Dini continuity of the first derivatives of generalized solutions to the Dirichlet problem for linear elliptic second order equations in nonsmooth domains

POLONICI MATHEMATICI LXIX.2 (1998)

Dini continuity of the first derivatives of

generalized solutions to the Dirichlet problem for linear elliptic second order equations in nonsmooth domains

by Michail Borsuk (Olsztyn)

Abstract. We consider generalized solutions to the Dirichlet problem for linear elliptic second order equations in a domain bounded by a Dini–Lyapunov surface and containing a conical point. For such solutions we derive Dini estimates for the first order generalized derivatives.

1. Introduction. We consider generalized solutions to the Dirichlet problem for a linear uniformly elliptic second order equation in divergence form

(DL)

 

 

 



 

 

 

∂

∂x

(a

(x)u

+ a

(x)u) + b

(x)u

+ c(x)u

= g(x) + ∂f

(x)

∂x

, x ∈ G, u(x) = ϕ(x), x ∈ ∂G

(summation over repeated indices from 1 to n is understood), where G ⊂ R

is a bounded domain with boundary ∂G and ∂G is a Dini–Lyapunov surface containing the origin O as a conical point. This last means that ∂G \ O is a smooth manifold but near O the domain G is diffeomorphic to a cone.

H¨older estimates for the first derivatives of generalized solutions to the problem (DL) are well known in the case where the leading coefficients a

(x) are H¨older continuous (see e.g. [5, 8.11] for smooth domains and [1]

for domains with angular points). Here we derive Dini estimates for the first derivatives of generalized solutions of the problem (DL) in a domain with a conical boundary point under minimal smoothness conditions on the leading coefficients (Dini continuity). It should be noted that interior Dini continuity

1991 Mathematics Subject Classification: 35J.

Key words and phrases : elliptic equations, nonsmooth domains, Dini continuous, smoothness of generalized solutions.

[129]

of the first and second derivatives of generalized solutions to the problem (DL) was investigated in [3, 7] under the condition of Dini continuity of the first derivatives of the leading coefficients.

We introduce the following notations and definitions:

• [l]: the integral part of l (if l is not an integer);

• r = |x| = ( P

i=1

x

)

;

• G

⋐ G: G

has compact closure contained in G;

• mes G: volume of G;

• S

: the unit sphere in R

;

• B

(x

): the open ball in R

of radius r centered at x

;

• ω

= 2π

/(nΓ (n/2)): the volume of the unit ball in R

;

• σ

= nω

: the area of the n-dimensional unit sphere;

• R

: the half-space x

> 0;

• Σ: the hyperplane {x

= 0};

• B

= B

∩ R

, where x

∈ R

;

• (r, ω): the spherical coordinates of x ∈ R

with pole O;

• Ω: a domain in S

with smooth (n − 2)-dimensional boundary ∂Ω;

• G

= G ∩ {(r, ω) | 0 ≤ a < r < b, ω ∈ Ω}: a layer in R

;

• Γ

= ∂G ∩ {(r, ω) | 0 ≤ a < r < b, ω ∈ ∂Ω}: the lateral surface of the layer G

;

• D

u = u

= ∂u/∂x

, D

u = u

= ∂

u/∂x

∂x

;

• ∇u ≡ u

= (u

, . . . , u

): the gradient of u(x);

• n = n(x) = {ν

, . . . , ν

}: the unit outward normal to ∂G at the point x;

• dΩ: the (n − 1)-dimensional area element of the unit sphere;

• dσ: the (n − 1)-dimensional area element of ∂G;

• ∆: the Laplacian in R

;

• ∆

: the Laplace–Beltrami operator on the unit sphere S

;

• d(x) = dist(x, ∂G \ O);

• Φ(x): any possible extension into G of a boundary function ϕ(x), i.e., Φ(x) = ϕ(x) for x ∈ ∂G;

• A(t): a function defined for t ≥ 0, nonnegative, increasing, continuous at zero, with lim

A(t) = 0.

Definition 1.1. The function A is called Dini continuous at zero if

Td

0

t

A(t) dt < ∞ for some d > 0.

Definition 1.2. The function A is called an α-function, 0 < α < 1, on (0, d] if t

A(t) is decreasing on (0, d], i.e.

(1.1) A(t) ≤ t

τ

A(τ ), 0 < τ ≤ t ≤ d.

In particular, setting t = cτ, c > 1, we have

(1.2) A(cτ ) ≤ c

A(τ ), 0 < τ ≤ c

d.

If an α-function A is Dini continuous at zero, then we say that A is an α-Dini function. In that case we also define the function B(t) =

Tt

0

(A(τ )/τ ) dτ . It is obvious that B is increasing and continuous on [0, d], and B(0) = 0. We integrate the inequality (1.1) over τ from 0 to t:

(1.3) A(t) ≤ αB(t).

Similarly from (1.1) we derive

d

\

δ

(A(t)/t

) dt =

d

\

δ

t

(A(t)/t

) dt ≤ δ

A(δ)

d

\

δ

t

dt ≤ (1 − α)

A(δ)/δ, whence by (1.3),

δ

d

\

δ

(A(t)/t

) dt ≤ (1 − α)

A(δ) (1.4)

≤ α(1 − α)

B(δ), ∀α ∈ (0, 1), 0 < δ < d.

Definition 1.3. The function B is called equivalent to A, written A ∼ B, if there exist positive constants C

and C

such that

C

A(t) ≤ B(t) ≤ C

A(t) for all t ≥ 0.

