and {x ∈ : |g| > 2t} ⊂ {x ∈ : |g − C| > t}. This proves (2.2). Let g(x) be a measurable function and a.e. finite in the cube Q 0 . For 0 < s < 1/2, let n o |{x ∈ Q r : |g − m g (Q r )| > tψ(r )}| [g]BMO0,s (Q ) = sup inf t : ≤ s , (2.3) 0 ψ |Q r | Q r ⊂Q 0 where Q r = Q r (y) is the cube centered at y of edge length 2r with the edges parallel to those of Q 0 . When ψ = 1, (2.3) becomes n o |{x ∈ Q : |g − m g (Q)| > t}| [g]BMO0,s (Q 0 ) = sup inf t : ≤s , (2.4) |Q| Q⊂Q 0 where the edges of Q are parallel to those of Q 0 . For g ∈ L p (Q 0 ) ( p > 0), let Z 1/ p 1 [g]BMO p (Q 0 ) = sup |g − m g (Q r )| p d x . ψ Qr Q r ⊂Q 0 ψ(r )

(2.5)

We state below a local version of a result in [St] which substantially weakens the definition condition for BMO. 2.1 Let 0 < s < 1/2. Assume K = [g]BMO0,s (Q 0 ) < ∞. Then g ∈ BMO(Q 0 ), and for LEMMA

6


any cube Q ⊂ Q 0 and t > 0, x ∈ Q : |g − m g (Q)| > t ≤ Cn e−(Cn t)/K |Q|,

(2.6)

where Cn is a dimensional constant. The proof of Lemma 2.1 is very similar to that of [St, Lemma 3.1], and we omit it. Similarly to [St, Lemma 3.5], one can show that if Q 1 and Q 2 are two cubes such that Q 1 ⊂ Q 2 ⊂ Q 0 and |Q 2 | ≤ 2k |Q 1 |, then |m g (Q 1 ) − m g (Q 2 )| ≤ 6k[g]BMO0,s (Q 0 ) . When [g]BMO0,s (Q ψ

0)

(2.7)

< ∞, one can expect a better estimate of the distribution

function of g than (2.6). LEMMA 2.2 R 2r Let Q 0 = Q r0 (x0 ) and 92r0 (τ ) = τ 0 ψ(t)/t dt. If K = [g]BMO0,s (Q ψ

0)

< ∞

for some 0 < s < 1/2, then [g]BMO p (Q 0 ) ≤ C K for p > 0 and for 92r0 (r0 ) ≤ ψ t/(bn K ) < 92r0 (0), in h x ∈ Q 0 : |g − m g (Q 0 )| > t ≤ 9 −1 t , (2.8) 2r0 b K n

−1 is the inverse function of 92r0 . where C and bn are constants and 92r 0

Proof By Lemma 2.1, for Q r ⊂ Q 0 we have Z 1/ p |g − m g (Q r )| p d x ≤ C[g]BMO0,s (Qr ) ≤ C[g]BMO0,s (Q ) ψ(r ). ψ

Qr

r

Therefore [g]BMO p (Q 0 ) ≤ C K . ψ

jk

To show (2.8), bisect Q 0 j times into 2n j parallel subcubes Q jk = Q r j with the edge length 2r j = 21− j r0 . According to (2.7), if Q 1 = Q r/2 ⊂ Q 2 = Q r ⊂ Q 0 , then |m g (Q 1 ) − m g (Q 2 )| ≤ 6n[g]BMO0,s (Q 2 ) ≤ 6nψ(r )[g]BMO0,s (Q ) . ψ

2

Therefore jk

|m g (Q 0 ) − m g (Q r j )| ≤

j−1 X h=0

6nψ(rh )[g]BMO0,s (Q ) . ψ

0

(2.9)


7

By (2.9), we obtain x ∈ Q 0 :|g − m g (Q 0 )| > t j−1 o X n X jk jk ≤ ψ(rh ) . x ∈ Q : |g − m g (Q )| > t − 6n K k

h=0

P j−1

Set t j = (6n + bn )K h=0 ψ(2−h r0 ). Then by Lemma 2.1, x ∈Q 0 : |g − m g (Q 0 )| > t j X x ∈ Q jk : |g − m g (Q jk )| > bn K jψ(2− j r0 ) ≤ k

X x ∈ Q jk : |g − m g (Q jk )| > bn j[g] ≤ BMO0,s (Q jk ) k

≤

2n j X

2−n j r nj = r nj .

