**On the solution of a class of** **partial differential equations**

**Alik M. Najafov**

^{1,2}

### and **Aygun T. Orujova**

^{B}

^{2}

1Azerbaijan University of Architecture and Construction, AZ 1141, Baku, Azerbaijan

2Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 9 B. Vahabzadeh St., Baku AZ 1141, Azerbaijan

Received 14 February 2017, appeared 22 May 2017 Communicated by Maria Alessandra Ragusa

**Abstract.**In the paper we study the solution and smoothness of the solution of one class
of partial differential equations of higher order in bounded domainG⊂**R**^{n} satisfying
the flexible*λ-horn condition.*

**Keywords:**flexible*λ-horn, generalized derivatives,generalized solution, smoothness of*
solution.

**2010 Mathematics Subject Classification:** 26A33, 46E30, 30H25, 35Q35.

**1** **Introduction**

In the paper [7], Besov generalized spaces of the form

n

\

i=0

L^{<}_{p}^{l}^{i}^{>}

i,θi (G), (1.1)

1 ≤ p^{i} ≤ _{∞, 1} ≤ *θ*^{i} ≤ _{∞,} (i = 0, 1, . . . ,n), l^{i} = (l_{1}^{i}, . . . ,l_{n}^{i}), l^{0}_{j} ≥ 0, l_{i}^{i} > 0, l^{i}_{j} ≥ 0, (j 6= i =
1, 2, . . . ,n), are introduced and studied. In the paper [13], the generalized spaces of Besov–

Morrey type

n

\

i=0

L^{<}_{p}^{l}^{i}^{>}

i,θ_{i},a,κ,τ(G,*λ*), (1.2)

with the finite norm

kfkTn
i=0L^{<}^{li}^{>}

pi,θi,a,κ,τ(G,λ) =

### ∑

n i=0kfk

L^{<}^{li}^{>}

pi,θi,a,κ,τ(G,λ), (1.3)

kfk

L^{<}^{li}^{>}

pi,θi,a,κ,τ(G,λ) =

Z _{h}_{0}

0

∆^{m}^{i} h* ^{λ}*;G,

*λ*D

^{k}

^{i}f

p^{i},a,κ,τ

h^{(}^{λ,l}^{i}^{−}^{k}^{i}^{)}

*θ*^{i}

dh h

1
*θ*i

, (1.4)

BCorresponding author. Emails: aygun.orucova@imm.az (Aygun T. Orujova), aliknajafov@gmail.com (Alik M.

Najafov).

kfk_{p}i,a,κ,τ;G=kfk_{L}

pi,a,κ,τ(G)=sup

x∈G

(
Z _{∞}

0

"

[t]^{−}

(κ,a) pi

1 kfk_{p}i,G_{t}κ(x)

#*τ*

dt t

)_{τ}^{1}

, (1.5)

where 1 ≤ p^{i} < _{∞}, 1 ≤ *θ*^{i} ≤ _{∞}, m^{i} ∈ _{N}^{n}, k^{i} ∈ _{N}^{n}_{0}, a ∈ [0, 1], κ ∈ (0,∞)^{n}, *τ* ∈ [1,∞],
l^{i} = (l_{1}^{i}, . . . ,l^{i}_{n}), l^{0}_{j} ≥ 0, l^{i}_{i} > 0, l^{i}_{j} ≥ 0, (j 6= i = 1, 2, . . . ,n) are introduced, differential and
difference-differential properties of functions from there spaces, determined inn-dimensional
domains and satisfying the flexible horn condition are studied. In the case l^{0} = (0, . . . , 0)_{,}
l^{i} = (0, . . . ,l_{i}, . . . , 0), p^{i} = p,*θ*^{i} =*θ* space (1.1) coincides with the spaceB^{l}_{p,θ}(G)studied in [2],
the space (1.2) coincides with the space B^{l}_{p,θ,a,}_{κ}_{,τ}(G) studied in [9], while in the case a = 0,
*τ*= _{∞}it coincides with space (1.1). Note that consideration of such a space enables to study
higher order differential equations of general form. In other words, the obtained imbedding
theorems in the form of Sobolev type inequality in spaces (1.1) and (1.2) enable to estimate
higher order generalized derivatives than in the case of spacesB^{l}_{p,θ}(G)_{and}B^{l}_{p,θ,a,}