An equivalence test is known [4, theorem of Sec. 1]: A ∼ B if and only if

(1.5) lim

t→0

A(2t)/A(t) > 1.

In some cases we shall consider functions A such that also

(1.6) sup

0<τ ≤1

A(τ t)/A(τ ) ≤ cA(t), ∀t ∈ (0, d],

with some constant c independent of t. Examples of α-Dini functions A which satisfy (1.5), (1.6) with c = 1 are:

t

, 0 ≤ t < ∞;

t

ln(1/t), t ∈ (0, d], d = min(e

, e

), e

< α < 1.

We will consider the following function spaces:

• C

(G): the Banach space of functions having all the derivatives of order at most l (if l = integer ≥ 0) and of order [l] (if l is noninteger) continuous in G and whose [l]th order partial derivatives are uniformly H¨older continuous with exponent l − [l] in G; |u|

is the norm of the element u ∈ C

(G); if l 6= [l] then

|u|

= X

j=0

sup

G

|D

u| + sup

|α|=[l]

sup

x,y∈G x6=y

|D

u(x) − D

u(y)|

|x − y|

.

• C

(G): the set of functions in C

(G) with compact support in G.

• C

(G): the set of bounded and continuous functions f on G with [f ]

= sup

x,y∈G x6=y

|f (x) − f (y)|

A(|x − y|) < ∞;

equipped with the norm

kf k

= |f |

+ [f ]

, this set is a Banach space. We also define the quantity

[f ]

= sup

y∈G\{x}

|f (x) − f (y)|

A(|x − y|) . It is not difficult to see that if A ∼ B then [f ]

∼ [f ]

.

If k ≥ 1 is an integer then we denote by C

(G) the subspace of C

(G) consisting of functions whose (k−1)th order partial derivatives are uniformly Lipschitz continuous and each kth order derivative belongs to C

(G); it is a Banach space with the norm

kf k

= |f |

+ X

|β|=k

[D

f ]

.

The interpolation inequality (see [8, (10.1)]) will be needed: if the domain has a Lipschitz boundary, then for any ε > 0 there exists a constant c(ε, G) such that for every f ∈ C

(G),

(1.7)

X

|D

f |

≤ ε X

[D

f ]

+ c(ε, G)|f |

.

• L

(G): the Banach space of p-integrable functions u on G (p ≥ 1) with norm u

.

Moreover, λ = λ(Ω) will stand for the smallest positive eigenvalue of the problem

(EVP)

( ∆

ψ + λ(λ + n − 2)ψ = 0, ω ∈ Ω ⊂ S

, ψ(ω) = 0, ω ∈ ∂Ω,

and c(. . .) will be different constants depending only on the quantities ap- pearing in parentheses.

Let T ⊂ ∂G be a nonempty set. Following [5, Sec. 6.2] and [8, Sec. 3]

we shall say that the boundary portion T is of class C

if for each point x

∈ T there are a ball B = B(x

), a one-to-one mapping ψ of B onto a ball B

and a constant K > 0 such that:

(i) B ∩ ∂G ⊂ T, ψ(B ∩ G) ⊂ R

; (ii) ψ(B ∩ ∂G) ⊂ Σ;

(iii) ψ ∈ C

(B), ψ

∈ C

(B

);

(iv) kψk

≤ K, kψ

k

≤ K.

It is not difficult to see that for y = ψ(x) one has

(1.8) K

|y − y

| ≤ |x − x

| ≤ K|y − y

|, ∀x, x

∈ B.

Lemma [8, Sec. 7, (iv)]. Let A be an α-function and f ∈ C

(B), ψ

∈ C

(B

). Then f ◦ ψ

∈ C

(B) and

(1.9) [f ◦ ψ

]

≤ K

[f ]

.

2. Dini estimates of the first derivatives for the generalized Newtonian potential (cf. [5, Ch. 4]). We shall consider the Dirichlet prob- lem for the Poisson equation

(PE)

 

 

 



∆v = G + X

D

F

, x ∈ G, v(x) = 0, x ∈ ∂G.

Let Γ (x − y) be the normalized fundamental solution of Laplace’s equa- tion. The following estimates are known (see e.g. [5, (2.12), (2.14)]):

(2.1)

|Γ (x − y)| = |x − y|

/(n(n − 2)ω

), n ≥ 3,

|D

Γ (x − y)| ≤ |x − y|

/(nω

),

|D

Γ (x − y)| ≤ |x − y|

/ω

,

|D

Γ (x − y)| ≤ C(n, β)|x − y|

. We define the functions

(2.2) z(x) =

\

G

Γ (x − y)G(y) dy, w(x) = D

\

G

Γ (x − y)F

(y) dy, assuming that the functions G(x) and F

(x), j = 1, . . . , n, are integrable on G. The function z is called the Newtonian potential with density func- tion G, and w is called the generalized Newtonian potential with density function div F. We now give estimates for these potentials.

Let B

= B

(x

), B

= B

(x

) be concentric balls in R

and z, w be Newtonian potentials in B

.

Lemma 1. Suppose G ∈ L

(B

), p > n/2, and F

∈ L

(B

), j = 1, . . . , n. Then

(2.3) |z|

≤ c(p)R

ln

(1/(2R)) G

, n = 2,

|z|

≤ c(p, n)R

G

, n ≥ 3,

(2.4) |w|

≤ 2R X

|F

|

, where 1/p + 1/p

= 1.