k=1

Obviously, j−1 X

ψ(2−h r0 ) ≤ (log 2)−1

Z

2r0

r j−1

h=0

ψ(t) dt = (log 2)−1 92r0 (r j−1 ). t

If t = bn K 92r0 (r ) with 0 < r ≤ r0 , choose j such that r j < r ≤ r j−1 . Therefore we obtain x ∈ Q 0 : |g − m g (Q 0 )| > t ≤ x ∈ Q 0 : |g − m g (Q 0 )| > t j ≤ r n ≤ r n . j Hence (2.8) follows immediately. 2.3 For 0 < δ ≤ 1 ≤ p < ∞, there exist two constants C1 and C2 such that COROLLARY

[g]BMO p (Q 0 ) ≤ C1 [g]BMO0,s (Q ψ

ψ

0)

≤ C2 [g]BMOδ (Q 0 ) . ψ

Remark 2.4 If ψ satisfies a Dini condition, then g is bounded and |g − m g (Q 0 )| ≤ bn [g]BMO0,s (Q ) 92r0 (0) in Q 0 . ψ

0

In general, (2.8) shows that g is in some Orlicz space. For example, if g ∈ BMOψ (Q 0 ) with ψ(t) = (log(1/t))−γ , 0 < γ < 1, then Z h C |g − m (Q )| 1/(1−γ ) i g 1 0 exp d x ≤ C2 [g]BMO0,s (Q ) Q0 ψ

0

8


(for more details, see [Sp]). 2.5 Let g ∈ BMOψ (Q ∗ ). Then for Q ⊂ Q ∗ with dist(Q, ∂ Q ∗ ) ≥ |h|, Z g(x) − g(x − h) d x ≤ Cψ(2h)[g] BMO1 (Q ∗ ) . LEMMA

ψ

Q

Proof Obviously, for any C ∈ R, Z Z g(x) − g(x − h) d x ≤ 2 Q r (x0 )

Q r +|h| (x0 )

|g(x) − C| d x.

It follows that for Q r (x0 ) ⊂ Q, Z n g(x) − g(x − h) d x ≤ 2[g] BMO1 (Q ∗ ) ψ(r + |h|)(r + |h|) . ψ

Q r (x0 )

Divide Q into nonoverlapping subcubes Q i = Q ri (xi ) such that |h|/2 ≤ ri ≤ |h|. Then Z r + |h| n X i g(x) − g(x − h) d x ≤ 2[g]BMO1 (Q ∗ ) ψ(ri + |h|) |Q i | ψ ri Q i

≤ Cψ(2|h|)[g]BMO1 (Q ∗ ) |Q|. ψ

This completes the proof. 2.6 If ψ(0) = 0, then the imbedding from BMOψ (Q 0 ) to L p (Q 0 ) ( p ≥ 1) is compact. COROLLARY

Proof It suffices to prove Corollary 2.6 for p = 1. Let {g j } be a bounded sequence in BMOψ (Q 0 ); that is, there exists a constant N such that [g j ]BMO1 (Q 0 ) ≤ N and ψ R 0 Q 0 |g j | d x ≤ N . By Lemma 2.5, {g j } is compact in any Q b Q 0 . By Corollary 2.3, Z Z |g j |2 d x ≤ 2

Q0

Therefore

Q0

Z Q0

\Q 0

|g j − (g j ) Q 0 |2 d x + 2|(g j ) Q 0 |2 ≤ C.

|g j | d x ≤ C|Q 0 − Q 0 |1/2 .

Combined with the compactness of {g j } in L 1 (Q 0 ), it yields that {g j } is compact in L 1 (Q 0 ).


9

3. Inequalities of Campanato type In this section we show that the Evans-Krylov estimate implies Campanato inequalities. In particular, the Campanato inequalities hold for concave or convex uniformly elliptic equations F(D 2 u) = C. We first prove the following interpolation lemma. LEMMA 3.1 R1 Let u ∈ BMOψ (B R ). Assume 0 ψ(t)/t < ∞. For 0 < ε ≤ R and 0 < δ < ∞, there exists a constant C = C(n, δ) such that Z ε Z 1/δ ψ(t) −n/δ ∞ |u| L (B R ) ≤ C [u]BMOψ (B R ) + Cε |u|δ d x . (3.1) t 0 BR

In particular, for ψ(t) = t α , 0 < α ≤ 1, |u| L ∞ (B R ) ≤ Cεα [u]C α (B R ) + Cε−n/δ

Z

|u|δ d x

1/δ

.