κ,τ(G)_{.}
**Example 1.1.** Let us consider an equation of the form

u^{(}_{x}^{5}2^{)}y^{3}+u^{(}_{x}^{4}2^{)}y^{2}+u^{(}_{x}^{3}2^{)}y^{1} +u^{(}_{xy}^{2}^{)}+u^{(}_{x}^{1}^{)}+u= f(x), (1.6)
in our case the solution of this equation is sought in the spaceL^{(}_{2}^{0,0}^{)}^{T}L^{(}_{2}^{3,1}^{)}^{T}L^{(}_{2}^{1,4}^{)}. One can
look for the solution of equation (1.6) in the space W_{2}^{(}^{6,3}^{)} (B^{(}_{2,2}^{6,3}^{)}), but then this solution will
require additional derivatives, in other words, in our case the solution belongs to a wider
class.

In this paper we study the existence, uniqueness and smoothness of one class of higher or- der partial differential equations. Earlier, a problem of smoothness of another kind equations was studied in [1,3–6,8,10–12].

Note that in this paper, as in the papers [9,12], unlike the previous papers for |*α*| = _{l}^{i}_{,}
(i=1, 2, . . . ,n) f*α* belongs to a wider class. Furthermore, as in the papers [5,6,8,10–12,14,15]

here the coefficients do not require smoothness.

**2** **Main results**

At first we give two theorems proved in the paper [13].

**Theorem 2.1** ([13]). Let the open set G ⊂ _{R}^{n} satisfy the flexible *λ-horn condition [2],* *λ* =
(*λ*_{1}, . . . ,*λ*_{n}), *λ*_{j} > 0, (j = 1, 2, . . . ,n), 1 ≤ p^{i} ≤ p ≤ _{∞,} 1 ≤ *θ*^{i} ≤ _{∞,} (i=0, 1, 2, . . . ,n);
*ν*= (*ν*_{1},*ν*_{2}, . . . ,*ν*_{n})_{,} *ν*_{j} ≥_{0}be entire(j=1, 2, . . . ,n)_{;}

1) *ν*_{j} ≥l^{0}_{j} (j=1, 2, . . . ,n);

2) *ν*_{j} ≥l^{i}_{j} (j6=i,j=1, 2, . . . ,n)_{,} *ν*_{i} <l_{i}^{i} (j=i, i=1, 2, . . . ,n);
1≤*τ*_{1} ≤*τ*_{2} ≤_{∞;}κ= cκ^{,} ^{1}_{c} =max_{j}=1,...,nκj

*λ*j; f ∈ ^{T}^{n}_{i}_{=}_{0}L^{<}^{l}^{i}^{>}

p^{i},θ^{i}a,κ,τ(G,*λ*)and
*µ*^{i} =

### ∑

n j=1

l^{i}_{j}*λ*_{j}−*ν*_{j}*λ*_{j}− *λ*_{j}−κja_{j}
1

p^{i} − ^{1}
p

>0 (i=1, . . . ,n).
Then D* ^{ν}* :

^{T}

^{n}

_{i}

_{=}

_{0}L

^{<}

^{l}

^{i}

^{>}

p^{i},θ^{i},a,κ,τ1(G,*λ*) ,→ L_{p,b,}_{κ}_{,τ}_{2}(G). Precisely, for f ∈ ^{T}^{n}_{i}_{=}_{0}L^{<}_{p}_{i}^{l}_{,θ}^{i}^{>}_{i}_{a,}