P r o o f. The estimates follow from inequalities (2.1), H¨older’s inequality and [5, Lemma 4.1].

In the following the D operator is always taken with respect to the x variable.

Lemma 2 [5, Lemmas 4.1, 4.2]. Let ∂G ∈ C

, G ∈ L

(G), p > n, F

∈ C

(G), where A is an α-function Dini continuous at zero. Then for any x ∈ G,

D

z(x) =

\

G

D

Γ (x − y)G(y) dy, (2.5)

D

w(x) =

\

G0

D

Γ (x − y)(F

(y) − F

(x)) dy (2.6)

− F

(x)

\

∂G0

D

Γ (x − y)ν

d

σ

(i = 1, . . . , n); here G

is any domain containing G for which the Gauss divergence theorem holds and F

are extended to vanish outside G.

Lemma 3 (cf. [5, Lemma 4.4]). Let G ∈ L

(B

), p > n, F

∈ C

(B

), where A is an α-function Dini continuous at zero. Then z, w ∈ C

(B

) and

kzk

≤ c(n, p, R, A

(2R)) G

, (2.7)

kwk

≤ c(n, p, α, R, A

(2R), B(2R)) X

kF

k

. (2.8)

P r o o f. Let x, x ∈ B

and G = B

. By formulas (2.5), (2.6), taking into account (2.1) and H¨older’s inequality and setting |x − y| = t, y − x = tω, dy = t

dt dΩ, we have

|D

z| ≤ (nω

)

\

B2

|x − y|

|G(y)| dy (2.9)

≤ (nω

)

G

n

B2

|x − y|

dy o

= p − 1

p − n (2R)

G

,

|D

w(x)| ≤ (nω

)

R

X

|F

(x)|

\

∂B2

d

σ

(2.10)

+ ω

X

[F

]

\

B2

A(x − y)

|x − y|

dy

≤ 2

X

|F

(x)| + n X

[F

]

0

A(t) t dt

≤ c(n)B(2R) X

(|F

(x)| + [F

]

).

Taking into account (2.5) we obtain by subtraction

|D

z(x) − D

z(x)| ≤

\

B2

|D

Γ (x − y) − D

Γ (x − y)| · |G(y)| dy.

We set δ = |x − x|, ξ =

(x − x) and write B

= B

(ξ) ∪ {B

\ B

(ξ)}. Then (2.11)

\

Bδ(ξ)

|D

Γ (x − y) − D

Γ (x − y)| · |G(y)| dy

≤

\

B_δ(ξ)

|D

Γ (x − y)| · |G(y)| dy +

\

B_δ(ξ)

|D

Γ (x − y)| · |G(y)| dy

≤ (nω

)

n

Bδ(ξ)

|x − y|

|G(y)| dy +

\

Bδ(ξ)

|x − y|

|G(y)| dy o

≤ 2(nω

)

\

B3δ/2(x)

|x − y|

|G(y)| dy

≤ 2(nω

)

G



B3δ/2(x)

|x − y|

dy 

≤ 2(nω

)

G

 3δ 2



{n + (1 − n)p

}

≤ 2(nω

)

(2R)

{n + (1 − n)p

}

· A(|x − x|)

A(2R) G

, 1/p + 1/p

= 1 (here we take into account that δ

≤ (2R)

A(δ)/A(2R) for all α > 0 by (1.1), since δ ≤ 2R). Similarly,

(2.12)

\

B2\B_δ(ξ)

|D

Γ (x − y) − D

Γ (x − y)| · |G(y)| dy

≤ |x − x|

\

B2\Bδ(ξ)

|DD

Γ (e x − y)| · |G(y)| dy

(for some e x between x and x)

≤ δω

\

|y−ξ|≥δ

|y − e x|

|G(y)| dy

≤ 2

δω

\

|y−ξ|≥δ

|y − ξ|

|G(y)| dy (since |y − ξ| ≤ 2|y − e x|)

≤ 2

δω

G



|y−ξ|≥δ

|y − ξ|

dy 

≤ 2

ω

(p − 1)

δ

G

≤ 2

ω

(p − 1)

(2R)

A(|x − x|)

A(2R) G

. From (2.11) and (2.12), taking into account (1.3), we obtain (2.13) |D

z(x) − D

z(x)|

≤ c(n, p, R)A

(2R) G

A(|x − x|)

≤ c(n, p, R)A

(2R) G

B(|x − x|), ∀x, x ∈ B

. The first of the required estimates, (2.7), follows from (2.3) and (2.13).

Now we derive the estimate (2.8).

By (2.6) for all x, x ∈ B

we have (2.14) D

w(x) − D

w(x)

= X

(F

(x)J

+ (F

(x) − F

(x))J

) + J

+ J

+ X

(F

(x) − F

(x))J

+ J

, where

J

=

\

∂B2

(D

Γ (x − y) − D

Γ (x − y))ν

(y) d

σ,

J

=

\

∂B2

D

Γ (x − y)ν

(y) d

σ,

J

=

\

Bδ(ξ)

D

Γ (x − y)(F

(x) − F

(y)) dy,

J

=

\

Bδ(ξ)

D

Γ (x − y)(F

(y) − F

(x)) dy,

J

=

\

B2\Bδ(ξ)

D

Γ (x − y) dy,

J

=

\

B2\Bδ(ξ)

(D

Γ (x − y) − D

Γ (x − y))(F

(x) − F

(y)) dy.

(Here we set again δ = |x − x|, ξ =

(x − x) and write B

= B

(ξ) ∪ {B

\ B

(ξ)}.)

