On the boundary value problems
for fully nonlinear elliptic equations
of second order
M.V. SAFONOV
127 Vincent Hall, University of Minnesota, Minneapolis, MN, 55455
Abstract
Fully nonlinear second-order, elliptic equations F (x, u, Du, D2 u) = 0 are considered
in a bounded domain Ω ⊂ Rn , n ≥ 2. The class of equations includes the Bellman
equations supm (Lm u + f m ) = 0, where the functions f m and the coefficients of the
linear operators Lm are bounded in the Hölder space C α (Ω), 0 < α < 1. We prove
the interior C 2,α -smoothness of solutions in Ω with some small α > 0. Under the
Dirichlet boundary condition u = φ on ∂Ω with φ ∈ C 2,α (Ω) and ∂Ω ∈ C 2,α , the
solutions u ∈ C 2,α (Ω). Under the oblique derivative condition b0 u + b · Du = φ
on ∂Ω, where b = (b1 , · · · , bn ) is not tangent to ∂Ω, the solutions u ∈ C 2,α (Ω) if
bi , φ ∈ C 1,α (Ω), and also ∂Ω ∈ C 1,α .
Contents
1. Introduction
2
2. The Hölder spaces
4
3. Formulation of main existence results
9
4. Interior C 2,α − estimates: the simplest nonlinear equations
12
5. Interior C 2,α − estimates: general equations
16
6. Some boundary estimates for solutions of linear elliptic equations
20
7. Boundary C 2,α − estimates: the Dirichlet problem
22
8. Boundary C 2,α − estimates: the oblique derivative problem
25
1
1. Introduction
In this paper we consider general nonlinear elliptic equations including the
Bellman equations
(1.1)
m
m
m
sup(Lm u + f m ) = sup(am
ij Dij u + bi Di u + c u + f ) = 0
m
m
(the summation over repeated indices is everywhere understood). Such equations are also important from the viewpoint of the applications to the theory
of controlled diffusion processes (see [10]).
We investigate the Dirichlet and the oblique derivative problems in a bounded domain Ω ⊂ Rn , n ≥ 2, for nonlinear elliptic equations in the Hölder space
C 2,α (Ω), 0 < α < 1. Leaving aside the simpler one- and two- dimensional
cases, we note that the interior C 2,α -smoothness of solutions to the equation
(1.1) was first proved in 1977 by Brézis and Evans [2] in the case when m
assumes only two values. In 1981, Krylov [11], [12] has established the C 2,α smoothness of the solutions of the Bellman elliptic and parabolic equations
in the higher-dimensional case, both in the interior of the domain and near
its boundary, under appropriate smoothness of the boundary and the boundary values of the solutions. At about the same time, Evans [7] (see also [9])
independently proved the C 2,α -smoothness of solutions of elliptic equations
(1.1) in the interior of the domain. Under the oblique derivative condition,
the C 2,α -smoothness of solutions near the boundary was proved in [16], [17].
In all those papers, and also in [4], [5], [9], [13], as they apply to (1.1), it
m m
m
is assumed that the functions am
are uniformly bounded, together
ij , bi , c , f
with all their first and second derivatives.
The C 2,α -estimates of solutions fo the Bellman equation (1.1) with coefm m
m
ficients am
in C α for some small α ∈ (0, 1), were first obtained
ij , bi , c , f
in [24], including the estimates near the boundary in the Dirichlet case. For
the oblique derivative problem, the corresponding result was proved in [1].
Some other extensions of the results in [24], both for the Dirichlet and for the
oblique derivative problem, were derived by Trudinger [28]. There are also
some close results in the papers of Caffarelli [3] and Wang [29], where they
treat both the C 2,α -estimates and the estimates in the Sobolev spaces W 2,p
when f m ∈ Lp , 1 < p < ∞.
Here we give an enlarged exposition of results in [24], [26], [1]. We introduce a class of nonlinear equations including the Bellman equaitons (1.1) with
coefficients in C α for some small α ∈ (0, 1), and we show that the Dirichlet
problem and the oblique derivative problem are solvable in C 2,α . Under minimal assumptions on the boundary and the boundary data, we receive also
the C 2,α -estimates near the boundary for solutions of these problems. These
results are formulated in Section 3 (Theorems 3.1–3.3). Notice that in the case
of linear elliptic equations, they turn into the classical Schauder-type results
(see [9], Ch.6; [19], Ch.3), even with an improvement: Theorem 3.3 states that
the solution of the oblique derivative problem in Ω still belongs to C 2,α (Ω) if
∂Ω ∈ C 1,α . For linear elliptic equations such an improvement was proved by
Lieberman [15].
2
The most essential part in the proof of Theorems 3.1–3.3 consists in the
appropriate a priori C 2,α -estimates of the solutions. The basic idea of deriving
such estimates is the “local” decomposition of the solution into “smooth” and
“small” terms. The “smooth” term is the solution of an auxiliary problem for
the simplest nonlinear equation corresponding to the case when in (1.1) we
m
m
have am
= const, and bm
= 0. For the technical realization of this
ij , f
i = c
idea, it is convenient to use some equivalent seminorms in C 2,α introduced by
Campanato [6]. In Section 2, we expose a simple approach to the Campanato
type of seminorms which are equivalent to the usual seminorms in “weighted”
Hölder spaces (Theorem 2.1).
For the completeness of the presentation, in Section 4 we prove the interior
C 2,α -estimates of Krylov [11] and Evans [7], in a particular case of the simplest
nonlinear equaitons. Our approach is new in some details, while it relies, as
well as [11], [7], on the results of [14], [22]. In Section 5, we extend the interior
C 2,α -estimates to the solutions of the general nonlinear equaitons. The next
Section 6 is devoted to the boundary behaviour of solutions to the linear elliptic
equations with measurable coefficients. These auxiliary results help us to get
the C 2,α -estimates near the boundary for soluitons of nonlinear equations; the
Dirichlet and the oblique derivative conditions are treated in Sections 7 and 8
correspondingly.
BASIC NOTATIONS. Rn is Euclidean space of dimension n, with standard basis {e1 , · · · , en }, and points x = (x1 , · · · , xn ) written in coordinates
relative to this
basis; (x, y) = xi yi is the inner product of x, y ∈ Rn ; |x| =
∑
n
(x, x)1/2 = ( x2i )1/2 ; R+
= {x ∈ Rn : xn > 0}, R0n = {x ∈ Rn : xn = 0}, S n
denotes the n(n+1)/2- dimensional space of all real symmetric n×n matrices.
We will identify R0n and Rn−1 .
∂Ω is the boundary of the set Ω ⊂ Rn , Ω = Ω ∪ ∂Ω; Bρ (x) = {y ∈ Rn :
|y − x| < ρ} is the ball of radius ρ > 0 centered at x ∈ Rn ,
n
Bρ+ (x) = R+
∩ Bρ (x), Bρ0 (x) = R0n ∩ Bρ (x), Ωρ (x) = Ω ∩ Bρ (x).
∑
l = (l1 , · · · , ln ) is a multi-index, i.e. li = integer ≥ 0, with |l| =
li .
We define xl = xl11 xl22 · · · xlnn , l! = l1 ! l2 ! · · · ln !. For functions u = u(x),
we set Di u = ∂u/∂xi , Dij u = ∂ 2 u/∂xi ∂xj ; Du = (D1 u, · · · , Dn u) is the
gradient of u, D2 u = [Dij u] is the Hessian matrix. Moreover, we define
the first and the second derivatives of u in the direction λ ∈ Rn as follows:
Dλ u = ∂u/∂λ = λi Di u, Dλλ u = ∂ 2 u/∂λ2 = λi λj Dij u. We will also use
the multi-index notation Dl u = ∂ |l| u/∂xl11 · · · ∂xlnn , with the understanding
D0 u = u. For integer k ≥ 0, Pk denotes the collection of all polynomials of
degree at most k. In particular, the Taylor polynomial of degree k for the
function u at the point y ∈ Rn is
∑
(1.2)
Ty,k u(x) =
Dl u(y) · (x − y)l /l! ∈ Pk .
|l|≤k
Throughout this paper, N will denote various positive constants. In the
intermediate calculations, we will usually omit the dependence of N on the
original quantities.
3
Acknowledgements
This work was completed while the author was visiting the Australian National
University during Spring 1994. A considerable part of the research was carried
out ealier, during the visit of the Dipartimento di Matematica Applicata dell’
Università di Firenze in 1990.
2. The Hölder spaces
Let Ω be a domain in Rn , n ≥ 1. For k = 0, 1, 2, · · · , we denote C k (Ω) the set
of functions u = u(x) whose derivatives Dl u for |l| ≤ k are continuous in Ω.
We set
(2.1)
|u|0 = |u|0;Ω = sup |u|,
[u]k,0 = [u]k,0;Ω = max |Dl u|0;Ω .
|l|=k
Ω
Definition 2.1. C k,0 (Ω) is the Banach space of functions u ∈ C k (Ω) with the
finite norm
(2.2)
|u|k = |u|k,0 = |u|k,0;Ω =
k
∑
[u]j,0;Ω , k = 0, 1, 2, · · · .
j=0
Further, we call u Hölder continuous with exponent α in Ω, if the quantity
(2.3)
[u]α = [u]α;Ω = sup |u(x) − u(y)|/|x − y|α , 0 < α ≤ 1
x,y∈Ω
is finite. We set
(2.4)
[u]k,α = [u]k,α;Ω = max[Dl u]α;Ω .
|l|=k
Definition 2.2. The Hölder space C k,α (Ω) is the Banach space of functions
u ∈ C k (Ω) with the finite norm
(2.5)
|u|k,α = |u|k,α;Ω = |u|k,0;Ω + [u]k,α;Ω , k = 0, 1, 2, · · · , 0 < α ≤ 1.
We will also use the similar notations for closed domains Ω and more
generally, for Ω ∪ Γ, where Γ ⊂ ∂Ω. Obviously, for bounded domain Ω we have
C k,0 (Ω) = C k (Ω). For simplicity we will write C 0,α = C α , if 0 < α < 1. From
the elementary inequality
|u(x)v(x) − u(y)v(y)| ≤ |u(x)| · |v(x) − v(y)| + |v(y)| · |u(x) − u(y)|
and (2.1), (2.3), it follows
(2.6)
[uv]α ≤ |u|0 · [v]α + |v|0 · [u]α for u, v ∈ C α (Ω), 0 < α ≤ 1.
The following lemma contains the well known interpolation inequalities (see
[9], Sec. 6.8).
4
Lemma 2.1. Suppose j+β < k+α, where j, k = 0, 1, 2, · · · , and 0 ≤ α, β ≤ 1.
Let u ∈ C k,α (Br ), where Br = Br (x0 ), r > 0. Then for any ε > 0 we have
(2.7)
rj+β [u]j,β;Br ≤ ε rk+α [u]k,α;Br + N (ε)|u|0;Br ,
with a constant N (ε) = N (ε, n, k, α, β). The similar inequalities are also true
n
for Br+ = Br+ (x0 ), x0 ∈ R+
= {x ∈ Rn : xn ≥ 0}.