κ,τ1(G,*λ*)in the
domain G there exists the generalized derivative D* ^{ν}*f , for which the the following inequalities are valid:

kD* ^{ν}*fk

_{p,G}≤C

_{1}

### ∑

n i=_{0}

T^{µ}^{i}kfk

L^{<}_{pi,θi a,}^{li}^{>}

κ,τ1

(G,λ), (2.1)

and

kD* ^{ν}*fk

_{p,b,}

_{κ}

_{,τ}

2;G ≤C_{2}kfkn

T i=0

L^{<}_{pi,θi a,}^{li}^{>}

κ,τ1

(G,λ) (p^{i} ≤ p< _{∞}). (2.2)
In particular,*µ*^{i,0} =_{∑}^{n}_{j}_{=}_{1}^{}_{l}^{i}_{j}*λ*_{j}−*ν*_{j}*λ*_{j}− *λ*_{j}−κja_{j} _{1}

p^{i}

>_{0,}(_{i}=_{1, . . . ,}_{n})_{, then D}* ^{ν}*f is continuous
on G and

sup

x∈G

|D* ^{ν}*f(x)| ≤C

_{1}

### ∑

n i=0T^{µ}^{i,0}kfk

L^{<}^{li}^{>}

pi,θi a,κ,τ1

(G,λ) (2.3)

where T is an arbitrary number from(_{0, min}(1,T0)], b= (_{b}_{1}_{,}_{b}_{2}_{, . . . ,}_{b}_{n})_{,} _{b}_{j}any numbers and satisfy
the conditions

0≤b_{j} ≤1, for*µ*^{i,0}>0,
0≤b_{j} <1, for*µ*^{i,0}=0,
0≤b_{j} <a_{j}+ ^{µ}

ip 1−a_{j}

n *λ*_{j}−κja_{j}, for*µ*^{i,0}<_{0,}

(2.4)

but withκ replaced byκ^{, C}1and C_{2} are constants independent of f , moreover C_{1} is independent also
of T.

Let*γ*be ann-dimensional vector.

**Theorem 2.2**([13]). Let the conditions of Theorem2.1be fulfilled. Then for *µ*^{i} >0(i= 1, 2, . . . ,n)
the derivative D* ^{ν}*f satisfies on G the Hölder condition in the metrics L

_{p}with the exponent

*σ, more*exactly,

k_{∆}(*γ,*G)D* ^{ν}*fk

_{p,G}≤CkfkTn i=

_{0}L

^{<}

^{li}

^{>}

pi,θi,a,κ,τ(G,λ)|*γ*|* ^{σ}*, (2.5)
here

*σ*is any number satisfying the inequalities:

0≤*σ*≤1, for *µ*_{0}
*λ*_{0} >1,
0≤*σ*<1, for *µ*_{0}

*λ*_{0} =1,
0≤*σ*≤ ^{µ}^{0}

*λ*_{0}, for *µ*_{0}
*λ*_{0} <1,

(2.6)

where*µ*_{0} = min*µ*^{i} (i= 1, 2, . . . ,n), *λ*_{0} = max*λ*_{j} (j=1, 2, . . . ,n), while C is a constant indepen-
dent of f and|*γ*|.

In particular, if *µ*^{i,0}>0(i=1, 2, . . . ,n), then
sup

x∈G

|_{∆}(*γ,*G)D* ^{ν}*f(x)| ≤CkfkT

_{n}i=0L

^{<}

^{li}

^{>}

pi,θi,a,κ^{,τ}(G,λ)|*γ*|^{σ}^{0}, (2.7)
*σ*^{0}satisfies the same conditions that*σ*satisfies, but with*µ*^{i} replaced by*µ*^{i,0}.