We estimate these integrals by analogy with [5, pp. 58–59]:

|J

| ≤ |x − x|

\

∂B2

|DD

Γ (e x − y)| d

σ

(for some point e x between x and x)

≤ |x − x|nω

\

∂B2

|e x − y|

d

σ

≤ n

2

|x − x|R

(since |e x − y| ≥ R for y ∈ ∂B

)

≤ n

2

A(|x − x|)R

δ/A(δ)

≤ n

2

A(|x − x|)/A(2R)

(since δ = |x − x| ≤ 2R and δ/A(δ) ≤ 2R/A(2R) by (1.1))

≤ n

2

αB(δ)/A(2R) (by (1.3)).

Next,

|J

| ≤ 2

,

|J

| ≤ ω

[F

]

\

Bδ(ξ)

|x − y|

A(|x − y|) dy

≤ ω

[F

]

\

B3δ/2(x)

|x − y|

A(|x − y|) dy

= n[F

]

\

0

t

A(t) dt

≤ (3/2)

n[F

]

B(δ) (by (1.2)).

By analogy with the estimate for J

we obtain

|J

| ≤ (3/2)

n[F

]

B(δ), |J

| ≤ 2

(see [5, p. 59]), and

|J

| ≤ |x − x|

\

B2\Bδ(ξ)

|DD

Γ (e x − y)| · |F

(x) − F

(y)| dy

(for some e x between x and x)

≤ |x − x|c(n)

\

|y−ξ|≥δ

|F

(x) − F

(y)| · |e x − y|

dy

≤ c(n)δ[F

]

\

|y−ξ|≥δ

A(|x − y|)|e x − y|

dy

≤ 2

c(n)δ[F

]

\

|y−ξ|≥δ

A((3/2)|ξ − y|)|ξ − y|

dy

(since |x − y| ≤ (3/2)|ξ − y| ≤ 3|x − e y|)

≤ 2

nω

c(n)(3/2)

δ[F

]

R\

δ

t

A(t) dt

(since A((3/2)t) ≤ (3/2)

A(t) by (1.2))

≤ α

1 − α (3/2)

nω

2

c(n)[F

]

B(δ) (by (1.4)).

Now from (2.14) and the above estimates we obtain (2.15) |D

w(x) − D

w(x)|

≤ c(n, α) X

(|F

(x)|A

(2R) + [F

]

+ [F

]

)B(|x − x|),

∀x, x ∈ B

. Finally, from (2.10) and (2.15) it follows that w ∈ C

(B

) and the estimate (2.8) holds. Lemma 3 is proved.

Theorem 1. Let v be a generalized solution of equation (PE) in B

with G ∈ L

(B

), F

∈ C

(B

), where A is an α-function satisfy- ing the Dini condition at zero, and let v = 0 on B

∩ Σ. Then v ∈ C

(B

) and

kvk

1

≤ c 

|v|

2

+ G

2

+ X

kF

k

2

 , where c = c(n, α, R, A

(2R), B(2R)).

Theorem 1 follows from (2.7), (2.8), representation of solutions of (PE) by means of the fundamental solution and by the same argument as in [5, 4.4–4.5] (see also [5, 8.11]).

3. Dini continuity near a smooth portion of the boundary Theorem 2 (cf. [5, Corollary 8.36]). Let A be an α-Dini function (0 <

α < 1) satisfying the condition (1.5). Let T ⊂ ∂G be of class C

. Let u ∈ W

(G) be a weak solution of the problem (DL) with ϕ ∈ C

(∂G).

Suppose the coefficients of the equation in (DL) satisfy the conditions

a

(x)ξ

ξ

≥ ν|ξ|

, ∀x ∈ G, ξ ∈ R

, a

, a

, f

∈ C

(G) (i, j = 1, . . . , n),

b

, c ∈ L

(G), g ∈ L

(G).

Then u ∈ C

(G ∪ T ) and for every G

⋐ G ∪ T , kuk

≤ c(n, T, ν, k, d

) 

|u|

+ g

(3.1)

+ X

i=1

kf

k

+ kϕk

 ,

where d

= dist(G

, ∂G \ T ) and k = max

i,j=1,...,n

{ka

, a

k

, |b

, c|

}.

P r o o f. We use the perturbation method. We freeze the leading coeffi- cients a

(x) at x

∈ G∪T by setting, without loss of generality, a

(x

) = δ

(see [5, Lemma 6.1]), and rewrite the equation of (DL) in the form (PE) for the function v(x) = u(x) − ϕ(x) with

G(x) = g(x) − b

(x)(D

v + D

ϕ) − c(x)(v(x) + ϕ(x)), (3.2)

F

(x) = (a

(x

) − a

(x))D

v − a

(x)D

ϕ (3.3)

− a

(x)(v(x) + ϕ(x)) + f

(x) (i = 1, . . . , n).