(3.2)

BR

Proof R For x ∈ B R , let u x,ρ = Bρ (x)∩B R u(y) dy. Obviously, for 0 < ρ < R, Z |u(y) − u x,2ρ | dy ≤ Cn [u]BMOψ (B R ) ψ(2ρ). |u x,ρ − u x,2ρ | ≤ Cn B2ρ (x)∩B R

We obtain |u(x) − u x,ε/2 | ≤

∞ X

|u x,ε2−(i+2) − u x,ε2−(i+1) |

i=0 ∞ ε X ψ i+1 2 i=0 Z ε ψ(t) ≤ Cn [u]BMOψ (B R ) . t 0

≤ Cn [u]BMOψ (B R )

Therefore for x ∈ B R , Z |u(x)| ≤ C 0

ε

ψ(t) [u]BMOψ (B R ) + t

Z B R ∩B ε (x)

|u(y)| dy.

(3.3)

2

For 1 ≤ δ < ∞, (3.1) follows immediately from (3.3) by the Hölder inequality. When 0 < δ < 1, by the Young inequality, Z Z 1/δ 1 |u(y)| dy ≤ kuk L ∞ (B R ) + Cδ |u(y)|δ dy . (3.4) 2 B R ∩Bε/2 (x) B R ∩Bε/2 (x) By (3.3) and (3.4), we obtain (3.1) for 0 < δ < 1.

10


THEOREM 3.2 Let F(D 2 u) = 0. Assume that F has Evans-Krylov estimate (1.5) with constants A and α. Then for p > 0, there exists C such that for 0 < ρ < R < 1, Z Z ρ n+αp inf |D 2 u − M| p . (3.5) inf |D 2 u − M| p ≤ C R M∈S B R M∈S Bρ

Proof If Br (x0 ) ⊂ B1 and 0 < ρ < r/2, by rescaling x = x0 + r y and v(y) = (1/r 2 )u(x0 + r y), it is easy to see that F(D 2y v(y)) = 0 in B1 . Therefore v satisfies (1.5) and ρ α osc Bρ (x0 ) D 2 u ≤ A osc Br/2 (x0 ) D 2 u. (3.6) r Let 0 < R < R0 < 1. For x0 , x ∈ B R , if |x − x0 | < (R0 − R)/2, by (3.6), |D 2 u(x) − D 2 u(x0 )| C ≤ osc B R0 D 2 u. α |x − x0 | (R0 − R)α

(3.7)

If |x − x0 | ≥ (R0 − R)/2, (3.7) is obviously valid with C ≥ 2α . It follows that for any M ∈ S , C [D 2 u]C α (B R ) ≤ kD 2 u − Mk L ∞ (B R0 ) . (R0 − R)α By Lemma 3.1, we have for 0 < ε ≤ R0 , n Z 1/ p o C α 2 −n/ p 2 p 2 ε [D u] |D u − M| . [D u]C α (B R ) ≤ α (B ) + ε C R0 (R0 − R)α B R0 We choose ε/(R0 − R) small enough and obtain 2

[D u]C α (B R )

1 C ≤ [D 2 u]C α (B R ) + 0 2 (R0 − R)α+n/ p

Z

|D 2 u − M| p

1/ p

.

B R0

By [H1, Proposition 5.1], for 0 < R < 1, 2

[D u]C α (B R )

C ≤ (1 − R)α+n/ p

Z

|D 2 u − M| p

Bρ

.

(3.8)

B1

Therefore by (3.8), for 0 < ρ < 1/2, Z Z inf |D 2 u − M0 | p ≤ Cρ n+αp inf M0 ∈S

1/ p

M∈S

|D 2 u − M| p . B1

By rescaling, we obtain (3.5) and Theorem 3.2 is proved. Note that (3.5) holds when F is concave since concave equations have Evans-Krylov estimates. Some efforts have been made concerning C 2,α -estimates for some nonconcave and nonconvex equations (see [CY], [H2], [Y]).