Let us consider the Dirichlet problem for a higher order partial differential equation, i.e.

consider a problem of the form

|*α*|≤|

### ∑

l^{i}|,

|*β*|≤|l^{i}|
i=1,2,...,n

D* ^{α}*(a

*(x)D*

_{αβ}*u(x)) =*

^{β}### ∑

|*α*|≤|l^{i}|,
i=1,2,...,n

D* ^{α}*f

*(x), (2.8)*

_{α}D* ^{ν}*u|

*=*

_{∂G}*ϕ*

*|*

_{ν}*, (2.9) where it is assumed that G is a bounded n-dimensional domain with piecewise-smooth boundary*

_{∂G}*∂G,*

*ν*= (

*ν*

_{1}, . . . ,

*ν*

_{n}), where |

*ν*| < |l

^{i}|, i = 1, 2, . . . ,n while

*α*= (

*α*

_{1}, . . . ,

*α*

_{n}),

|*α*|=*α*_{1}+*α*_{2}+. . .+*α*_{n},*β* = (*β*_{1}, . . . ,*β*_{n});*α*_{j},*β*_{j} ≥0 are integer (j=1,n). We assume that the
coefficientsa* _{αβ}*(x)are bounded measurable functions in the domain G, a

*(x) ≡ a*

_{αβ}*(x)and for*

_{βα}*ξ*∈ R

^{n}

|*α*|≤|

### ∑

l^{i}|,

|*β*|≤|l^{i}|
i=1,2,...,n

(−1)^{|}^{α}^{|}a* _{α,β}*(x)

*ξ*

_{α}*ξ*

*≥C*

_{β}_{0}

### ∑

|*α*|≤|l^{i}|
i=1,2,...,n

|*ξ** _{α}*|

^{2}, C

_{0}=const>0, (2.10)

we also assume that f*α*∈ L2(G), for all*α*= (*α*_{1}, . . . ,*α*n).

The function u ∈ ^{T}^{n}_{i}_{=}_{0}L^{<}_{2}^{l}^{i}^{>}(_{G})is called a generalized solution of problem (2.8)–(2.9) in
the domainG, if D* ^{ν}*u−

*ϕ*

*∈*

_{ν}^{T}

^{n}

_{i}

_{=}

_{0}L

^{˚}

^{<}

_{2}

^{m}

^{i}

^{>}(G), when|m

^{i}+

*ν*|<|l

^{i}|,(i=1, 2, . . . ,n)and for any function

*ϑ*(x)∈

^{T}

^{n}

_{i}

_{=}

_{0}L

^{˚}

^{<}

_{2}

^{l}

^{i}

^{>}(G)the following integral identity is valid:

|*α*|≤|

### ∑

l^{i}|,

|*β*|≤|l^{i}|,
i=1,2,...,n

Z

Ga* _{αβ}*(x)D

*u(x)D*

^{β}

^{α}*ϑ*(x)dx =

### ∑

|*α*|≤|l^{i}|

(−1)^{|}^{α}^{|}

Z

G f* _{α}*D

^{α}*ϑ*(x)dx. (2.11)

The space ^{T}^{n}_{i}_{=}_{0}L˚^{<}_{2}^{l}^{i}^{>}(G) is completion of C_{0}^{∞}(G) in the metric ^{T}^{n}_{i}_{=}_{0}L^{<}_{2}^{l}^{i}^{>}(G). Prove that
there exists a unique generalized solution of problems (2.8) and (2.9). Consider for *φ,ψ* ∈

T_{n}

i=_{0}L^{<}_{2}^{l}^{i}^{>}(G)the bilinear functional
F(* _{φ,}ψ*) =

### ∑

|*α*|≤|l^{i}|,

|*β*|≤|l^{i}|,
i=1,2,...,n

(−_{1})^{|}^{α}^{|}

Z

G

a* _{αβ}*(x)D

^{β}*φ*(x)D

^{α}*ψ*(x)dx

−

### ∑

|*α*|≤|l^{i}|

(−1)^{|}^{α}^{|}

Z

G

f* _{α}*D

^{α}*ψ*(x)dx−

### ∑

|*β*|≤|l^{i}|

(−1)^{|}^{β}^{|}

Z

G

f* _{β}*D

^{β}*φ*(x)dx

= I(*φ,ψ*)−(f* _{α}*,

*ψ*)−(f

*,*

_{β}*φ*), where

I(*φ,φ*) = I(*φ*)≥ k*φ*k^{2}_{T}_{n}

i=0L^{<}_{2}^{li}^{>}(G).