Further, let a subset Γ ⊂ ∂Ω be given, Γ ̸= ∂Ω (the case Γ = ∅ is not
excluded). For k = 0, 1, 2, · · · , 0 ≤ α ≤ 1, γ ∈ R1 , and u ∈ C k (Ω ∪ Γ), we set
(2.8)
(γ)
(γ)
[u]k,α = [u]k,α;Ω∪Γ = sup dk+α+γ (x) · [u]k,α;Ω(x) ,
x∈Ω∪Γ
where
(2.9)
1
d(x) = dist(x, ∂Ω \ Γ),
2
Ω(x) = Ωd(x) (x) = Ω ∩ Bd(x) (x).
Definition 2.3. For Γ ⊂ ∂Ω, k = 0, 1, 2, · · · , and γ ∈ R1 , C k;γ (Ω ∪ Γ) =
C k,0;γ (Ω ∪ Γ) is the Banach space of functions u ∈ C k (Ω ∪ Γ) with the finite
norm
(γ)
(γ)
∥u∥k,0 = ∥u∥k,0;Ω∪Γ =
(2.10)
k
∑
(γ)
[u]j,0;Ω∪Γ .
j=0
Definition 2.4. For Γ ⊂ ∂Ω, k = 0, 1, 2, · · · , 0 < α ≤ 1, and γ ∈ R1 ,
the weighted Hölder space C k,α;γ (Ω ∪ Γ) is the Banach space of functions u ∈
C k (Ω ∪ Γ) with the finite norm
(2.11)
(γ)
(γ)
(γ)
(γ)
∥u∥k,α = ∥u∥k,α;Ω∪Γ = ∥u∥k,0;Ω∪Γ + [u]k,α;Ω∪Γ .
We will consider only very special cases of Γ: either Γ = ∅ and Ω is an
n
arbitrary domain in Rn , or Γ ⊂ ∂Ω ∩ R0n and Ω ⊂ R+
. Therefore, in (2.9) we
+
n
. So we can rewrite
have either Ω(x) = Bd(x) (x) or Ω(x) = Bd(x) (x), x ∈ R+
(2.7) in the form
dj+β (x)[u]j,β;Ω(x) ≤ ε dk+α (x)[u]k,α;Ω(x) + N (ε)|u|0;Ω(x) .
Multiplying both sides of this onequality by dγ (x), and then taking the sup
over x ∈ Ω∪Γ, we arrive at the following interpolation inequalities for weighted
Hölder spaces.
Corollary 2.1. Suppose j + β < k + α, and let u ∈ C k,α;γ (Ω ∪ Γ), γ ∈ R1 .
Then for and ε > 0 we have
(2.12)
(γ)
(γ)
(γ)
[u]j,β;Ω∪Γ ≤ ε [u]k,α;Ω∪Γ + N (ε, n, k, α, β) · |u|0,0;Ω∪Γ .
The following lemma is related to the approximation of a function u by
means of its Taylor polynomial Ty,k u defined in (1.2).
5
Lemma 2.2. Let u ∈ C k,α (Ω), 0 < α ≤ 1. Then for any x, y ∈ Ω such that
the segment [x, y] ⊂ Ω, we have
|u(x) − Ty,k u(x)| ≤ N (n)[u]k,α · |x − y|k+α .
(2.13)
Proof: By Taylor’s formula,
u(x) = Ty,k−1 u(x) +
∑
Dl u(ξ) · (x − y)l /l!,
|l|=k
where ξ ∈ [x, y]. Further, from (2.4) it follows
max |Dl u(ξ) − Dl u(y)| ≤ [u]k,α · |ξ − y|α ≤ [u]k,α · |x − y|α .
|l|=k
Therefore,
∑
l
l
l
|u(x) − Ty,k u(x)| = (D u(ξ) − D u(y)) · (x − y) /l! ≤ N [u]k,α · |x − y|k+α ,
|l|=k
that completes the proof.
Corollary 2.2. Let u ∈ C k,α (Ωρ ), where Ωρ = Bρ (x0 ), x0 ∈ Rn , or Ωρ =
n
Bρ+ (x0 ), x0 ∈ R+
. Then
(2.14)
Ek [u; Ωρ ] = inf sup |u − p| ≤ N (n)[u]k,α ρk+α .
p∈Pk Ωρ
Lemma 2.3. Let k = 0, 1, 2, · · · , 0 < α ≤ 1, and u ∈ C k,α (Bρ ), Bρ = Bρ (x0 ).
Then for any ε > 0 we have
(2.15)
ρ−α max osc Dl u ≤ ε[u]k,α;Bρ + N (ε, n, k, α) · ρ−k−α Ek [u; Bρ ],
|l|=k
Bρ
where osc f = sup f − inf f . The similar inequalities are also true for Bρ+ =
n
.
Bρ+ (x0 ), x0 ∈ R+
Proof: Using the elementary inequality osc f ≤ 2 sup |f | and (2.7) with
r = ρ, j = k, β = 0, we have
1 −α
ρ max osc Dl u ≤ ρ−α [u]k,0;Bρ ≤ ε[u]k,α;Bρ + N (ε)ρ−k−α sup |u|.
|l|=k Bρ
2
Bρ
For arbitrary p ∈ Pk , the left-hand side of this inequality and [u]k,α remain the
same if we replace u by u − p. After the replacement, we take the infimum of
the right-hand side over p ∈ Pk . On redefining ε, this will give us the desired
estimate.
The next theorem is similar to Theorem 2.1 in [26] (see also [6]).
6
Theorem 2.1. Let k = 0, 1, 2, · · · , 0 < α ≤ 1, γ ∈ R1 , and u ∈ C k (Ω ∪ Γ)
(γ)
has a finite seminorm [u]k,α in (2.8). Set
ωk (x, ρ) = max osc Dl u,
(2.16)
(2.17)
|l|=k Ωρ (x)
(γ)
Ωρ (x) = Ω ∩ Bρ (x),
(γ)
M̂k,α = M̂k,α [u; Ω ∪ Γ] = sup dk+α+γ (x)
x∈Ω∪Γ
sup
ρ−α ωk (x, ρ),
ρ∈(0,d(x)]
(2.18)
(γ)
(γ)
Mk,α = Mk,α [u; Ω ∪ Γ] = sup dk+α+γ (x)
x∈Ω∪Γ
sup
ρ−k−α Ek [u; Ωρ (x)],
ρ∈(0,d(x)]
where d(x) = 12 dist(x, ∂Ω\Γ), Ek is defined in (2.14). Then all the seminorms
(γ)
(γ)
(γ)
[u]k,α , M̂k,α , and Mk,α are equivalent :
(2.19)
N1−1 [u]k,α ≤ M̂k,α ≤ N2 [u]k,α ,
(2.20)
N3−1 [u]k,α ≤ Mk,α ≤ N4 [u]k,α ,
(γ)
(γ)
(γ)
(γ)
(γ)
(γ)
where the constants N depend only on n, k, α, γ.
Proof: To prove the first inequality in (2.19), we fix x0 ∈ Ω∪Γ, d = d(x0 ) =
\ Γ), |l| = k, and x, y ∈ Ωd (x0 ), such that
1
dist(x0 , ∂Ω
2
(2.21)
(γ)
[u]k,α ≤ 2dk+α+γ |Dl u(x) − Dl u(y)|/|x − y|α .
We consider separately the cases (a) ρ = |x − y| < d/2 and (b) ρ ≥ d/2.
In the case (a), we have x, y ∈ Ωd/2 (y) ⊂ Ω3d/2 (x0 ). Since d/2 ≤ d(y) ≤ 3d/2,
from (2.21) it follows
(2.22)
[u]k,α ≤ N dk+α+γ (z) · ρ−α osc Dl u
(γ)
Ωρ (z)
with N = N (k, γ), z = y. Obviously, this inequality is also true in the case
(b) for z = x0 , ρ = d, and N = 21+α . In any case, we have (2.22), where
0 < ρ ≤ d(z), that yields the first inequality in (2.19) with N1 = N1 (k, γ).
The second inquality is trivial with N2 = 2α .
(γ)
(γ)
(γ)
Further, from Lemma 2.3 and the definition of [u]k,α , M̂k,α , and Mk,α , we
get the inequailty
(γ)
(γ)
(γ)
M̂k,α ≤ ε[u]k,α + N (ε, n, k, α) · Mk,α .
Setting ε = (2N1 )−1 and using (2.19), we obtain the first inequality in (2.20).
Finally, the last inequality follows immediately from Corollary 2.2 with N4 =
N4 (n).
We will investigate the boundary value problems in a bounded domain Ω
under some natural restrictions on the boundary ∂Ω. The following assumptions are usually called the Lipschitz condition on ∂Ω.
7
Assumptions 2.1. There exist positive constants r0 and K0 , such that for
each x0 ∈ ∂Ω we have
(2.23)
Ωr0 (x0 ) = Ω ∩ Br0 (x0 ) = {x = (x′ , xn ) ∈ Rn : xn > ψ0 (x′ )} ∩ Br0 (x0 )
in an orthonormal system centered at x0 , where ψ0 is defined on the projection
Br00 of Br0 (x0 ) onto R0n , and
(2.24)
|ψ0 (x′ ) − ψ0 (y ′ )| ≤ K0 · |x′ − y ′ | for all x′ , y ′ ∈ Br00 .
Lemma 2.4. Let Ω be a bounded domain in Rn , satisfying Assumptions 2.1,
and let 0 < α < 1. Then for any function f (x) on Ω with a finite seminorm
(−α)
[f ]1,0;Ω , we have
(−α)
[f ]α;Ω ≤ N (n, r0 , K0 , diam Ω, α) · [f ]1,0;Ω .
(2.25)
Proof. Let us fix x1 , x2 ∈ Ω, and set r = |x1 − x2 |. From the geometrical
properties of Ω it follows that we can choose x0 ∈ Ω such that
Br/N (x0 ) ⊂ Ω, |xk − x0 | ≤ N r, k = 1, 2.
Further, we can connect x0 with xk by means of a smooth path in Ω,
{x = hk (s) : 0 ≤ s ≤ sk }, hk (0) = x0 , hk (sk ) = xk ,
parametrised by the arc length s in such a manner that
(2.26)
1
0 ≤ sk ≤ N r, d(hk (s)) = dist(hk (s), ∂Ω) ≥ (sk − s)/N, 0 ≤ s ≤ sk .
2
(−α)
Since supΩ d1−α (x) maxi |Di f (x)| ≤ [f ]1,0 , we get
(2.27)
∫ sk
∫
(−α)
k
|f (x ) − f (x0 )| ≤
|Df (hk (s))| ds ≤ n[f ]1,0 ·
0
sk
dα−1 (hk (s)) ds.
0
By virtue of (2.26), the last integral does not exceed
∫ sk
N
(sk − s)α−1 ds ≤ N sαk ≤ N rα = N |x1 − x2 |α ,
0
so we obtain:
(−α)
|f (x1 ) − f (x2 )| ≤ |f (x1 ) − f (x0 )| + |f (x2 ) − f (x0 )| ≤ N [f ]1,0 · |x1 − x2 |α .
This estimate with arbitrary x1 , x2 ∈ Ω implies (2.25).
Remark 2.1. In the standard approach to the Schauder interior estimates
(γ)
(see [9], Ch.6) the notation [u]k,α is used for
(2.28)
A = max sup dk+α+γ
x,y
|l|=k x,y∈Ω
|Dl u(x) − Dl u(y)|
= sup δ k+α+γ [u]k,α;Ωδ ,
|x − y|α
δ>0
8
where 0 < α ≤ 1, k + α + γ ≥ 0, dx,y = dist ({x, y}, ∂Ω) , and
(2.29)
Ωδ = {x ∈ Ω : dist(x, ∂Ω) > δ}.