It is not difficult to observe that the conditions on the coefficients of the equation and on T are invariant under maps of class C

. Therefore after a preliminary rectification of T by means of a diffeomorphism ψ ∈ C

it is sufficient to prove the theorem in the case T ⊂ Σ. This is carried out using Theorem 1 in a standard way (see [5, Chs. 6, 8]). In this connection we use the following estimates for the functions (3.2), (3.3):

(3.4) G

2

≤ g

2

+ k  X

i=1

|D

v|

2

+ |v|

2

+ X

|D

ϕ|

2

+ |ϕ|

2



≤ g

2

+ k  ε

X

[D

v]

2

+ c

|v|

2

+ |ϕ|

2



(by (1.7)),

(3.5) X

kF

k

2

≤ nkA(2R)k∇vk

2

+ k X

|D

v|

2

+ c(k)(|v|

2

+ kϕk

2

) + X

kf

k

2

. Taking into account once more the inequality (1.7) and the condition (1.5) that ensures the equivalence [ ]

∼ [ ]

, from (3.4)–(3.5) we finally obtain (3.6) G

2

+ X

kF

k

2

≤ k(ε + nA(2R))kvk

2

+ c

(k)(|v|

2

+ kϕk

2

) +

X

kf

k

2

+ g

2

for all ε > 0.

Since A is continuous, choosing ε, R > 0 sufficiently small we obtain the desired assertion and the estimate (3.1) in a standard way from (2.16) and (3.6).

4. Dini continuity near the conical point. We consider the problem (DL) under the following assumptions:

(i) ∂G is a Dini–Lyapunov surface with conical point O;

(ii) the uniform ellipticity holds:

νξ

≤ a

(x)ξ

ξ

≤ µξ

, ∀x ∈ G, ξ ∈ R

, where ν, µ = const > 0 and a

(0) = δ

(i, j = 1, . . . , n);

(iii) a

, a

∈ C

(G) (i, j = 1, . . . , n) where A is an α-Dini function on (0, d], α ∈ (0, 1), satisfying the conditions (1.5)–(1.6) and also

(4.1) sup

0<̺≤1

̺

/A(̺) ≤ const,

|x|  X

(b

(x))



+ |x|

|c(x)| ≤ A(|x|);

(iv) g ∈ L

(G), ϕ ∈ C

(∂G), f

∈ C

(G), j = 1, . . . , n;

(v)

\

G

r

H

(r)g

(x) dx < ∞,

\

G

r

H

(r)  X

j=1

|f

|

+ |∇Φ|

+ r

Φ



dx < ∞,

where H is a continuous increasing function satisfying the Dini condition at

t = 0.

Theorem 3. Let u be a generalized solution of (DL) and suppose that assumptions (i)–(v) are satisfied. Then there exist d > 0 and a constant c > 0 independent of u and depending only on parameters and norms of the given functions appearing in assumptions (i)–(v), such that

|u(x)| ≤ c|x|A(|x|) 

g

+ X

kf

k

+ kϕk

(4.2)

+ n

G

 r

H

(r)g

(x) + r

H

(r)

× X

i=1

|f

(x)|

+ r

H

(r)|∇Φ|

+ |u|

+ |∇u|



dx o



, ∀x ∈ G

,

|∇u(x)| ≤ cA(|x|) 

g

+ X

i=1

kf

k

+ kϕk

(4.3)

+ n

G

 r

H

(r)g

(x) + r

H

(r)

× X

i=1

|f

(x)|

+ r

H

(r)|∇Φ|

+ |u|

+ |∇u|



dx o



, ∀x ∈ G

.

P r o o f. We use Kondrat’ev’s method of layers: we move away from the conical point of ̺ > 0 and work in G

; after the change of variables x = ̺x

the layer G

takes the position of a fixed domain G

with smooth boundary.

1

. We consider a solution u in the domain G

with some positive d ≪ 1;

then u is a weak solution in G

of the problem

(DL)

 

 

 



 

 

 

∂

∂x

(a

(x)u

+ a

(x)u) + b

(x)u

+ c(x)u

= g(x) + ∂f

∂x

, x ∈ G

, u(x) = ϕ(x), x ∈ Γ

⊂ ∂G

.

We make the change of variables x = ̺x

and set v(x

) = ̺

A

(̺)u(̺x

),

̺ ∈ (0, d), 0 < d ≪ 1. Then v satisfies in G

the problem

 

 

 

 

 

 

 



∂

∂x

(a

(̺x

)v

+ ̺a

(̺x

)v) + ̺b

(̺x

)v

+ ̺

c(̺x

)v

= A

(̺) X

∂f

(̺x

)

∂x

+ ̺A

(̺)g(̺x

), x

∈ G

, v(x

) = ̺

A

(̺)ϕ(̺x

), x

∈ Γ

.

To solve this problem we use Theorem 2. We check its assumptions. Since under assumption (ii), A is increasing, ̺ ∈ (0, d) and 0 < d ≪ 1, from the inequality ̺

|x − y| ≥ |x − y| for ̺ ∈ (0, d) it follows that

A(|x

− y

|) = A(̺

|x − y|) ≥ A(|x − y|) and by (iii) we have

X

i,j

ka

(̺ ·)k

1/4

+ ̺ X

i

ka

(̺ ·)k

1/4

≤ X

i,j

ka

k

̺/4

+ d X

i

ka

k

̺/4

< ∞.

Further, let Φ be a regularity preserving extension of the boundary func- tion ϕ into a domain G

for ε > 0 (such an extension exists; see e.g. [5, Lemma 6.38]).

Since ϕ ∈ C

(∂G) we have kΦk

̺/4

≤ c(G)kϕk

̺/4

≤ const.