The variational problem is stated as follows: it is required to find the function *φ* ∈

T_{n}

i=0L^{<}_{2}^{l}^{i}^{>}(G), l = (l_{1}, . . . ,ln), l_{j} > 0, j = 1, . . . ,n are entire, that gives the least value to
the integralF(*φ*)and is unique. Equation (2.8) is the Euler equation for the considered varia-
tional problem.

F(*φ,φ*) =F(*φ*) = I(*φ*)−(f* _{α}*+ f

*,*

_{β}*φ*)

≥ I(*φ*)−

### ∑

|*α*|≤|l^{i}|

(−1)^{|}^{α}^{|}
_{Z}

G

|f* _{α}*|

^{2}dx+

Z

G

|D^{α}*φ*(x)|^{2}dx

−

### ∑

|*α*|≤|l^{i}|

(−1)^{|}^{β}^{|}
_{Z}

G

|f* _{β}*|

^{2}dx+

Z

G

|D^{β}*φ*(x)|^{2}dx

≥ k*φ*kT_{n}

i=0L^{<}_{2}^{li}^{>}(G)−2k*φ*kT_{n}

i=0L^{<}_{2}^{li}^{>}(G)−d=−d_{1}.

Then means that F(*φ*) is lower bounded on ^{T}_{i}^{n}_{=}_{0}L_{2}^{<}^{l}^{i}^{>}(G). Show that there exists *φ*0 ∈

T_{n}

i=0L^{<}_{2}^{l}^{i}^{>}(G)such thatF(*φ*_{0}) =min

*φ*∈^{T}^{n}_{i}_{=}_{0}L^{<}_{2}^{li}^{>}(G)F(*φ*). Indeed, letk=min_{Φ}_{∈}Tn

i=0L^{<}_{2}^{li}^{>}(G)F(*φ*).
Fix some sequence *φ*_{m} ∈ ^{T}^{n}_{i}_{=}_{0}L^{<}_{2}^{l}^{i}^{>}(G) such that lim_{m}→_{∞}F(*φ*_{m}) = k and let *ε* > 0. Choose
m*ε* so that for m ≥ m*ε* and *µ* = 0, 1, 2, . . . would hold F(*φ*m+*µ*) < k+*ε. Then noting*

1

2(*φ*_{m}+*µ*+*φ*_{m}) ∈ ^{T}^{n}_{i}_{=}_{0}L^{<}_{2}^{l}^{i}^{>}(G), we have F ^{φ}^{m}^{+}^{µ}_{2}^{+}^{φ}^{m}

≥ k. Further, by direct calculations
we show that I ^{φ}^{m}^{+}^{µ}_{2}^{−}^{φ}^{m}

< 4ε. From the ellipticity condition (2.10) it follows that
k*φ*_{m}+*µ*−*φ*_{m}kT_{n}

i=0L^{<}_{2}^{li}^{>}(G) < 2q

*ε*

c_{0}. It means that the sequence {*φ*_{m}} in fundamental in the
space ^{T}^{n}_{i}_{=}_{0}L^{<}_{2}^{l}^{i}^{>}(G). Therefore, because of completeness in the space ^{T}^{n}_{i}_{=}_{0}L^{<}_{2}^{l}^{i}^{>}(G) there
exists the function *φ*_{0}∈^{T}^{n}_{i}_{=}_{0}L^{<}_{2}^{l}^{i}^{>}(G)such that lim_{m}→_{∞}k*φ*_{m}−*φ*_{0}kTn

i=0L^{<}_{2}^{li}^{>}(G) =0. By [7, The-
orem 1], it is proved that |F(*φ*_{m})−F(*φ*_{0})| ≤Ck*φ*_{m}−*φ*_{0}kT_{n}

i=0L^{<}_{2}^{li}^{>}(G), and hence it follows that
k = _{lim}_{m}_{→}_{∞}F(*φ*_{m}) = F(*φ*_{0}). Show that the function delivering minimum to the functions
F(*φ*)in the space^{T}^{n}_{i}_{=}_{0}L^{<}_{2}^{l}^{i}^{>}(G)is unique and satisfies