11

4. Estimates in BMOψ In this section we prove the main theorem, Theorem A. As a consequence, BMOψ estimates hold for the class of concave equations. An example is given at the end of the paper. Proof of Theorem A Our argument is based on a nonlinear perturbation. Let 0 < ρ < ρ0 ≤ 1, 0 < r < R < ρ0 − ρ, and x0 ∈ Bρ . Let w ∈ C 2 (B R ) ∩ C(B R ) be the solution to the problem ( Fx0 ,R (D 2 w) = f x0 ,R in B R (x0 ), w=u

on ∂ B R (x0 ).

By rescaling, (1.5), and Theorem 3.2, we have for p > 0, Z Z r n+α0 p 2 p inf |D w − M| ≤ C inf |D 2 w − M| p . R M∈S Br M∈S B R

(4.1)

By uniform ellipticity, P − (M2 − M1 ) ≤ Fx0 ,R (M2 ) − Fx0 ,R (M1 ) ≤ P + (M2 − M1 ),

where P + and P − are Pucci extremal operators defined by P + (M) = sup trace(AM), A∈Aλ,3

P (M) = −

inf trace(AM).

A∈Aλ,3

Here Aλ,3 denotes all symmetric matrices whose eigenvalues belong to [λ, 3]. Therefore v = u − w is a strong solution to the following:  − 2 2 2   P (D v) ≤ Fx0 ,R (D u) − F(D u, x) + f (x) − f x0 ,R ≤ P + (D 2 v), in B R (x0 ),   v = 0, on ∂ B (x ). R

0

By the W 2,δ -estimates and the Aleksandrov-Bakelman-Pucci (ABP) maximum principle (see [CC]), there exists 0 < δ = δ(n, λ, 3) < 1 such that 1/δ Z |D 2 v|δ d x B R/2 (x0 )

≤C ≤C

Z B R (x0 )

Z B R (x0 )

Fx

0 ,R (D

2

1/n n u) − F(D 2 u, x) + | f (x) − f x0 ,R |n

β n (x, x0 , R)|D 2 u|n

1/n

+ C[ f ]BMOψ (Bρ0 ) ψ(R).

(4.2)

12


Therefore for r < R/2, by (4.1), Z inf |D 2 u − M|δ M∈S Br (x0 ) Z Z r n+α0 δ ≤C inf |D 2 w − M|δ + inf |D 2 v − M|δ R/2 M∈S B R/2 (x0 ) M∈S Br (x0 ) Z Z r n+α0 δ ≤C inf |D 2 u − M|δ + C inf |D 2 v|δ . R M∈S B R/2 (x0 ) M∈S B R/2 (x0 ) It follows, by (4.2), that Z Z r n+α0 δ 2 δ inf |D u − M| ≤ C inf |D 2 u − M|δ R M∈S Br (x0 ) M∈S B R/2 (x0 ) Z δ/n +C β n (x, x0 , R)|D 2 u|n d x R n + C[ f ]δBMOψ (Bρ ) ψ δ (R)R n . B R (x0 )

0

(4.3) Now we estimate

R

B R (x0 ) β

n (x, x

0,

R)|D 2 u|n d x.

Case I. If ψ satisfies a Dini condition, then by (1.6) and Lemma 3.1, for 0 < ε < ρ0 − ρ, Z Z β n (x, x0 , R)|D 2 u|n d x ≤ β n (x, x0 , R) · kD 2 uknL ∞ (Bρ −ρ (x0 )) 0 B R (x0 ) B R (x0 ) Z 1/δ on n Z ε ψ(t) ≤ Cψ n (R) [D 2 u]BMOψ (Bρ0 ) + ε−n/δ |D 2 u|δ . t Bρ0 0 (4.4) Case II. If ψ fails to satisfy the Dini condition, then by the Hölder inequality, Z β n (x, x0 , R)|D 2 u|n d x B R (x0 ) Z ≤C β n (x, x0 , R) d x · |(D 2 u)x0 ,R |n B R (x0 ) nZ on/(n+σ ) +C β n+σ (x, x0 , R) B R (x0 ) nZ oσ/(n+σ ) × |D 2 u − (D 2 u)x0 ,R |n(n+σ )/σ . B R (x0 )

By [H1, Proposition 2.7], Z Z 1 ψ(t) 2 2 −n |(D u)x0 ,R | ≤ C [D u]BMOψ (Bρ0 ) + C(ρ0 − ρ) |D 2 u|. t Bρ0 R

(4.5)

(4.6)