By definition of the norm in C

we obtain

(4.4) sup

x,y∈G^2̺

̺/4

x6=y

|∇Φ(x) − ∇Φ(y)|

A(|x − y|) ≤ kΦk

̺/4

≤ c(G)kϕk

̺/4

. Now we show that by (v) and the smoothness of ϕ,

(4.5) |ϕ(x)| ≤ c|x|A(|x|), |∇Φ(x)| ≤ cA(|x|), ∀x ∈ G

. Indeed, from

ϕ(x) − ϕ(0) =

1

\

0

d

dτ Φ(τ x) dτ = x

\

0

∂Φ(τ x)

∂(τ x

) dτ by H¨older’s inequality we have

(4.6) |ϕ(x) − ϕ(0)| ≤ r|∇Φ|.

From (iv) it follows that (4.7)

\

G^̺₀

(r

|∇Φ|

+ r

|ϕ|

) dx

=

\

G^̺₀

(r

H

(r)|∇Φ|

+ r

H

(r)|ϕ|

)(r

H(r)) dx

≤ const ̺

H(̺).

Since |ϕ(0)| ≤ |ϕ(x)| + |ϕ(x) − ϕ(0)|, by (4.6) we obtain

|ϕ(0)| ≤ |ϕ(x)| + r|∇Φ|.

Squaring both sides, multiplying by r

and integrating over G

we obtain (4.8) ϕ

(0)

\

G^̺₀

r

dx ≤ 2

\

G^̺₀

(r

ϕ

(x) + r

|∇Φ|

) dx < ∞

by (4.7). Since

\

G^̺₀

r

dx = mes Ω

̺

\

0

dr r = ∞,

the assumption ϕ(0) 6= 0 contradicts (4.8). Thus ϕ(0) = 0. Then from (4.4) we have

|∇Φ(x) − ∇Φ(y)| ≤ const A(|x − y|)kϕk

̺/4

, ∀x, y ∈ G

,

|∇Φ(y)| ≤ |∇Φ(x) − ∇Φ(y)| + |∇Φ(x)|

≤ cA(|x − y|)kϕk

̺/4

+ |∇Φ(x)|.

Hence considering y to be fixed in G

and x variable, we get

|∇Φ(y)|

\

G^2̺_̺/4

r

dx ≤ 2c

kϕk

̺/4

\

G^2̺_̺/4

r

A

(|x − y|) dx

+ 2

\

G^2̺_̺/4

r

|∇Φ(x)|

dx

or by (4.7),

̺

|∇Φ(y)|

≤ c(mes Ω, k

)(̺

A

(̺) + ̺

H(̺)), ∀y ∈ G

.

Hence the assumption (4.1) yields the second inequality of (4.5). Now the

first inequality of (4.5) follows from (4.6) and ϕ(0) = 0. Thus (4.5) is proved.

Now we obtain

(4.9) ̺

A

(̺)kϕ(̺ ·)k

1/4

≤ c̺

A

(̺)kΦ(̺ ·)k

1/4

= c̺

A

(̺)

 sup

x^′∈G²

1/4

|Φ(̺x

)| + sup

x^′∈G²

1/4

|∇

Φ(̺x

)|

+ sup

x^′,y^′∈G²_1/4 x^′6=y^′

|∇

Φ(̺x

) − ∇

Φ(̺y

)|

A(|x

− y

|)



≤ c

+ cA

(̺) sup

x,y∈G^2̺_̺/4

|∇Φ(x) − ∇Φ(y)|

A(̺

|x − y|)

= c

+ c[∇Φ]

̺/4

A

(̺) sup

0<t<4̺

A(t) A(̺

t)

≤ const, ∀̺ ∈ (0, d), by (4.5), since by (1.6),

sup

0<t<4̺

A(t)

A(̺

t) = sup

0<τ <4

A(τ ̺)

A(τ ) ≤ cA(̺).

In the same way we have (4.10) A

(̺)kf

k

1/4

= A

(̺)



|f

|

̺/4

+ sup

x,y∈G^2̺_̺/4 x6=y

|f

(x) − f

(y)|

A(̺

|x − y|)

 .

Since f

∈ C

(G), we get

|f

(x) − f

(y)| ≤ ec

A(|x − y|), ∀x, y ∈ G

, (4.11)

\

G^̺₀

r

|f

(x)|

dx =

\

G^̺₀

(r

H

(r)|f

(x)|

)(H(r)r

) dx (4.12)

≤ const ̺

H(̺) by (v). Now fix y in G

. Then

|f

(y)| ≤ |f

(x)| + |f

(x) − f

(y)| ≤ |f

(x)| + ec

A(|x − y|).

Hence

|f

(y)|

\

G^2̺_̺/4

r

dx ≤ 2

\

G^2̺_̺/4

r

|f

(x)|

dx + 2ec

\

G^2̺_̺/4

r

A

(|x − y|) dx.

Calculations and (4.12) give

̺

|f

(y)|

≤ c(ec

, k

, mes Ω)(̺

A

(̺) + ̺

H(̺)), ∀y ∈ G

. Hence by the assumption (4.1) it follows that

(4.13) |f

(x)| ≤ c

A(̺), ∀x ∈ G

, j = 1, . . . , n.

Further, in the same way as in the proof of (4.9), sup

x,y∈G^2̺_̺/4 x6=y

|f

(x) − f

(y)|

A(̺

|x − y|) ≤ [f

]

̺/4

sup

0<t<4̺

A(t) A(̺

t) (4.14)

≤ cA(̺)[f

]

̺/4

. Now from (4.10), (4.13) and (4.14) we obtain

(4.15) A

(̺)

X

|f

|

1/4

≤ const.