D* ^{ν}*u|

*=*

_{∂G}*ϕ*

*|*

_{ν}*. Indeed,*

_{∂G}*φ*∈

^{T}

^{n}

_{i}

_{=}

_{0}L

^{<}

_{2}

^{l}

^{i}

^{>}(G)andF(

*φ*

_{0}) =k, we have:

0≤ I

*φ*−*φ*_{0}
2

= ^{1}

2F(*φ*) + ^{1}

2F(*φ*_{0})−F

*φ*+*φ*_{0}
2

≤ ^{k}
2+ ^{k}

2−k=_{0,}
I(*φ*−*φ*_{0}) =_{0,}

then again by the ellipticity condition (2.10),k*φ*_{m}−*φ*_{0}kTn

i=0L^{<}_{2}^{li}^{>}(G)
m→0

−→0, hence it follows that
*φ*coincides with*φ*_{0}as an element of^{T}^{n}_{i}_{=}_{0}L^{<}_{2}^{l}^{i}^{>}(G). By Theorem 1 in [7] we have:

kD* ^{ν}*(

*φ*

_{m}−

*φ*

_{0})|

*k*

_{∂G}_{L}

2(* _{∂G}*) ≤ k

*φ*

_{m}−

*φ*

_{0}kTn

i=0L^{<}_{2}^{li}^{>}(G) →0,
m→_{∞,} |*ν*|<|l^{i}|(i=1, 2, . . . ,n), as

kD^{ν}*φ*_{m}|* _{∂G}*−

*φ*

*ν*|

*k*

_{∂G}_{L}

2(G) →0,
m→_{∞,} |*ν*| ≤ |l^{i}|(i=1, 2, . . . ,n), therefore

kD^{ν}*φ*_{0}|* _{∂G}*−

*φ*

*|*

_{ν}*k*

_{∂G}_{L}

2(G)=0,

|*ν*| ≤ |l^{i}|(i = 1, 2, . . . ,n). Taking into account the conditions _{dλ}^{d} F(*φ*_{0}+*λψ*)^{}

*λ*=0 = 0, show
that the function *φ*_{0} ∈ ^{T}^{n}_{i}_{=}_{0}L_{2}^{<}^{l}^{i}^{>}(G), minimizing the integral F(*φ*), satisfies the following
equation:

I(*φ*_{0},*ψ*)−(f* _{α}*,

*ψ*) =0. (2.12) Now prove that the function

*φ*

_{0}∈

^{T}

^{n}

_{i}

_{=}

_{0}L

^{<}

_{2}

^{l}

^{i}

^{>}(G), minimizing the integralF(

*φ*)is the solution (generalized) of problem (2.8)–(2.9). For that we suppose thata

*(x)are bounded in absolute value in the domain G together with its derivatives and the function f*

_{α,β}*α*has derivatives be- longing to the space L

_{2}(G). Denote byΘ(t) some monotonically decreasing function on the interval

^{1}

_{2}≤t ≤1, and possessing the following properties:

Θ 1

2+0

=1; Θ(1−0) =−1;

Θ^{(}^{s}^{)}
1

2 +_{0}

=_{Θ}^{(}^{s}^{)}(_{1}−_{0}) =_{0} _{for any}s>_{0.}

The function

(Θ^{0}(t), ^{1}_{2} ≤t≤1,

0, −_{∞}<t ≤ ^{1}_{2}, 1≤t <_{∞}

is infinitely differentiable and finite over the whole axis. Let*η*>0 andG* _{η}* ={y:

*ρ*(y,R

^{n}\G)>

*η*}, x be an arbitrary point of domain G and r = *ρ*(x,y). Following S. L. Sobolev [16] we
introduce the function

*ψ*(x) =*γ*
r

h_{1}

−*γ*
r

h_{2}

.