In particular, in the case γ = −k − α we have A = [u]k,α;Ω . One can show
(γ)
that for Lipschitz domains Ω seminorms [u]k,α in (2.8) and A in (2.28) are
equivalent, if k + α + γ ≥ 0.
However, in the case k + α + γ < 0 we have A < ∞ only for polynomials
(γ)
of degree at most k (and then A = 0), while [u]k,α < ∞ for more general class
of functions. For example, if k + α + γ < 0 ≤ k + 1 + γ and u ∈ C k+1 (B1 ),
then by the mean value theorem we obtain
(γ)
(γ)
[u]k,α ≤ N1 [u]k+1,0 ≤ N2 [u]k+1 < ∞.
3. Formulation of main existence results
Let Ω be a bounded domain in Rn and constants K, K1 ≥ 0, ν ∈ (0, 1],
α ∈ (0, 1) be fixed. We will consider nonlinear elliptic equations
(3.1)
F [u] = F (x, u, Du, D2 u) = 0 in Ω,
where F (x, u, ui , uij ) is defined on Ω×R1 ×Rn ×S n and satisfies the following
conditions:
Assumptions 3.1. (F0) The function F (x, u, ui , uij ) is lower convex with
respect to [uij ] ∈ S n ; (F1) (the ellipticity condition)
ν|ξ|2 ≤ F (x, u, ui , uij + ξi ξj ) − F (x, u, ui , uij ) ≤ ν −1 |ξ|2 for all ξ ∈ Rn ;
(F2) F (x, u, ui , uij ) is nonincreasing with respect to u, and
∑
|F (x, u, ui , uij ) − F (x, u, ui , uij )| ≤ K · (|u − u| +
|ui − ui |)
i
for all x, u, u, ui , ui , uij ; (F3) |F (x, 0, 0, 0)| ≤ K1 for all x ∈ Ω;
(F4) for any fixed (u, ui , uij ) ∈ R1 × Rn × S n , the seminorm in C α ,
∑
∑
[F (·, u, ui , uij )]α;Ω ≤ K · (|u| +
|ui | +
|uij |) + K1 .
i
ij
Remark 3.1. It is easy to see that if F satisfies the additional condition
(F∗ ) F (x, u, ui , uij ) is infinitely differentiable with respect to (u, ui , uij ) ∈
R1 × Rn × S n , then conditions (F0)–(F2) can be rewritten as follows:
(F0∗ ) ∂ 2 F/∂uij ∂upq · ηij ηpq ≥ 0 for all [ηij ] ∈ S n ; (F1∗ ) the functions
aij = ∂F/∂uij satisfy
(3.2)
aij = aji , ν|ξ|2 ≤ aij ξi ξj ≤ ν −1 |ξ|2 for all ξ ∈ Rn ;
(F2∗ ) the functions bi = ∂F/∂ui , c = ∂F/∂u satisfy
(3.3)
|bi | ≤ K,
−K ≤ c ≤ 0.
9
We notice that our equations (3.1) include the Bellman equations (1.1), if
m
α
aij = am
ij satisfy (3.2), c ≤ 0 for all m, and the norms in C (Ω),
m m
|am
ij , bi c |α;Ω ≤ K,
|f m |α;Ω ≤ K1 .
Consequently, the following Theorems 3.1–3.3 can be viewed as generalizations
of known Schauder-type results for linear equations (see [9], Theorems 6.13,
6.14, 6.31; [19], Ch.3).
Theorem 3.1. Let Ω be a bounded Lipschitz (satisfying Assumptions 2.1)
domain in Rn , d0 = diam Ω ≤ R0 = const < ∞, φ ∈ C(Ω), and let
F (x, u, ui , uij ) satisfy Assumptions 3.1. Then the Dirichlet problem
(3.4)
F [u] = 0 in Ω, u = φ on ∂Ω
has a unique solution u ∈ C 2,α;0 (Ω) ∩ C(Ω), provided 0 < α < α for some
constant α = α(n, ν) ∈ (0, 1). Moreover, we have
(3.5)
U0 = sup |u| ≤ sup |φ| + N (n, ν, K, d0 ) · K1 ,
Ω
(3.6)
∂Ω
(0)
∥u∥2,α;Ω ≤ N (n, ν, K, α, R0 ) · (U0 + d2+α
K1 ).
0
We will use the following definition for the classification of the boundaries
∂Ω having higher then the Lipschitz smoothness.
Definition 3.1. The boundary ∂Ω of a bounded domain Ω ⊂ Rn belongs
to the class C k,α for k = 1, 2, · · · , 0 < α ≤ 1, if there exists a function
Ψ(x) ∈ C k,α (Rn ) such that
Ω = {x ∈ Rn : Ψ(x) > 0} and |DΨ| ≥ 1 on ∂Ω.
Theorem 3.2. Let Ω be a bounded domain in Rn with the boundary ∂Ω ∈
C 2,α , φ ∈ C 2,α (Ω), and let F (x, u, ui , uij ) satisfy Assumptions 3.1. Then the
Dirichlet problem (3.4) has a unique solution u ∈ C 2,α (Ω), provided 0 < α < α
for some constant α = α(n, ν) ∈ (0, 1). Moreover, we have (3.5) and
(3.7)
|u|2,α;Ω ≤ N (n, ν, K, α, Ω) · (U0 + K1 + |φ|2,α;Ω ).
Theorem 3.3. Let Ω be a bounded domain in Rn with the boundary ∂Ω ∈
C 1,α , the functions b0 , b1 , · · · , bn , g ∈ C 1,α (Ω), and let F (x, u, ui , uij ) satisfy
Assumptions 3.1. Suppose that for some constant ν0 > 0,
(3.8)
b0 ≥ ν0 , b · N =
n
∑
bi Ni ≥ ν0 |b| > 0 on ∂Ω,
i
where N = −|DΨ|−1 DΨ is the outward unit normal on ∂Ω. Then the oblique
derivative problem
(3.9)
F [u] = 0 in Ω, Bu = b0 u + b · Du = g on ∂Ω
10
has a unique solution u ∈ C 2,α (Ω), provided 0 < α < α for some constant
α = α(n, ν, ν0 ) ∈ (0, 1). Moreover, we have
|u|2,α;Ω ≤ N · (U0 + K1 + |g|1,α;Ω ),
(3.10)
where the constant N depends only on n, ν, K, ν0 , domain Ω (with ∂Ω ∈ C 1,α ),
and on the norms of the functions b0 , b1 , · · · , bn in C 1,α (Ω).
Theorems 3.1, 3.2 are similar to Theorems 1.1, 1.2 from [26], Theorem 3.3
for Bellman equations (1.1) in Ω with ∂Ω ∈ C 2,α was proved in [1]. Analogous
results are also true in the parabolic case, with the same modifications as in
the theory of linear equations ([18], Theorems 5.1–5.3 in Ch.4).
Below, in Sections 4–8, we describe the technique of deriving C 2,α -estimates
for solutions of the problems (3.4), (3.9) by given constants n, K, K1 , α, · · · .
On the grounds of the C 2,α -estimates, one can prove Theorems 3.1–3.3 by the
standard continuity method (see [9], Ch.17; [13], Sec. 1.3). Therefore, we will
only supplement these sections with some remarks concerning the continuity
method applied to our problems.
Our approach to C 2,α -estimates in general case uses analogous estimates
in the case of simplest nonlinear equations and the comparison principle for
nonlinear equations. The simplest nonlinear equations are defined as follows.
Definition 3.2. The simplest nonlinear elliptic equation has the form F0 [u] =
F0 (D2 u) = 0, where the function F0 (uij ) satisfies (F0), (F1) on S n with some
constant ν ∈ (0, 1], and F0 (0) = 0. We denote F(ν) the class of all such
functions F0 (uij ).
The comparison principle is based on the next simple lemma.
Lemma 3.1. Let F (x, u, ui , uij ) satisfies (F0)–(F2) with some constants
K ≥ 0, ν ∈ (0, 1], and let u, v ∈ C 2 (Ω). Then F [u] − F [v] = L(u − v) in Ω,
where the linear elliptic operator L = aij Dij + bi Di + c has the coefficients
satisfying (3.2), (3.3) on Ω. Furthermore, in F0 (uij ) ∈ F (ν), then
F0 [u] − F0 [v] = L0 (u − v) = aij Dij (u − v) in Ω,
(3.11)
where aij = aij (x) satisfy (3.2) on Ω. In particular (v = 0),
(3.12)
F0 [u] = L0 u = aij Dij u in Ω.
The coefficients aij , bi , c can be constructed directly (see [26], Lemma 1.1)
or through approximation of F by smooth functions. For example, (3.11)
follows from the equality
(3.13)
F0 (uij ) − F0 (vij ) = aij · (uij − vij ) for all [uij ], [vij ] ∈ S n ,
where aij (depending on uij , vij ) satisfy (3.2). If F0 (uij ) ∈ F(ν) is smooth,
then we can take in (3.13)
∫ 1
aij =
aij (θuij + (1 − θ)vij ) dθ, where aij (uij ) = ∂F0 (uij )/∂uij ,
0
11
and (3.11) holds by virtue of (F1∗ ).
From this lemma and the classical maximum principle (see [9], Theorem
3.7; [19], Ch.3, §1), we get
Corollary 3.1. Let F satisfy (F0)–(F2), and let u, v ∈ C 2 (Ω) ∩ C(Ω). Then
sup |u − v| ≤ sup |u − v| + N (n, ν, K, diam Ω) · sup |F [u] − F [v]|.
Ω
Ω
∂Ω
In particular, under the assumptions of Theorems 3.1 and 3.2, there exists at
most one solution of the problem (3.4).
Taking in this corollary v = 0, we arrive at the following:
Corollary 3.2. Under the assumptions of Theorems 3.1 and 3.2, the estimate
(3.5) holds.
Corollary 3.3. If u ∈ C 2 (Ω) is a solution of the simplest nonlinear equaiton
F0 [u] = 0 in Ω, then supΩ |u| = sup∂Ω |u|.
4. Interior C 2,α − estimates: the simplest nonlinear equations
In this section we will obtain the interior C 2,α -estimates for solutions of simplest nonlinear equations F0 [u] = F0 (D2 u) = 0. In the next section, these
estimates will be applied to the proof of similar estimates for solutions of
general nonlinear equations.
Theorem 4.1. Let ν ∈ (0, 1], x0 ∈ Rn , r > 0, Br = Br (x0 ), φ ∈ C(Br ) , and
the function F0 (uij ) ∈ F (ν). Then the problem
(4.1)
F0 [v] = F0 (Dij v) = 0 in Br , v = φ on ∂Br
has a unique solution v ∈ C 2,α;0 (Br ) ∩ C(Br ) , and
(4.2)
(0)
∥v∥2,α;Br ≤ N · sup |φ|,
∂Br
where the constants α ∈ (0, 1] and N > 0 depend only on n, ν.
This theorem (for more general equations) was proved independently by
N.V. Krylov [11] and L.C. Evans [7] (see also [9], Section 17.4). We give here
another proof which we precede with three lemmas. We will use the following
lemma from [14], [22] (see also [13], [9], Sec. 9.8).