It remains to verify the finiteness of ̺A(̺)

g(̺x

)

1/4

. We have

̺A

(̺) 

G²

1/4

|g(̺x

)|

dx



= ̺

A

(̺) 

G^2̺_̺/4

|g(x)|

dx 

≤ d

A

(d) 

G^2̺

̺/4

|g(x)|

dx 

≤ const, ∀̺ ∈ (0, d),

by the condition (1.1). Thus the conditions of Theorem 2 are satisfied.

By this theorem we have (4.16) kvk

1/2

≤ c{n, ν, G, max

i,j=1,...,n

(ka

(̺ ·)k

1/4

, ̺ka

(̺ ·)k

1/4

), A(2̺)}

× 

|v|

1/4

+ ̺

A

(̺)kϕ(̺ ·)k

1/4

+ ̺A

(̺) g(̺ ·)

1/4

+ A

(̺) X

kf

(̺ ·)k

1/4

 , ∀̺ ∈ (0, d).

2

. To estimate |v|

1/4

we use the local estimate at the boundary of the maximum of the modulus of a solution [5, Theorem 8.25]. We check the assumptions of that theorem. To this end, set

z(x

) = v(x

) − ̺

A

(̺)Φ(̺x

) and write the problem for the function z:

 

 

 



 

 

 

∂

∂x

(a

(̺x

)z

j

+ ̺a

(̺x

)z) + ̺b

(̺x

)z

i

+ ̺

c(̺x

)z

= G(x

) + ∂F

(x

)

∂x

, x

∈ G

, z(x

) = 0, x

∈ Γ

,

where

G(x

) ≡ ̺A

(̺)g(̺x

) − A

(̺)b

(̺x

)Φ

i

(̺x

) (4.17)

− ̺A

(̺)c(̺x

)Φ(̺x

),

F

(x

) ≡ A

(̺)f

(̺x

) − ̺

A

(̺)a

(̺x

)Φ

(̺x

) (4.18)

− A

(̺)a

(̺x

)Φ(̺x

) (i = 1, . . . , n).

First we verify the smoothness of the coefficients (see the remark at the end of [5, 8.10]). Let q > n. We have

\

G²_1/4

|̺a

(̺x

)|

dx

= ̺

\

G^2̺

̺/4

|a

(x)|

dx (4.19)

≤ c

(G)d

ka

k

, ∀̺ ∈ (0, d).

By (iii) we also obtain

\

G²

1/4

|̺b

(̺x

)|

dx

= ̺

\

G^2̺_̺/4

|b

(x)|

dx ≤ 4

̺

\

G^2̺_̺/4

|rb

(x)|

dx (4.20)

≤ 4

̺

\

G^2̺_̺/4

A

(r) dx ≤ 2

\

G^2̺_̺/4

r

A

(r) dx

= 2

mes Ω

2̺

\

̺/4

A

(r) r dr

≤ 2

mes Ω · A

(2d)

2d\

0

A(r) r dr,

\

G²_1/4

|̺

c(̺x

)|

dx

= ̺

\

G^2̺_̺/4

|c(x)|

dx

(4.21)

≤ 4

̺

\

G^2̺

̺/4

|r

c(x)|

dx

≤ 2

\

G^2̺

̺/4

r

A

(r) dx

≤ 2

mes Ω · A

(2d)

2d

\

0

A(r) r dr, for q > n and all ̺ ∈ (0, d).

In the same way from (4.17) we get (4.22) ̺A

(̺) G(x

)

1/4

= ̺A

(̺) 

G^2̺_̺/4

̺

n

|g(x)|

+  X

i=1

|b

(x)| 

|∇Φ|

+ |c(x)|

|Φ(x)|

o dx 

. By (iv) setting q = n/(1 − α) > n and applying H¨older’s inequality we obtain

(4.23) ̺A

(̺) 

G^2̺_̺/4

̺

|g(x)|

dx 

≤ c̺

A

(̺) 

G^2̺

̺/4

̺

|g(x)|

dx 

≤ c̺

A

(̺) g

̺/4

(mes Ω ln 8)

≤ c(d, α, q, mes Ω, A(d)) g

̺/4

, since by (1.1), ̺

A

(̺) ≤ d

A

(d) for all ̺ ∈ (0, d). Similarly, (4.24) ̺A

(̺) 

G^2̺

̺/4

r

× n X

i=1

|b

(x)| 

|∇Φ|

+ |c(x)|

|Φ(x)|

o dx 

≤ c(mes Ω)

kϕk

̺/4

A

(̺)

2̺

\

̺/4

A(r)

r dr.

From (4.22)–(4.24) we obtain (4.25) G(̺ ·)

1/4

≤ const



q, α, d, mes Ω, A(d),

2̺

\

̺/4

A(r) r dr



× ( g

̺/4

+ kϕk

̺/4

), q = n/(1 − α) > n.

Finally, in the same way from (4.18) we have (4.26)

X

\

G²

1/4

|F

(x

)|

dx

≤ c 

q, G, max

j=1,...,n

n X

i=1

ka

k

, X

i=1

ka

k

o

×

\

G^2̺

̺/4

r

A

(r)  X

i=1

|f

(x)|

+ |∇Φ|

+ |Φ|

 dx.

It follows from (4.5) as ̺ → +0 that |∇Φ(0)| = 0. Therefore

|∇Φ(x)| = |∇Φ(x) − ∇Φ(0)| ≤ A(|x|)kϕk

̺/4

, ∀x ∈ G

, and hence

|Φ(x)| ≤ r|∇Φ| ≤ |x|A(|x|)kϕk

̺/4

, ∀x ∈ G

.