For 0< h_{1} < h_{2} < *η. Obviously,ψ*is infinitely differentiable finite function with support
of annular domain ^{h}_{2}^{1} <r< ^{h}_{2}^{2}. Therefore,*ψ*∈^{T}^{n}_{i}_{=}_{0}L^{<}_{2}^{l}^{i}^{>}(G), and D^{(}^{s}^{)}*ψ*|* _{∂G}* =0 for anys >0.

Then from the expression (2.12) by definition of the generalized derivative it follows that, Z

G*ω*
r

h_{1}

*φ*(x)dx=

Z

G*ω*
r

h_{2}

*φ*(x)dx,

where
*ω*

r
h_{i}

=D^{β}

a* _{α,β}*(x)D

^{α}*γ*r

h_{i}

−(−1)^{|}^{α}^{|}f*α*D^{α}*γ*
r

h_{i}

, (i=1, 2).
The function *ω* _{h}^{r}

i

possesses all the properties of a kernel. Then for the function *φ*_{0} (the
solution of the variational problem) we can construct the Sobolev averaging*φ*_{0,h}_{i}(x), i = 1, 2
over the ballh_{i}, (i=1, 2)centered at the pointx:

*φ*_{0,h}_{i}(x) = ^{1}
*σ*_{n}h^{n}_{i}

Z

**R**^{n}*ω*

|z−x|
h_{i}

*φ*_{0}(z)dz, i=1, 2.

Then we can rewrite equality (2.12) in the form *φ*_{0,h}_{1}(x) =*φ*_{0,h}_{2}(x). Consequently, forh <*η*
*φ*_{0,h}(x) =*φ*_{0}(x).

As the average function *φ*_{0,h}(x)is continuous and has any order continuous derivatives,
then *φ*_{0}(x) also possesses these properties. Making integration in parts in the equality
I(*φ*_{0,h},*ψ*)−(f* _{α}*,

*ψ*) =0, in the limiting case

Z

G

### ∑

|*α*|≤|l^{i}|,

|*β*|≤|l^{i}|
i=1,2,...,n

*ψ*(x)^{h}D^{α}

a* _{α,β}*(x)D

^{β}*φ*

_{0}(x)

^{}−D

*f*

^{α}*α*(x)

^{i}dx =0.

Hence, by arbitrariness of the function*ψ*(x)it follows

|*α*|≤|

### ∑

l^{i}|,

|*β*|≤|l^{i}|
i=1,2,...,n

D^{α}

a* _{α,β}*(x)D

^{β}*φ*

_{0}(x)

^{}=

### ∑

|*α*|≤|l^{i}|,
i=1,2,...,n

D* ^{α}*f

*(x).*

_{α}Thus the solution of the variational problem from the class^{T}^{n}_{i}_{=}_{0}L^{<}_{2}^{l}^{i}^{>}(G)is also the solu-
tion of problem (2.8)–(2.9) and this solution is unique.

Assume also that f* _{α}* ∈ L

_{2,α,}

_{κ}(G)for|

*α*|= |l

^{i}|(i= 1, 2, . . . ,n), 0<d <1, d= const,b≤d, x0 ∈G

_{d};G

_{d}is a subdomain of the domain Gsuch that

G_{d}={x^{00} :|x^{00}−x^{0}_{j}|>d^{λ}^{j}, x^{0} ∈ *∂G,* j=1, 2, . . . ,n},
and

Πb(x_{0}) ={x :|x_{j}−x_{j,0}|>b^{λ}^{j}, j=1, 2, . . . ,n}.