Lemma 4.1. Let ν ∈ (0, 1], ρ > 0, µ ∈ (0, 1], V ∈ C 2 (B2ρ ), and suppose that
V ≥ 0, aij Dij V ≤ 0 in B2ρ , where the functions aij = aij (x) satisfy
(4.3)
aij = aji , ν|ξ|2 ≤ aij ξi ξj ≤ ν −1 |ξ|2 for all ξ ∈ Rn .
Moreover, let the Lebesgue measure |{x ∈ Bρ : V (x) ≥ 1}| ≥ µ · |Bρ |. Then
inf V ≥ β = β(n, ν, µ) > 0.
Bρ
12
The next lemma is standard in the study of Hölder spaces.
Lemma 4.2. Let constants q > 1, α > 0, ρ0 > 0 be given, and let ω(ρ) be a
positive non-decreasing function on (0, ρ0 ] satisfying the inequality
(4.4)
q α ω(ρ) ≤ ω(qρ) for all ρ ∈ (0, ρ0 /q).
Then
(4.5)
ρ−α ω(ρ) ≤ q α ρ−α
0 ω(ρ0 ) for all ρ ∈ (0, ρ0 ].
Proof. By virtue of monotony of ω(ρ), (4.5) is evident for ρ ∈ (ρ0 /q, ρ0 ]. If
ρ < ρ0 /q, then q k ρ ∈ (ρ0 /q, ρ0 ] for some natural k, and using (4.4), we obtain:
ρ−α ω(ρ) ≤ (qρ)−α ω(qρ) ≤ · · · ≤ (q k ρ)−α ω(q k ρ) ≤ q α ρ−α
0 ω(ρ0 ).
Lemma 4.3. For v ∈ C 2 (Bρ ), let us set
(4.6)
ω = ω(ρ) = max osc Dij v,
Bρ
i,j
∫
and
∗
∗
ω = ω (ρ) =
osc Dλλ v dsλ ,
Λ
Bρ
where the surface integral over Λ = ∂B1 (0) is considered. Then
(4.7)
N −1 ω ≤ ω ∗ ≤ N ω,
where N = N (n).
Proof. Consider the function Q(λ, x) = Dλλ v(x) = λi λj Dij v(x) on the
set Rn × Bρ . Notice that for all i, j, λ, x,
2 Dij v(x) = Q(λ + ei + ej , x) − Q(λ + ei , x) − Q(λ + ej , x) + Q(λ, x).
Integrating this inequality over λ ∈ B1 (0), we obtain:
{∫
∫
∫
∫
2 |B1 | · Dij v(x) =
−
−
+
B1 (ei +ej )
B1 (ei )
B1 (ej )
}
Q(λ, x) dλ.
B1 (0)
In the right-hand side we have four integrals over unit balls with centers at
ei + ej , ei , ej , 0 ∈ Rn . All these balls are contained in B3 (0), therefore,
∫
osc Q(λ.x) dλ.
(4.8)
2 |B1 | · ω ≤ 4
B3 (0)
x∈Bρ
Further, Q(rλ, x) ≡ r2 Q(λ, x) for all r > 0. Hence by passing to the
spherical coordinates, the integral in (4.8) can be rewritten as follows:
∫ 3
∫
∫ 3 ∫
3n+2 ∗
n+1
osc Q(λ, x) dsλ =
r dr
osc Dλλ v · dsλ =
ω .
dr
x∈Bρ
x∈Bρ
n+2
0
Λ
0
∂Br (0)
13
This relation together with (4.8) give us the first inequality in (4.7). The
second inequality is obvious because for all λ ∈ Λ,
∑
ω∑ 2
osc Dλλ v ≤ ω ·
|λi λj | ≤
(λi + λ2j ) = nω.
x∈Bρ
2
i,j
i,j
Proof of Theorem 4.1. Now we will prove only the estimate (4.2) assuming
that the problem (4.1) is solvable in C 2,α;0 (Br ) ∩ C(Br ). This assumption will
be substantiated below, in Remark 5.1.
Step 1. Notice that each function F0 (uij ) ∈ F (ν) can be easy approximated
by smooth functions F0δ (uij ) ∈ F (ν), δ > 0, such that
(4.9)
|F0 (uij ) − F0δ (uij )| ≤ δ for all [uij ] ∈ S n , δ > 0.
If v δ is the solution of the problem
F0δ [v δ ] = 0 in Br ,
v δ = φ on ∂Br ,
then by Corollary 3.1 and (4.9) we obtain
sup |v δ − v| ≤ N · sup |F0 [v δ ] − F0 [v]| ≤ N · sup |F0 [v δ ] − F0δ [v δ ]| ≤ N δ.
Br
Br
Br
Therefore, if v δ satisfy the estimate (4.2) for all δ > 0, then this estimate
remains valid for v, the solution of initial problem (4.1). Thus we can assume without loss of generality that F0 (uij ) is smooth. Then automatically
v ∈ C ∞ (Br ) (see [13], Lemma I.3.2; [9], Lemma 17.16). Next, replacing r by
r − ε and then letting ε → 0+, we can assume v ∈ C 4 (Br ).
Step 2. Let us fix
(4.10)
1
z ∈ Br , 0 < ρ ≤ d(z) = dist(z, ∂Br ), ω = ω(ρ) = max osc Dij v.
i,j Bρ (z)
2
Relying on the smoothness of F0 (uij ) and v(x), we differentiate the equality
F0 [u] = 0 twice in the direction λ ∈ Λ. Using (F0∗ ), (F1∗ ) in Remark 3.1, we
have
(4.11)
aij Dij Dλλ v = −∂ 2 F0 /∂uij ∂upq · Dij Dλ v · Dpq Dλ v ≤ 0 in Br ,
where aij = ∂F0 /∂uij . Hence the functions
V λ (x) = Dλλ v(x) − inf Dλλ v
B2ρ (z)
satisfy the relations
(4.12)
V λ ≥ 0, aij Dij V λ ≤ 0 in B2ρ (z).
Step 3. Next, we fix x ∈ Bρ (z). From the definition of ω in (4.10) it follows
that there exists a point y ∈ Bρ (z) such that
ω = ω(ρ) ≥ max |Dij u(x) − Dij u(y)| ≥ ω/2.
i,j
14
By virtue of (3.13), we have aij ·(Dij v(x)−Dij v(y)) = 0, where aij (depending
on x, y) satisfy (4.3). Therefore, if µ1 ≤ µ2 ≤ · · · ≤ µn are eigenvalues of the
matrix M = [Mij ] = [Dij v(x) − Dij v(y)], then µn ≥ 2µω for some constant
µ = µ(n, ν) > 0. Moreover, we can assume µ = µ(n, ν) > 0 to be suitable
chosen so that the Lebesgue measure on Λ = ∂B1 (0),
|{λ ∈ Λ : Mij λi λj ≥ µω}| ≥ 2µ · |Λ|.
Since x ∈ Bρ (z) can be taken arbitrarily, and
Mij λi λj = Dλλ u(x) − Dλλ u(y) ≤ V λ (x),
we arrive at the estimate
|{λ ∈ Λ : V λ (x) ≥ µω}| ≥ 2µ · |Λ| for all x ∈ Bρ (z).
(4.13)
Step 4. Now we set
Γ = {(λ, x) ∈ Λ × Bρ (z) : V λ (x) ≥ µω} ⊂ Λ × Bρ (z),
Γλ = {x ∈ Bρ (z) : V λ (x) ≥ µω} for λ ∈ Λ,
Γx = {λ ∈ Λ : V λ (x) ≥ µω} for x ∈ Bρ (z),
Λ0 = {λ ∈ Λ : |Γλ | ≥ µ · |Bρ |}.
(4.14)
By (4.13) and Fubini’s theorem, the product-measure on Λ × Bρ (z),
∫
|Γ| =
|Γx | dx ≥ 2µ · |Λ| · |Bρ |.
Bρ (z)
On the other hand, by definition of Λ0 ,
∫
|Γλ | dsλ ≤ |Λ \ Λ0 | · µ|Bρ | ≤ µ|Λ| · |Bρ |,
Λ\Λ0
therefore,
∫
∫
|Λ0 | · |Bρ | ≥
|Γλ | dsλ = |Γ| −
Λ0
|Γλ | dsλ ≥ µ · |Λ| · |Bρ |,
Λ\Λ0
so we obtain |Λ0 | ≥ µ · |Λ|.
Step 5. The relations (4.12) allow us to apply Lemma 4.1 to the functions
V (x) = (µω)−1 V λ (x), λ ∈ Λ0 . This gives us
osc Dλλ v − osc Dλλ v ≥ inf Dλλ v − inf Dλλ v = inf V λ ≥ βµ · ω(ρ)
B2ρ (z)
Bρ (z)
Bρ (z)
B2ρ (z)
Bρ (z)
for all λ ∈ Λ0 . Taking into account (4.14) and using Lemma 4.3, we obtain:
]
∫ [
∗
∗
ω (2ρ)−ω (ρ) =
osc Dλλ v − osc Dλλ v dsλ ≥ |Λ0 |·βµ·ω(ρ) ≥ µ1 ·ω ∗ (ρ)
Λ
B2ρ (z)
Bρ (z)
15
for some constant µ1 = µ1 (n, ν) > 0.
Step 6. Taking α = α(n, ν) = log2 (1 + µ1 ) > 0 and ρ0 = d(z), we see that
2α ω ∗ (ρ) = (1 + µ1 )ω ∗ (ρ) ≤ ω ∗ (2ρ) for all ρ ∈ (0, ρ0 /2].
Using Lemma 4.2 with q = 2 and then again Lemma 4.3, we get
(4.15)
ρα ω(ρ) ≤ N (n, ν)ρ0−α ω(ρ0 ) for all ρ ∈ (0, ρ0 ].
This estimate implies
ρ2+α
ρ−α ω(ρ) ≤ N ρ20 ω(ρ0 ) ≤ N · [v]2,0;Br .
0
(0)
Taking the sup over z ∈ Br and ρ ∈ (0, ρ0 ] = (0, d(z)], and applying Theorem
2.1, we get
(0)
(0)
(0)
[v]2,α ≤ N M̂2,α ≤ N [v]2,0 .
Finally, from the interpolation inequalities (2.12) and Corollary 3.3 it follows
(0)
∥v∥2,α ≤ N sup |v| ≤ N sup |φ|.
Ω
∂Ω
So we have proved the desired estimate (4.2).
Remark 4.1. Theorem 4.1 is the only point where the convexity condition
(F0) is used. It is an open problem whether it remains valid without (F0). In
the case n = 2, this is so and follows from well-known result of Nirenberg [21]
on C 1,α -smoothness of solutions of linear elliptic equations with measurable
coefficients. Therefore, in the case n = 2 all our results, including Theorems
3.1–3.3, remain in effect without (F0). It was shown in [27] that the result of
Nirenberg fails for n = 3, and hence the convexity condition (F0) is essential
in our considerations when n ≥ 3.
5. Interior C 2,α − estimates: general equations
We will use the interior estimate (4.2) in C 2,α;0 for solutions of simplest nonlinear equations in the proof of estimates in C 2,α;0 , 0 < α < α for solutions of
general nonlinear equations.