Similarly it follows from (4.13) as ̺ → +0 that f

(0) = 0 for j = 1, . . . , n.

Therefore we have for all x ∈ G

,

|f

(x)| = |f

(x) − f

(0)| ≤ A(r)[f

]

̺/4

.

Consequently, estimating the right side of (4.26) and taking into account the inequalities obtained, we have

X

F

1/4

≤ c 

q, G, max

j=1,...,n

n X

i=1

ka

k

, X

ka

k

o

(4.27)

× mes Ω ·  X

i=1

kf

k

̺/4

+ kϕk

̺/4



.

So all conditions of [5, Theorem 8.25] are satisfied. By this theorem we get

(4.28) sup

x^′∈G¹_1/2

|z(x

)|

≤ c  z

1/4

+ G

+ X

F

1/4



≤ c  z

1/4

+ g

̺/4

+ X

kf

k

̺/4

+ kϕk

̺/4

 .

Setting w(x) = u(x) − ϕ(x) we have for w(x) the problem

(DL)

 

 

 



 

 

 

∂

∂x

(a

(x)w

+ a

(x)w) + b

(x)w

+ c(x)w

= G(x) + ∂F

∂x

, x ∈ G

, w(x) = 0, x ∈ Γ

⊂ ∂G

, where

G(x) = g(x) − b

(x)Φ

− c(x)Φ(x), F

(x) = f

(x) − a

(x)Φ

− a

(x)Φ(x).

Moreover, by assumptions (i), (ii),

|a

(x) − δ

| ≤ ka

k

A(|x|), x ∈ G.

By [6, Theorem 1] there is a constant c > 0 independent of w, G, F

such that

\

G^̺₀

r

|∇w|

dx ≤ c̺

\

G^2d₀

n |w(x)|

+ |∇w|

+ G

(x) + X

|F

(x)|

(4.29)

+ r

H

(r)G

(x) + r

× H

(r) X

|F

(x)|

o

dx, ∀̺ ∈ (0, d).

Our assumptions guarantee that the integral on the right side is finite. Since z(x

) = ̺

A

(̺)w(̺x

) we obtain from (4.29),

\

G²_1/4

|∇

z|

dx

≤ 2

̺

A

(̺)

\

G^2̺_̺/4

r

|∇w|

dx (4.30)

≤ c̺

A

(̺)

\

G

n

|w|

+ |∇w|

+ G

(x)

+ X

|F

(x)|

+ r

H

(r)G

(x)

+ r

H

(r) X

|F

(x)|

o dx.

By assumptions (i)–(iv) we have

(4.31)

|G(x)|

≤ c{|g|

+ A

(r)(r

|∇Φ|

+ r

Φ

)}, X

i=1

|F

(x)|

≤ c n X

i=1

|f

(x)|

+ max

i,j=1,...,n

(ka

k

, ka

k

)(|∇Φ|

+ Φ

) o . Applying now the Friedrichs inequality and taking into account (4.1), we obtain from (4.30), (4.31),

z

1/4

≤ c

∇

z

(4.32)

≤ c̺

A

(̺)

\

G

n

|w|

+ |∇w|

+ g

(x)

+ X

|f

(x)|

+ |∇Φ|

+ Φ

+ r

H

(r)g

(x)

+ r

H

(r) X

|f

(x)|

+ r

H

(r)|∇Φ|

+ r

A

(r)|∇Φ|

o dx

≤ const n

g

+ X

kf

k

+ kϕk

+

\

G

 |w|

+ |∇w|

+ r

H

(r)g

(x)

+ r

H

(r) X

|f

(x)|

+ r

H

(r)|∇Φ|



dx o

by assumptions (iii)–(v). By the definition of z(x

), inequalities (4.28), (4.32) and assumptions (i)–(v) we finally obtain

|v|

1/4

≤ |z|

1/4

+ ̺

A

(̺)|ϕ(̺ ·)|

(4.33)

≤ c 

g

+ X

i=1

kf

k

+ kϕk

+ n

G



|w|

+ |∇w|

+ r

H

(r)g

(x)

+ r

H

(r) X

|f

(x)|

+ r

H

(r)|∇Φ|



dx o

 .

3

. Returning to the variables x, u(x), we now obtain from inequalities (4.16), (4.33),

(4.34) ̺

A

(̺) sup

x∈G^̺_̺/2

|u(x)| + A

(̺) sup

x∈G^̺_̺/2

|∇u(x)|

+ sup

x,y∈G^̺_̺/2 x6=y

|∇u(x) − ∇u(y)|

A(̺)B(|x − y|)

≤ c 

g

+ X

kf

k

+ kϕk

+ n

G



|u|

+ |∇u|

+ r

H

(r)g

(x)

+ r

H

(r) X

|f

(x)|

+ r

H

(r)|∇Φ|



dx o

 .

Setting |x| = 2̺/3 we deduce from (4.34) the validity of (4.2), (4.3). This completes the proof of Theorem 3.

Remark . As an example of A that satisfies all the conditions of Theo- rem 3, besides the function r

, one may take A(r) = r

ln(1/r), provided λ ≥ 1 + α. In the case of A(r) = r

the result of [1] follows from Theorem 3 for a single equation and the estimate (4.2) coincides with [6, (10)].

5. Global regularity and solvability

Theorem 4. Let A be an α-Dini function (0 < α < 1) that satisfies

the conditions (1.5), (1.6), (4.1). Let G \ {O} be a domain of class C

, and