13

By the Hölder inequality, Z |D 2 u − m D 2 u (Bρ0 )| Bρ0

≤

nZ

|D u − m D 2 u (Bρ0 )|

2−δ

2

Bρ0

Since |m D 2 u (Bρ0 )|δ ≤ Cn,δ Z

R Bρ0

Z

2

|D u| ≤ Bρ0

Bρ0

o1/2 n Z Bρ0

o1/2

.

|D 2 u|δ , by (2.1), we have

|D 2 u − m D 2 u (Bρ0 )| + |m D 2 u (Bρ0 )|

≤ C ψ(ρ0 )[D 2 u]BMOψ (Bρ0 ) +C

|D 2 u − m D 2 u (Bρ0 )|δ

Z

|D 2 u|δ

1/δ

1−δ/2

Z

|D 2 u|δ

1/2

Bρ0

.

(4.7)

Bρ0

By the Young inequality, from (4.6) and (4.7) one obtains |(D 2 u)x0 ,R | ≤ C

Z

1 R

ψ(t) 2 n/δ [D u]BMOψ (Bρ0 ) + Cρ0 (ρ0 − ρ)−2n/δ t

Z

|D 2 u|δ Bρ0

1/δ

.

(4.8)

By (4.8) and (1.8), it follows from (4.5) that if R ≤ R ∗ , Z β n (x, x0 , R)|D 2 u|n d x B R (x0 )

Z

≤

Cε0n ωn (R)

1 R

ψ(t) 2 [D u]BMOψ (Bρ0 ) + C(ρ0 − ρ)−2n/δ t

+ Cε0n ωn (R)[D 2 u]nBMOψ (Bρ ) ψ n (R) 0 Z n n 2 n −2n 2 /δ ≤ Cε0 ψ (R) [D u]BMOψ (Bρ ) + C(ρ0 − ρ) 0

Z

2

|D u|

δ

1/δ n

Bρ0

n/δ |D u| . 2

δ

(4.9)

Bρ0

By (4.4) and (4.9), it follows from (4.3) that Z Z r n+α0 δ inf |D 2 u − M|δ ≤ C inf |D 2 u − M|δ R M∈S Br (x0 ) M∈S B R (x0 ) Z + C R n ψ δ (R) ε0δ [D 2 u]δBMOψ (Bρ ) + C(ρ0 − ρ)−2n |D 2 u|δ 0

Bρ0

+ C[ f ]δBMOψ (Bρ ) ψ δ (R)R n 0 for x0 ∈ Bρ , 0 < ρ < ρ0 ≤ 1, and 0 < r < R ≤ min{ρ0 − ρ, R ∗ }.

(4.10)

14


By [H1, Proposition 2.1], Z Z ψ δ (r )r n 2 δ inf inf |D 2 u − M|δ |D u − M| ≤ C δ ψ (R)R n M∈S B R (x0 ) M∈S Br (x0 ) Z n δ δ 2 δ −2n + Cr ψ (r ) ε0 [D u]BMOψ (Bρ ) + C(ρ0 − ρ) |D 2 u|δ + [ f ]δBMOψ (Bρ ) 0

0

Bρ0

for x0 ∈ Bρ , 0 < ρ < ρ0 ≤ 1, and 0 < r < R ≤ min{ρ0 − ρ, R ∗ }. Therefore, if ε0 is small enough and by Corollary 2.3, we obtain for 0 < ρ < ρ0 ≤ 1, [D 2 u]δBMOψ (Bρ ) ≤ C

1 1 1 + δ ∗ + n ∗ n 2n − ρ)(ρ0 − ρ) ψ (R )(R ) (ρ0 − ρ) 1 |D 2 u|δ + [D 2 u]δBMOψ (Bρ ) + C[ f ]δBMOψ (Bρ ) . 0 0 2

ψ δ (ρ0 Z

× Bρ0

Then by [G, Lemma 3.1], for 0 < ρ < 1, [D 2 u]BMOψ (Bρ ) ≤

1/δ C 1 + ψ δ (R ∗ )(R ∗ )n (1 − ρ)2n

Z

|D 2 u|δ

1/δ

B1

+ C[ f ]BMOψ (B1 ) . This completes the proof of Theorem A. COROLLARY 4.1 Let u be a L n -viscosity solution of (1.1) in B1 . Assume that F(M, x) is concave in M and satisfies (1.2) and that F(0, x) = 0 in B1 . Suppose that f ∈ BMOψ (B1 ) and that ρ α /ψ(ρ) is almost increasing for some 0 < α < α0 = α0 (n, λ, 3), where α0 (n, λ, 3) is the constant in the Evans-Krylov estimate for concave equations. Assume that β(x, x0 , R) satisfies (1.6) if ψ satisfies a Dini condition and that β(x, x0 , R) satisfies (1.8) if the Dini condition for ψ fails. Then D 2 u ∈ BMOψ (B1/2 ) and (1.7) holds.