∑ In the case of linear equations, we can consider
linear equations L0 v =
aij Dij v = 0 with constant coefficients in place of
simplest nonlinear equations F0 [v] = 0, so (4.2) is true with α = 1, and hence
for general linear equations with coefficients in C α , we can take arbitrary
α ∈ (0, 1).
Theorem 5.1. Let Ω be a bounded domain in Rn with d0 = diam Ω ≤ R0 =
const < ∞, and let F (x, u, ui , uij ) satisfy Assumptions 3.1 with some constants
K, K1 ≥ 0, ν ∈ (0, 1], α ∈ (0, α), where α = α(n, ν) is the constant in Theorem
4.1. Suppose u ∈ C 2,α;0 (Ω) ∩ C(Ω) is a solution of the equation F [u] = 0 in
Ω. Then the estimate (3.6) holds.
16
Proof. Throughout the proof we will denote by N different constants depending only on n, ν, K, α, R0 . We will also use the brief notations
(0)
(0)
U2,α = [u]2,α , Uk = [u]k,0 .
Step 1. Let us fix
y ∈ Ω, d = d(y) =
1
dist(y, ∂Ω), ρ ∈ (0, d], and ε ∈ (0, 1/2].
2
We set r = ρ/ε and consider separately the cases (a) r ≤ d and (b) r > d.
In the case (a), we take
φ = u − Ty,2 u, u = u(y), ui = Di u(y), uij = Dij u(y),
and F0 (uij ) = F (y, u, ui , uij + uij ). Since F0 (0) = F [u](y) = 0, F0 ∈ F(ν).
Next, we determine v as the solution of the problem
(5.1)
F0 [v] = F0 (Dij v) = 0 in Br = Br (y), v = φ on ∂Br .
From Theorem 4.1 it follows v ∈ C 2,α;0 (Br ) ∩ C(Br ) and
r2+α [v]2,α;Br/2 ≤ N · sup |φ|.
∂Br
Having in mind that ρ = εr ≤ r/2, applying Corollary 2.2 to v in Bρ , and
then Lemma 2.2 to u in Br , we obtain (with different constants N ):
ρ−2−α E2 [v; Bρ ] ≤ N ρα−α [v]2,α;Br/2 ≤ N ρα−α r−2−α sup |φ|
Br
≤ N ρα−α rα−α [u]2,α;Br = N εα−α [u]2,α;Br .
(0)
Since r ≤ d, by definition of U2,α = [u]2,α in (2.8), we get
(5.2)
d2+α ρ−2−α E2 [v; Bρ ] ≤ N εα−α U2,α .
Step 2. Further,we will evaluate F0 [φ] on Br = Br (y). For x ∈ Br , we
set u = u(x), ui = Di u(x), uij = Dij u(x). Notice that Dij φ(x) = uij − uij .
Therefore,
|F0 [φ](x)| = |F0 [φ](x) − F [u](x)| = |F (y, u, ui , uij ) − F (x, u, ui , uij )|
≤ |F (y, u, ui , uij ) − F (y, u, ui , uij )| + |F (y, u, ui , uij ) − F (x, u, ui , uij )|
in Br . Since |x − y| < r ≤ d ≤ d0 ≤ R0 , we have
∑
|u − u| +
|ui − ui | ≤ N r · |u|2,0;Br ≤ N rα · |u|2,0;Br .
i
By virtue of (F2) and (F4),
A = r−α sup |F0 [φ]| ≤ N · (|u|2,0;Br + K1 ) ,
Br
17
hence
d2+α A ≤ N · (U2 + U1 + U0 + d2+α
K1 ).
0
(5.3)
Now we proceed to evaluate φ − v on Br . By Lemma 3.1,
F0 [φ] = F0 [φ] − F0 [v] = L0 (φ − v) = aij Dij (φ − v) in Br
with aij satisfying (3.2). The functions φ − v together with
w(x) =
Arα 2
(r − |x − y|2 )
2nν
satisfy the relations
L0 w ≤ −Arα ≤ −|L0 (φ − v)| in Br = Br (y),
w = φ − v = 0 on ∂Br .
By the comparison principle, we get
sup |φ − v| ≤ sup |φ − v| ≤ sup |w| =
Bρ
Br
Br
A 2+α
r .
2nν
Using the equality r = ρ/ε and (5.3), we obtain the estimate
d2+α ρ−2−α sup |φ − v| ≤ N ε−2−α (U2 + U1 + U0 + d2+α
K1 ).
0
(5.4)
Bρ
Step 3. Now we will combine together (5.2) and (5.4). Obviously
E2 [u; Bρ ] ≤ E2 [v; Bρ ] + E2 [φ − v; Bρ ] ≤ E2 [v; Bρ ] + sup |φ − v|,
Bρ
so we receive
(5.5)
K1 ).
d2+α ρ−2−α E2 [u; Bρ ] ≤ N εα−α U2,α +N ε−2−α (U2 +U1 +U0 +d2+α
0
We have considered the case (a) r = ρ/ε ≤ d. In the case (b) r = ρ/ε > d,
we have d2+α ρ−2−α < ε−2−α , and E2 [u; Bρ ] ≤ supBd |u| ≤ U0 , so the left hand
side of (5.5) does not exceed ε−2−α U0 . Since y ∈ Ω and 0 < ρ ≤ d = d(y) are
chosen in an arbitrary manner, we get the following estimate for the seminorm
in Theorem 2.1:
(5.6)
M2,α ≤ N εα−α U2,α + N ε−2−α (U2 + U1 + U0 + d2+α
K1 )
0
(0)
(0)
for all ε > 0. By this theorem, (5.6) remains valid with U2,α in place of M2,α .
Choosing then ε = ε( n, ν, K, α, R0 ) > 0 such that the coefficient of U2,α would
be less then 1/2, we get
K1 ).
U2,α ≤ N · (U2 + U1 + U0 + d2+α
0
(5.7)
Finally, from (5.7) and the interpolation inequalities (2.12),
U2 + U1 ≤ ε U2,α + N (ε)U0 ,
ε > 0,
it follows
(0)
K1 ),
∥u∥2,α = U2,α + U2 + U1 + U0 ≤ N · (U0 + d2+α
0
completing the proof of theorem.
18
Remark 5.1. We relied on Theorem 4.1, though its proof given in Section
4 was not quite complete because we assumed the solvability of the problem
(4.1). This gap can be removed with the help of some variant of the method
of continuation with respect to the parameter ν. By virtue of (3.12), in
∑ the
case ν = 1 the class F(ν) = F(1) consists of the only function tr[uij ] =
uii
corresponding to the Laplace operator ∆, so the standard results of the linear
theory of elliptic equations yield all the statements of Theorem 4.1 with α = 1.
Setting out from ν ′ = 1, F ′ (uij ) = tr[uij ], and using (3.12), we notice that
the functions F0 (uij ) ∈ F (ν), 0 < ν < 1, satisfy the estimate
|F0′ (uij ) − F0 (uij )| ≤ γ · max |uij |,
(5.8)
i,j
where the constant γ = γ(n, ν) → 0 as ν → 0. In the proof of Theorem 5.1
with ν close to ν ′ = 1, one can consider the solution v of the problem
F0′ [v] = F0 (Dij v) = 0 in Br = Br (y),
v = φ on ∂Br
instead of the problem (5.1). Then the estimate (5.2) remains valid. By virtue
of (5.8) we have
sup |F0′ [φ] − F0 [φ]| ≤ γ[φ]2,0;Br ≤ γrα [φ]2,α;Br ,
Br
A′ = r−α sup |F0′ [φ]| ≤ γ[φ]2,α;Br + A.
Br
The last estimate together with (5.3) imply
d2+α A′ ≤ γU2,α + N · (U2 + U1 + U0 + d2+α
K1 ).
0
Hence in the right hand sides of (5.4)–(5.6) we will have an additional term
N ε−2−α γU2,α . If ν is close enough to 1, then γ is small and the coefficient of
U2,α in (5.6) still can be made less then 1/2.
Thus, starting from ν ′ = 1, we see that Theorem 5.1 remains true for some
ν = ν1 < 1. As was pointed out in Section 3, Theorem 3.1 can be obtained on
the grounds of the estimates provided by Corollary 3.2 and Theorem 5.1. In
turn, Theorem 4.1 is a special case of Theorem 3.1 with Ω = Br and F = F0 .
So, all Theorems 3.1, 4.1, and 5.1 are true for ν = ν1 .
Further moving past ν ′ = ν1 , we can approximate F0 (uij ) ∈ F(ν) by the
functions
F0′ (uij ) = θδij + (1 − θ)F0 (uij ), 0 < θ < 1.
Moreover, if ν < ν1 is close to ν1 , then for small θ > 0 we have F0′ (uij ) ∈
F(ν1 ) and the constant γ in (5.8) will also be small. Hence all the previous
considerations are valid for some ν = ν2 < ν1 . Continuing this procedure, we
can embrace the arbitrary ν ∈ (0, 1].
19
6. Some boundary estimates for solutions of linear elliptic equations
We will essentially use the following Lemma 6.1 announced in [23] (see also
[25]). For applications to nonlinear equations, the same results can be obtained by Krylov’s method (see [12] and [13], comments to §1 of Ch.5), which
implies the consideration of auxiliary degenerate elliptic or parabolic equation
for V (x)/xn .
n
Lemma 6.1. Let ν ∈ (0, 1], x0 ∈ R0n , r > 0, and Br+ = R+
∩Br (x0 ). Suppose
2
+
+
that V ∈ C (Br ) ∩ C(Br ) be a solution of the equation
aij Dij V = 0 in Br+ ,
(6.1)
where aij = aij (x) satisfy the conditions (4.3), and moreover,
V = 0 on Γ = R0n ∩ Br (x0 ).
(6.2)
Then the function ω(ρ) = osc
V (x)/xn satisfies the estimate
+
Bρ
ρ−α ω(ρ) ≤ N r−α ω(r) for all ρ ∈ (0, r],
(6.3)
where the constants α ∈ (0, 1], N > 0 depend only on n, ν.
Proof. By Lemma 4.2, for the proof of (6.3) it is sufficient to obtain, for
example, the estimate
10α ω(ρ) ≤ ω(10ρ) for ρ ∈ (0, r/10].
(6.4)
Using the transformation x −→ ρ−1 · (x − x0 ), we can consider only the case
x0 = 0, ρ = 1. In addition, replacing, if necessary, V (x) by one of the functions
λxn ± V (x), λ = const, we assume that
+
0 ≤ x−1
n · V (x) ≤ ω = ω(10) in B10 ,
(6.5)
and moreover,
(6.6)
|{x ∈ B2 (x∗ ) : x−1
n · V (x) ≥ ω/2}| ≥ |B2 |/2,
+
where x∗ = (0, · · · , 0, 4). Since V ≥ 0 and aij Dij V = 0 in B4 (x∗ ) ⊂ B10
,
and by virtue of (6.6),
|{x ∈ B2 (x∗ ) : V (x) ≥ ω}| ≥ |B2 |/2,
from Lemma 4.1 it follows that V (x) ≥ βω on B2 (x∗ ), where β = β(n, ν) > 0.
We fix y = (y1 , · · · , yn ) ∈ B1+ and set
y ∗ = (y1 , · · · , yn−1 , 4),
w(x) = (|x − y ∗ |−γ − 4−γ ) · βω,
where the constant γ = γ(n, ν) > 0 is so large that aij Dij w(x) ≥ 0 for
+
, and
all x ̸= y ∗ . Moreover, we have V (x) ≥ 0 = w(x) on ∂B4 (y ∗ ) ⊂ B10
20
V (x) ≥ βω ≥ w(x) on ∂B1 (y ∗ ) ⊂ B2 (x∗ ). Consequently, by the classical
maximum principle, V (x) ≥ w(x) on B4 (y ∗ ) \ B1 (y ∗ ). In particular,
V (y) ≥ w(y) = ((4 − yn )−γ − 4−γ )βω ≥ β1 ω · yn ,
where β1 = β1 (n, ν) > 0. Since y ∈ B1 can be selected in an arbitrary manner,
we get V (x)/xn ≥ β1 ω for all x ∈ B1+ . This estimate together with (6.5)
yield ω(1) ≤ (1 − β1 ) · ω(10). Taking α = α(n, ν) = − log10 (1 − β1 ) > 0, we
obtain the desired inequality (6.3).
Remark 6.1. By standard barrier technique, one can show that if (6.1), (6.2)
+
are valid for B2r
in place of Br+ , then
sup |V (x)/xn | ≤ N (n, ν) · r−1 sup |V |.
Br+
+
B2r
Therefore, in this case we have
ρ−α ω(ρ) ≤ N r−1−α sup |V | for all ρ ∈ (0, r].
+
B2r
We observe that since V (x0 ) = 0, the estimate (6.3) yields the existence of
the derivative
Dn V (x0 ) = lim ρ−1 · V (x0 + ρen ).
ρ→0+
Lemma 6.1 can be applied to each point y0 ∈ Γ in place of x0 , hence there
exists Dn V on Γ and moreover, the following assertions hold.
Corollary 6.1. Under the assumptions of Lemma 6.1, we have
0
[Dn V ]α;Br/2
≤ N r−α ω(r),
(6.7)
0
where Br/2
= Br/2 (x0 ) ∩ R0n . In addition, if Dn V ∈ C(Br+ ), then
0
[Dn V ]α;Br/2
Dn V.
≤ N r−α osc
+
(6.8)
Br
Proof. Applying (6.3) to different half-balls Bρ+ (y0 )), we get (6.7). Further,
if Dn V ∈ C(Br+ ), then
∫ xn
−1
V (x)/xn = xn
Dn V (x1 , · · · , xn−1 , t) dt,
0
Therefore,
V (x)/xn ≤ osc
Dn V,
ω(r) = osc
+
+
(6.9)
Br
Br
that yields (6.8).
Corollary 6.2. Let the assumptions of Lemma 6.1 be satisfied and suppose
that Dn V ∈ C(Br+ ). Then
ρ−1−α E1 [V ; Bρ+ ] ≤ N r−α osc
Dn V
+
Br
21
for all ρ ∈ (0, r].
Proof. Since ρ/xn > 1 in Bρ+ , from Lemma 6.1 applied to the functions
V (x) − λxn , λ ∈ R1 , we obtain:
ρ−1−α E1 [V ; Bρ+ ] ≤ ρ−1−α inf1 sup |V (x) − λxn |
λ∈R
Bρ+
1
≤ ρ−α inf1 sup |V (x)/xn − λ| = ρ−α ω(ρ) ≤ N r−α ω(r).
λ∈R B +
2
ρ
By virtue of (6.9), the desired estimate holds.
7. Boundary C 2,α − estimates: the Dirichlet problem
In order to obtain C 2,α -estimates of solutions near the boundary, with certain
boundary conditions, we need appropriate extensions of Theorem 4.1. The
following result of N.V. Krylov [12] can be treated as such an extension in the
case of the Dirichlet boundary condition.
Theorem 7.1. Let ν ∈ (0, 1], x0 ∈ R0n , r > 0, Br+ = Br+ (x0 ), φ ∈ C(Br+ ),
φ = 0 on Γ = R0n ∩ Br (x0 ) , and the function F0 (uij ) ∈ F (ν). Then the
problem
(7.1)
F0 [v] = F0 (Dij v) = 0 in Br+ , v = φ on ∂Br+
has a unique solution v ∈ C 2,α;0 (Br+ ∪ Γ) ∩ C(Br+ ) , and
(7.2)
(0)
∥v∥2,α;B + ∪Γ ≤ N · sup |φ|,
r
∂Br+
where the constants α ∈ (0, 1], N > 0 depend only on n, ν.
Our proof of this theorem is different from [12] and is based on Lemma 6.1.
We precede it with the following auxiliary statement.
Lemma 7.1. In addition to the assumption of Theorem 7.1, let the functions
F0 = F0 (uij ) and v = v(x) in (7.1) be smooth. Then for any ε > 0 and
0 < ρ ≤ r, we have
(7.3)
ρ−α ω(x0 , ρ) ≤ ε[v]2,α;Br+ + N (ε, n, ν)r−α ω(x0 , r),
where
(7.4)
ω(x, ρ) = max osc
Dij v,
+
i,j
Bρ (x)
Br+ = Br+ (x0 ), and α = α(n, ν) ∈ (0, 1] is the constant in Lemma 6.1.
Proof. For k = 1, 2, · · · , n − 1, set V k = Dk v. Since v = φ = 0 on Γ ⊂ R0n ,
we have also V k = 0 on Γ. Further, differentiating the equality F0 [v] = 0
with respect to xk , we obtain: aij Dij V k = 0 in Br+ , where aij = ∂F0 /∂uij .
22
Applying to V k first Lemma 2.3 with k = 1, and then Corollary 6.2, we find
that
ρ−α osc
Dik v = ρ−α osc
Di V k
+
+
Bρ
≤ ε[V
(7.5)
Bρ
k
]1,α;Bρ+
+ N (ε)ρ−1−α E1 [V k ; Bρ+ ]
≤ ε[v]2,α;Br+ + N (ε)r−α ω(x0 , r)
for all ε > 0, 0 < ρ ≤ r, and i = 1, 2, · · · , n.
Since k ≤ n − 1, on the left hand side of (7.5) there can appear any second
derivative of v except Dnn v. Further, by virtue of (3.11), for each x, y ∈ Bρ+
we have
aij · (Dij v(x) − Dij v(y)) = 0
with aij satisfying (3.2). Since ann ≥ ν > 0, we get
∑
D
v
≤
N
(n,
ν)
Dik v.
osc
osc
nn
+
+
Bρ
i+k≤2n−1
Bρ
From the last relation and (7.5), after redefining ε, the desired estimate follows.
Proof of Theorem 7.1. Let α = α(n, ν) ∈ (0, 1] be the smaller of the
constants α in Theorem 4.1 and Lemma 6.1. As in the proof of Theorem 4.1, we
will assume without loss of generality that F0 (uij ) is smooth and the problem
(7.1) has a solution v ∈ C 2,α;0 (Br+ ∪ Γ) ∩ C(Br+ ). The last assumption can
be substantiated by the method of continuation with respect to the parameter
ν, which is outlined in Remark 5.1. Notice that from the smoothness of F0 it
follows v ∈ C 3 (Br+ ∪ Γ) (see [9], Sec. 17.8).
We first prove that ω(x, ρ) in (7.4) satisfies
(7.6)
d2+α (x)ρ−α ω(x, ρ) ≤ εV2,α + N (ε)V2
for all x ∈ Br+ ∪ Γ, ρ ∈ (0, d(x)], and ε > 0, where in accordance with (2.9),
1
(0)
(0)
d(x) = dist(x, ∂Ω \ Γ), V2,α = [v]2,α;B + ∪Γ , V2 = [v]2,0;B + ∪Γ .
r
r
2
We will divide the proof of (7.6) into several cases.
(a) x ∈ Γ, 0 < ρ ≤ d(x). In this case (7.6) follows immediately from
Lemma 7.1 with x0 = x, r = d(x).
(b) d(x)/4 ≤ ρ ≤ d(x). Since dα (x)ρ−α < 4α < 4, the left hand side of
(7.6) does not exceed 4d2 (x)ω(x, d(x)) ≤ 8V2 , so this estimate is true even
with ε = 0.
(c) xn ≤ ρ ≤ d(x)/4, where x = (x′ , xn ), x′ ∈ Γ ⊂ R0n . In this case
+
(x′ ). Moreover, (7.6) is valid for x = x′ , hence 2ρ <
we have Bρ+ (x) ⊂ B2ρ
d(x)/2 < d(x′ ), and in view of (a) we get
d2+α (x)ρ−α ω(x, ρ) ≤ 22+2α d2+α (x′ )(2ρ)−α ω(x′ , ρ) ≤ 22+2α · [εV2,α + N (ε)V2 ],
which, after redefining ε, also leads to (7.6).
23
(d) 0 < ρ < ρ0 = min(d(x)/4, xn ). First we apply the estimate (4.15); this
gives us ρα ω(x, ρ) ≤ N ρ−α
0 ω(x, ρ0 ), and then (7.6) follows from (b) or (c),
depending whether ρ0 = d(x)/4 or ρ0 = xn < d(x)/4.
We have proved (7.6). By virtue of (2.19), we get
N1−1 V2,α ≤ M̂2,α [v; Br+ ∪ Γ] ≤ εV2,α + N (ε)V2 .
(0)
Setting ε = (2N1 )−1 , we get V2,α ≤ N V2 . Finally, using the interpolation
inequalities (2.12), and then Corollary 3.3, we obtain
(0)
∥v∥2,α;B + ∪Γ ≤ N · sup |v| = N · sup |φ|,
r
Br+
∂Br+
so the desired inequality (7.2) is true with α = α = α(n, ν) ∈ (0, 1].
Theorem 7.2. Let F (x, u, ui , uij ) satisfy Assumptions 3.1 with Ω = Br+0 (x0 )
and some constants K, K1 ≥ 0, ν ∈ (0, 1], α ∈ (0, α), where x0 ∈ R0n ,
r0 ∈ (0, 1], and α = α(n, ν) is the constant in Theorem 7.1. Let u0 ∈ C 2,α (Γ),
where Γ = R0n ∩ Br0 (x0 ) is identified with a ball in R0n = Rn−1 . Then for any
function u ∈ C 2,α;0 (B + ∪ Γ), satisfying the equalities
F [u] = 0 in B + , u = u0 on Γ,
(7.7)
we have
(7.8)
[
(0)
∥u∥2,α;B + ∪Γ
≤ N (n, ν, K, α) · sup |u| +
B+
r02+α (K1
]
+ |u0 |2,α;Γ ) .
Proof. Setting
û = u − u0 ,
F̂ (x, u, ui , uij ) = F (x, u + u0 (x), ui + Di u0 (x), uij + Dij u0 (x)),
one can see that the equalities (7.7) are equivalent to
F̂ [û] = 0 in B + , û = 0 on Γ.
Moreover, F̂ satisfies Assumptions 3.1 with a new constant K̂1 = N · (K1 +
|u0 |2,α;Γ ) in place of K1 . Thus the proof of (7.8) is reduced to the case u0 = 0.
As in the proof of Theorem 5.1, we introduce the notations
(0)
(0)
U2,α = [u]2,α;B + ∪Γ , Uk = [u]k,0;B + ∪Γ ,
and then we fix
y = (y ′ , yn ) ∈ B + ∪ Γ, d = d(y) =
1
dist(y, ∂B + \ Γ), ρ ∈ (0, d], ε ∈ (0, 1/2].