Proof Let ρε (x) be a standard mollifier, let ε j → 0, and let Z F j (M, x) = F(M, x − y)ρε j (y) dy. |y|≤ε

It is easy to check that F j is concave and uniformly satisfies the same structure conditions as F. Moreover, since ω(0) = 0, very similarly to the proof of Lemma 2.5, 2,n one can show that for any φ ∈ Wloc (B1 ), F j D 2 φ(x), x −→ F D 2 φ(x), x in L nloc (B1 ).


15

Choose f j ∈ C ∞ (B 1 ) such that f j −→ f in L n (B1 ) and [ f j ]BMOψ ≤ C[ f ]BMOψ . Let u j ∈ C 2,α (B3/4 ) ∩ C(B 3/4 ) be the solution to the problem ( F j (D 2 u j , x) = f j (x) in B3/4 , on ∂ B3/4 .

uj = u

By Theorem A, the W 2,δ -estimates, and the ABP maximum principle, {D 2 u j } is bounded in BMOψ (Br ) for 0 < r < 3/4. Therefore, by uniqueness of strong solutions in the class of L n -viscosity solutions, and because the limit of L n -viscosity solutions is an L n -viscosity solution of the limit problem, we get D 2 u ∈ BMOψ (B1/2 ), and (1.7) holds. Example 4.2 Let u be an L n -viscosity solution to the Bellman equation inf aαi j (x)Di j u = f (x) in B1 , α∈I

ij

ij

where aα (x) satisfies the uniformly elliptic condition (i.e., aα (x) ∈ Aλ,3 ). Let β∗ (x, x0 , r ) = sup |aαi j (x) − (aαi j )x0 ,r |. α∈I

By Corollary 4.1, (i) If f (x) ∈ BMO(B1 ) and if for R ≤ R ∗ , Z 1/(n+σ ) 2 β∗n+σ (x, x0 , R) d x ≤ ε0 log−1 , R B R (x0 ) (ii)

(iii)

then D 2 u ∈ BMO(B1/2 ). If f (x) ∈ BMOlog−q (2/t) (B1 ) with 0 < q < 1 and if for R ≤ R ∗ , Z 1/(n+σ ) 2 β∗n+σ (x, x0 , R) d x ≤ ε0 log−1 , R B R (x0 ) then D 2 u ∈ BMOlog−q (2/t) (B1/2 ). In particular, if f (x) = log1−q (2/|x|) with 0 < q < 1, then by the Poincaré inequality, f (x) ∈ BMOlog−q (2/t) (B1 ), and hence D 2 u ∈ BMOlog−q (2/t) (B1/2 ). If f (x) ∈ BMOlog−q (2/t) (B1 ) with q > 1 and Z 1/n 2 β∗n (x, x0 , R) d x ≤ C log−q , R B R (x0 ) then D 2 u ∈ BMOlog−q (2/t) (B1/2 ).

16


In particular, if f (x) = log−τ (2/|x|) with τ = q − 1 > 0, then f (x) ∈ BMOlog−(1+τ ) (2/t) (B1 ) by the Poincaré inequality , and hence D 2 u ∈ BMOlog−(1+τ ) (2/t) (B1/2 ). By [H1, Proposition 2.9], D 2 u ∈ C(B1/2 ) and |D 2 u(x) − D 2 u(y)| ≤ C log−τ (2/(|x − y|)). References [A]

P. ACQUISTAPACE, On B M O regularity for linear elliptic systems, Ann. Mat. Pura

[C]

L. A. CAFFARELLI, Interior a priori estimates for solutions of fully nonlinear

[CC]

L. A. CAFFARELLI and X. CABRE´ , Fully Nonlinear Elliptic Equations, Amer. Math.

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Caffarelli Department of Mathematics, University of Texas at Austin, Austin, Texas 78712, USA; [email protected] Huang Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435, USA; [email protected]