2
Then in the cases (a) ρ/ε ≤ min{yn , d/8} and (b) ρ/ε > d/8, quite analogously to (5.5), we obtain the estimate
(7.9)
d2+α ρ−2−α E2 [u; Bρ+ (y)] ≤ N εα−α U2,α + N ε−2−α (U2 + U1 + U0 + r02+α K1 ).
24
In the remained case (c) yn < ρ/ε ≤ d/8, we take d′ = 12 dist{y ′ , ∂B + \ Γ},
r = 4ρ/ε, φ = u − Ty′ ,2 u, u = u(y ′ ), ui = Di u(y ′ ), uij = Dij u(y ′ ), and
F0 (uij ) = F (y ′ , u, ui , uij + uij ) ∈ F (ν). It is easy to see that
+
Bρ+ (y) ⊂ Br/2
(y ′ ),
r ≤ d/2 ≤ d′ .
Moreover, since u = 0 on Γ and y ∈ Γ, also φ = 0 on Γ. Define v as the
solution of the problem (7.1) in Br+ = Br+ (y ′ ). By analogy to (5.2), relying on
Theorem 7.1 in place of Theorem 4.1, we have
d2+α ρ−2−α E2 [v; Bρ+ (y)] ≤ N εα−α U2,α .
The other points of the proof of Theorem 5.1, yielding the estimate (7.9)
in the case (c) and the desired estimate (7.8), are valid with minimal modifications.
Remark 7.1. Theorem 7.2 together with Theorem 3.1 yield Theorem 3.2.
Indeed, upon dividing ∂Ω into a finite number of small subsets and “flattening
of the boundary”, Theorem 7.2 gives C 2,α - estimates near ∂Ω for solutions of
the problem (3.4). These estimates and interior C 2,α -estimate (3.6) constitute
the estimate (3.7) in Theorem 3.2.
8. Boundary C 2,α − estimates: the oblique derivative problem
The formulation of the following Theorems 8.1 and 8.2 are similar to ones of
Theorems 7.1 and 7.2, only instead of the Dirichlet condition on Γ we now
have Dn u = 0 on Γ.
Theorem 8.1. Let ν ∈ (0, 1], x0 ∈ R0n , r > 0, Br+ = Br+ (x0 ), φ ∈ C(Br+ ) be
given, and the function F0 (uij ) ∈ F (ν). Then the equation
(8.1)
F0 [v] = F0 (Dij v) = 0 in Br+
with the boundary conditions
(8.2)
Dn v = 0 on Γ = R0n ∩ Br (x0 ), v = φ on ∂Br+ \ Γ,
has a unique solution v ∈ C 2,α;0 (Br+ ∪ Γ) ∩ C(Br+ ) , and
(8.3)
(0)
∥v∥2,α;B + ∪Γ ≤ N · sup |φ|,
r
∂Br+
where the constants α ∈ (0, 1], N > 0 depend only on n, ν.
Proof. We choose α ∈ (0, α) (for example, α = α/2), where α = α(n, ν) ∈
(0, 1] be the smaller of the constants α in Theorem 4.1 and Lemma 6.1. Under
such choice of α, we will prove (8.3).
Step 1. All the reasonings concerning the existence of the solution of the
problem (8.1), (8.2) are quite similar to ones in the previous section related
to the problem (7.1). Therefore, we will assume the existence of solution v,
25
and moreover, we will consider smooth F0 (uij ), so that v ∈ C 3 (Br+ ∪ Γ). By
Corollary 3.1, using the equalities (8.2), we get
V0 = sup |v| = sup |v| ≤ sup |φ|.
(8.4)
Br+
∂Br+ \Γ
∂Br+
Following the lines of the proof of Theorem 7.1, we notice that it suffices
to prove the estimate
d2+α (y)ρ−α ω(y, ρ) ≤ N · (V2 + V0 )
(8.5)
for all y ∈ Br+ ∪ Γ, 0 < ρ ≤ d(y) = 12 dist(y, ∂Ω \ Γ). This estimate is similar
to (7.6), and the cases (b)–(d)) of its proof remain valid, so we will consider
only the case (a) y ∈ Γ.
Step 2. Let us fix y ∈ Γ and d = d(y). Differentiating the equality (8.1)
with respect to xn , we obtain aij Dij Dn v = 0 in Br+ , where aij = ∂F0 /∂uij .
Since Dn v = 0 on Γ, we can apply Corollary 6.1 to the function V = Dn v in
Bd+ (y) ⊂ Br+ , that gives us the estimate
(8.6)
−α
0 (y) ≤ N d
[Dnn v]α;Bd/2
osc
Dnn v ≤ N d−2−α V2 .
+
Bd (y)
Besides this, we have Din v = 0 on Γ for i = 1, · · · , n − 1. Therefore,
0
setting Ω = Bd/2
(y) ⊂ R0n = Rn−1 , we see that the function v0 (x′ ) =
v0 (x1 , · · · , xn−1 ) = v(x′ , 0) satisfies the equality
F0 (Dij v0 (x′ ), 0, · · · , 0, Dnn v(x′ , 0)) = 0 in Ω.
By virtue of (8.6), the corresponding function
F (x′ , uij ) = F0 (uij , 0, · · · , 0, Dnn v(x′ , 0)) = 0 on S n−1
satisfies Assumptions 3.1 with K = 0, K1 = N d−2−α V2 . This enables us to
use Theorem 5.1, yielding the estimate
(8.7)
(0)
∥v0 ∥2,α;B 0
d/2
(y)
≤ N · (V0 + K1 d2+α ) ≤ N · (V0 + V2 ).
+
0
Step 3. Now we can apply Theorem 7.2 with B + = Bd/2
(y), Γ = Bd/2
(y).
This gives us
(
)
(0)
(0)
2+α
2+α
d [v]2,α;B + (y) ≤ 4 [v]2,α;B + ∪Γ ≤ N · V0 + ∥v0 ∥2,α;Γ ≤ N · (V0 + V2 ).
d/4
The last estimate contains (8.5) for 0 < ρ ≤ d/4. If d/4 < ρ ≤ d = d(y), then
d2+α ρ−α ω(y, ρ) ≤ 4α d2 ω(y, ρ) ≤ N V2 ,
so (8.5) is true for all ρ ∈ (0, d].
Theorem 8.2. Let ν ∈ (0, 1], x0 ∈ R0n , r0 > 0, B + = Br+0 (x0 ), Γ = R0n ∩
Br0 (x0 ), and let F (x, u, ui , uij ) satisfy Assumptions 3.1 with Ω = B + and some
26
constants K, K1 ≥ 0, α ∈ (0, 1). Then for any function u ∈ C 2,α;0 (B + ∪ Γ)
satisfying the equalities
F [u] = 0 in B + , Dn u = 0 on Γ,
(8.8)
we have
(0)
∥u∥2,α;B + ∪Γ
(8.9)
(
)
2+α
≤ N · sup |u| + r0 K1 ,
B+
where N = N (n, ν, K, α), provided 0 < α < α for some constant α =
α(n, ν) ∈ (0, 1).
Proof. We take as α = α(n, ν) ∈ (0, 1) the constant α in Theorem 8.1.
Then we can reproduce almost literary the proof of Theorem 7.2, only in the
case (c) we define v as the solution of the problem (8.1), (8.2), and accordingly,
we rely on Theorem 8.1 instead of Theorem 7.1
Repeating the reasonings in Step 2 of the proof of Theorem 5.1, we now
have
F0 [φ] = F0 [φ] − F0 [v] = L0 (φ − v) = aij Dij (φ − v) in Br+ (y ′ )
with aij satisfying (3.2), and the function v(x) together with
φ(x) = u(x) − Ty′ ,2 u(x), w(x) =
Arα 2
(r − |x − y ′ |2 )
2nν
satisfy the relations
|L0 (φ − v)| ≤ rα A = rα sup |F0 [φ]| ≤ −L0 w in Br+ (y ′ ),
Br+ (y ′ )
Dn (φ − v)| = Dn w = 0 on Γ ∩ Br (y ′ ),
φ − v = w = 0 on ∂Br+ (y ′ ) \ Γ.
Therefore, we can apply the comparison principle yielding
sup |φ − v| ≤ sup |φ − v| ≤ sup |w| =
Bρ+ (y ′ )
Br+ (y ′ )
Br+ (y ′ )
A 2+α
r .
2nν
The remained part of the proof is almost the same as in the proof of
Theorem 5.1 , so we obtain the estimate (8.9).
We will use the estimate (8.9) in the proof of the C 2,α -estimate (3.10) in
the formulation of the Theorem 3.3. However, “flattening of the boundary”
∂Ω ∈ C 1,α in the general case would deteriorate C 2,α -smoothness of solutions.
Therefore, we first consider a special case of the boundary conditions which is
reduced to (8.8). For this purpose, we need some auxiliary results concerning
the extension of functions from ∂Ω to Ω. The following lemma is contained in
[8], Lemma 2.3.
27
n
Lemma 8.1. Let r > 0, Br+ = R+
∩ Br (0), Br0 = R0n ∩ Br (0), and Φ0 ∈
1,α
0
C (Br ), 0 < α < 1. Then there exists a function Φ ∈ C ∞ (Br+ ) ∩ C 1,α (Br+ )
such that Φ = Φ0 on Br0 ,
|Φ|1,α;Br+ ≤ N (n, α) · |Φ0 |1,α;Br0 ,
(8.10)
and for any k = 2, 3, · · · , we have
(8.11)
max sup ynk−1−α |Dl Φ(y)| ≤ N (k, n, α) · [Φ0 ]1,α;Br0 .
|l|=k
Br+
Let us fix an arbitrary point x0 ∈ ∂Ω ∈ C 1,α . Using Definition 3.1 and the
implicit theorem, we cam choose an orthonormal coordinate system centered
at x0 and r0 ∈ (0, 1] such that Ωr0 (x0 ) = Ω ∩ Br0 (x0 ) is represented in the
form (2.23) with ψ0 ∈ C 1,α (Br00 ), where Br00 is the projection of Br0 (x0 ) onto
R0n .
Without loss of generality we now take x0 = 0. Applying Lemma 8.1
with Φ0 = ψ0 , 0 < r ≤ r, we obtain the existence of a smooth function
ψ ∈ C ∞ (Br+ ) ∩ C 1,α (Br+ ) such that ψ = ψ0 on Br0 , and (8.10), (8.11) are true
for Φ = ψ, Φ0 = ψ0 . Moreover, replacing ψ(y) by ψ(y) + N yn if necessary,
we may assume that Dn ψ ≥ 1 on Br+ . Now we introduce the new coordinates
x = x(y) ∈ C 1,α (Br+ ) by the mapping
x′ = y ′ , xn = ψ(y), y ∈ Br+ .
We have det ∂x/∂y = Dn ψ ≥ 1, therefore, the inverse mapping y = y(x),
where
y ′ = x′ , yn = η(x), x ∈ Ωr = x(Br+ ),
has the same smoothness as x = x(y). It is easy to see that
1
N −1 yn ≤ d(x) = dist(x, ∂Ω) ≤ N yn = N η(x), x ∈ Ωr ,
2
with N = N (Ω). Therefore, for yn = η(x) we have
(8.12)
max |Dl η(x)| ≤ N (k, α, Ω) · yn1+α−k , x ∈ Ωr , k = 2, 3, · · · .
|l|=k
Using all these properties, one can obtain the following lemma as a consequence of Lemma 8.1.
Lemma 8.2. Let Ω be a bounded domain in Rn with ∂Ω ∈ C 1,α , 0 < α < 1.
Under the previous assumptions, let r ∈ (0, r] be chosen small enough, so that
Ωr = x(Br+ ) ⊂ Ω, and let a function ϕ0 ∈ C 1,α (Ωr ) be given. Then there
exists a function ϕ ∈ C ∞ (Ωr ) ∩ C 1,α (Ωr ) such that ϕ = ϕ0 on γr = ∂Ω ∩ ∂Ωr ,
|ϕ|1,α;Ωr ≤ N (α, Ω) · |ϕ0 |1,α;Ωr ,
(8.13)
and for any k = 2, 3, · · · , we have
(8.14)
max sup dk−1−α (x)|Dl ϕ(x)| ≤ N (k, α, Ω) · [ϕ0 ]1,α;Ωr .
|l|=k Ωr
28
Notice that in this construction, we can take r > 0 independent on x0 ∈ ∂Ω.
Using then the standard partition of unity (see [9], Sec. 6.9), we arrive at the
following statement.
Corollary 8.1. If ϕ0 ∈ C 1,α (Ω), then there exists a function ϕ ∈ C ∞ (Ω) ∩
C 1,α (Ω) such that ϕ = ϕ0 on ∂Ω,
|ϕ|1,α;Ω ≤ N (α, Ω)|ϕ0 |1,α;Ω ,
(8.15)
and for any k = 2, 3, · · · , we have
(−1−α)
(8.16)
[ϕ]k,o;Ω
≤ N (k, α, Ω)[ϕ0 ]1,α;Ω .
The following two lemmas serve as the intermediate steps in the proof of the
estimate (3.10). As before, we fix x0 ∈ ∂Ω ∈ C 1,α , 0 < α < 1, and a suitable
C 1,α -mapping x = x(y), so that some portion of Ω near x0 is represented in
the form Ωr = x(Br+ ), 0 < r ≤ r0 , and γr = ∂Ω ∩ ∂Ωr = x(Γ), Γ = Br0 .
Lemma 8.3. Let F (x, u, ui , uij ) satisfy Assumptions 3.1 with some constants
K, K1 ≥ 0, 0 < α < 1. Then for any function u ∈ C 2,α;0 (Ωr ∪ γr ) satisfying
the equalities
(8.17)
F0 [u] = 0 in Ωr ,
we have
(8.18)
(0)
∥u∥2,α;Ωr ∪γr
Dn u = 0 on γr = ∂Ω ∩ ∂Ωr ,
(
)
2+α ∗
≤ N (n, ν, K, α, Ω) · sup |u| + r K1 ,
Ωr
where K1∗ = K1 + |u|2,0;Ωr , provided 0 < α < α for some constant α =
α(n, ν, Ω) ∈ (0, 1).
Proof. Under the C 1,α -diffeomorphism
(8.19)
x ∈ Ωr ←→ y ∈ Br+ , where x′ = y ′ , xn = ψ(y), yn = η(x),
let us define û(y) = u(x). We have
Di u(x) = Dk û(y) · Di yk (x),
(8.20)
Dij u(x) = Dkm û(y) · Di yk (x) · Dj ym (x) + Dn û(y) · Dij η(x).
From (8.17) it folows
F̂ [û(y)] = F̂ (y, û, Dk û, Dkm û) = 0 in Br+ ,
Dn û = 0 on Γ = Br0 ,
where
F̂ (y, u, uk , ukm ) = F (x, u, fik uk , fik fjm ukm + gij ),
fik = fik (y) = Di yk (x), gij = gij (y) = Dn û(y) · Dij η(x).
29
Since Dn û(y ′ , 0) = 0, by the mean value theorem we have |Dn û(y)| ≤
yn [û]2,0 . Together with (8.12), this gives us the estimates
|gij (y)| ≤ N ynα · [û]2,0 , |Dgij (y)| ≤ N ynα−1 · [û]2,0 .
Applying Lemma 2.4, we get
(−α)
[gij ]α ≤ N [gij ]1,0 ≤ N [û]2,0 ,
hence
|fik |α ≤ N,
|gij |α ≤ N [û]2,0 .
Relying on this estimates, it is easy to show that Assumptions 3.1 on
the function F yield the similar assumptions on F̂ , with Ω = Br+ and some
constants
ν̂ ≥ ν/N, K̂ ≤ N K, K̂1 ≤ N · (K1 + [û]2,0 )
in place of ν, K, K1 . By Theorem 8.2 we can assert
(
)
(0)
2+α
∥û∥2,α;B + ∪Γ ≤ N · sup |û| + r0 K̂1 .
B+
Furthermore, (8.20) brings us to the estimates
(0)
(0)
[u]2,0 ) ≤ N · [û]2,0 , ∥u∥2,α;Ωr ∪γr ≤ N · ∥û∥2,α;B + ∪Γ .
Since the mapping x = x(y) has the same properties as y = y(x), we also
have [û]2,0 ≤ N [u]2,0 . These inequalities provide us the estimate (8.18).
Lemma 8.4. In the formulation of Lemma 8.3, replace (8.17) with the equalities
(8.21)
F0 [u] = 0 in Ωr , Dn u = g0 on γr ,
where g0 ∈ C 1,α (Ωr ). Then the estimate (8.18) remains valid with
K1∗ = K1 + |u|2,0;Ωr + G0 ,
where G0 = |g0 |1,α;Ωr .
Proof. Combining Lemma 8.2 with the standard extension lemmas (see [9],
Sec. 6.9), we can construct a function g1 defined in a wider domain
Qr = {x = (x′ , xn ) ∈ Rn : x′ ∈ Br0 , ψ0 (x′ ) < xn < h} ⊃ Ωr , where h = const,
so that g1 ∈ C ∞ (Qr ) ∩ C 1,α (Qr ),
g1 = g0 on γr = ∂Ω ∩ ∂Ωr = {x ∈ Rn : x′ ∈ Br0 , xn = ψ0 (x′ )} ⊂ ∂Qr ,
and for any k = 2, 3, · · · , we have
(8.22)
max sup dk−1−α (x)|Dl g1 (x)| ≤ N (k, α, Ω)G0 .
|l|=k Qr
30
We also have
N −1 d(x) ≤ xn − ψ0 (x′ ) ≤ N d(x),
(8.23)
x ∈ Qr .
Now we define
′
∫
h
u0 (x) = u0 (x , xn ) = −
(8.24)
g1 (x′ , t) dt, x ∈ Qr .
xn
We state that
(8.25)
|Dl u0 (x)| ≤ N dα−1 (x)G0 for all |l| = 3, x ∈ Qr .
If |l| = 3, ln > 0, then Dl u0 (x) = Dij Dn u0 (x) = Dij g1 (x) for some i, j, and
(8.25) gives us (8.22). If |l| = 3, ln = 0, then (8.24), (8.22) yield
∫ h
∫ h
l
l
′
|D u0 (x)| = D g1 (x , t) dt ≤ N G0
dα−2 (x′ , t) dt,
xn
xn
and (8.25) follows from (8.23). Finally, applying Lemma 2.4 and using (8.25),
we obtain:
[u0 ]2,α = max[Dij u0 ]α ≤ N max[Dij u0 ]−α
1,0 ≤ N G0 .
i,j
i,j
Moreover, Dn u0 = g1 = g0 on γr . By setting û = u − u0 , as in the proof
of Theorem 7.2, this lemma is reduced to Lemma 8.3.
Theorem 8.3. Under the assumptions of Theorem 3.3, let u ∈ C 2,α (Ω) be a
solution of the problem (3.9). Then the estimate (3.10) holds.
Proof. Let us fix x0 ∈ ∂Ω. In the previous construction, we can choose
an orthonormal coordinate system with b(x0 ) = (b1 (x0 ), · · · , bn (x0 )) directed
along the positive xn - axis. By virtue of (3.8), we can impose the restriction
N −1 ≤ det ∂x/∂y ≤ N , where N = N (n, ν0 ), on the C 1,α -diffeomorphism
(8.19). Therefore, the constant α in Lemma 8.3 depends only on n, ν, ν0 .
Dividing both sides of the condition bi Di u + b0 u = g by bn (x0 ) > 0, we
can reduce it to the case bi (x0 ) = δin . Next, we rewrite it in the form
Dn u = g0 = g − (bi − δin )Di u − b0 u on γr = ∂Ω ∩ ∂Ωr .
Since |bi (x) − δin | = |bi (x) − bi (x0 )| ≤ N rα in Ωr , by virtue of (2.6) we get
|g0 |1,α;Ωr ≤ |φ|1,α + N rα U2,α + N |u|2,0 .
where U2,α = [u]2,α;Ω . Then Lemma 8.4 gives us
(8.26)
(0)
∥u∥2,α;Ωr ∪γr ≤ N r2+2α U2,α + N (r) · (K1 + |φ|1,α + |u|2,0 ) .
Further, let us fix δ = 1/N > 0 such that
dist (ωr , ∂Ωr \ γr ) ≥ δr, where ωr = Ω ∩ Bδr (x0 ).
31
The estimate (8.26) yields
(8.27)
[u]2,α;ωr ≤ N0 rα U2,α + N (r) · (K1 + |φ|1,α + |u|2,0 ) ,
where N0 does not depend on r > 0.
Using the last estimate with arbitrary x0 ∈ ∂Ω, we will show that
(8.28)
U2,α = [u]2,α;Ω ≤ N · (K1 + |φ|1,α + |u|2,0 ) .
By definition of [u]2,α , we can choose x, y ∈ Ω, and i, j, such that
(8.29)
U2,α ≤ 2 |Dij u(x) − Dij u(y)|/|x − y|α .
We consider separately three cases.
(a) |x − y| < δr/3, d(x) = dist(x, ∂Ω) < δr/3. In this case for some
x0 ∈ ∂Ω we have |x − x0 | = 2d(x) < 2δr/3, hence x, y ∈ Ω ∩ Bδr (x0 ) = ωr .
Now let us fix r > 0 such that N0 rα < 1/4 in (8.27). Since the right hand
side in (8.29) does not exceed [u]2,α;ωr , from (8.27) we obtain (8.28).
(b) |x − y| < δr/3, d(x) ≥ δr/3. We have y ∈ B(x) = Bd(x) (x), hence
(8.28) follows from the interior estimate (3.6).
(c) |x − y| ≥ δr/3. Directly from (8.29) it follows U2,α ≤ N |u|2,0 .
We have proved (8.28). Finally, using the interpolation inequalities which
are true even for Lipschitz domains (see [20], Ch.5, Sec.33), we can replace
|u|2,0 by U0 = supΩ |u| in (8.28), so that (8.28) turns into (3.10).
Remark 8.1. Under the assumptions of Theorem 3.3, U0 = supΩ |u| is easy
estimated by the comparison principle (see [9] the proof of Theorem 6.31).
Hence we have a priori estimates of solutions to the oblique derivative problem
(3.9) in C 2,α (Ω), depending only on the prescribed constants. On the grounds
of these C 2,α -estimates, the solvability of the problem (3.9) can be stated by
means of the satndard continuity method (see [9], Sec. 17.9).
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