**Theorem 2.3.** If ^{|}^{λ}_{2}^{|} ≤ (*λ,*l^{i}) (i= 1, 2, . . . ,n), then any generalized solution of equation(2.8) from
T_{n}

i=0L^{<}_{2}^{l}^{i}^{>}(G) is continuous in G and satisfies the Hölder condition in any subdomain compactly
imbedded into G.

Proof. Let at first all a* _{αβ}*(x) ≡ 0, except the ones for which |

*α*| = |

*β*| = |l

^{i}| (i = 1, 2, . . . ,n) and the left hand side equals zero. For any Θ(x)∈

_{Π}

_{b}(x

_{0}), such thatΘ≡1 in the vicinity of

*∂*Πb(x_{0})any polynomial

P(x) =

### ∑

|*α*|=|l^{i}|,
i=1,2,...,n

C* _{α}*x

^{α}and for arbitrary solutionu(x)from the variational principle it follows that Z

Πb(x0)

### ∑

|*α*|=|*β*|=|l^{i}|,
i=1,2,...,n

(−1)^{|}^{α}^{|}a* _{αβ}*(x)D

*(*

^{β}_{Θ}(x)(u(x)−P(x)))D

*(*

^{α}_{Θ}(x)(u(x)−P(x)))dx

≥

Z

Πb(_{x}_{0})

### ∑

|*α*|=|*β*|=|l^{i}|,
i=1,2,...,n

(−_{1})^{|}^{α}^{|}a* _{αβ}*(x)D

*((u(x)−P(x)))D*

^{β}*((u(x)−P(x)))dx*

^{α}= A(u(x)−P(x),Πb(x_{0})), (2.13)

moreover,*θ*(x) = 1−_{∏}^{n}_{i}_{=}_{1}*ω*_{j} ^{x}^{−}^{x}^{0}

b^{λ}^{j}

,x ∈ G, where*ω*_{j}(t)∈ C^{∞}(** _{R}**)is such that

*ω*

_{j}(t)≡ 1 for

|t| <2^{−}^{λ}^{j},*ω*_{j}(t)≡ 0 for|t|> 1, 0≤*ω*_{j}(t)≤ 1. It is seen that*θ*(x)≡ 0 inΠ^{b}

2 (x_{0}),*θ*(x)≡1
in the vicinity of *∂*Πb(x_{0}), and the coefficientsP(x)are chosen so that

Z

(_{Π}_{b}(x_{0}))\ _{Π}_{b}

2

(x_{0})(u−p(x))x* ^{α}*dx=0.

By means of (2.1) and (2.2) we get
A(u(x)−P(x)_{,}_{Π}_{b}(x_{0}))

≤ A

u(x)−P(x);Πb(x0)\^{}_{Π}b

2 (x0)^{}

+

Z

(_{Π}_{b}(x0))\ _{Π}_{b}

2

(x0)

### ∑

|*α*|<|l^{i}|

b^{2}^{|}^{α}^{|−}^{2}^{|}^{l}^{i}^{|}D* ^{α}*(u(x)−P(x))

^{2}dx

≤qA

u(x)−P(x), Πb(x_{0})\^{}_{Π}b

2 (x_{0})^{}. (2.14)

AsA(u(x)−p(x),G) =A(u(x),G), then in view of (2.14) by induction we get A

u(x),Π^{b}

2k (x_{0})

≤

1− ^{1}
q

k

A(u(x),Πb(x_{0})).

Let 0<*δ* < ^{b}

2^{k},*ζ* =1−^{1}_{q}, then
A(u(x),Π*δ*(x0))≤

*δ*
b

|^{ln}_{ln 2}* ^{ζ}*|

^{−}|

_{lnb/δ}

^{lnζ}|

A(u(x),G) =
*δ*

b
*ξ*−*σ*

A(u(x),G) (2.15)
for any*δ*≤ b, and consequently,

Z _{1}

0

*η*^{−}^{ξ}

Z

Π*η*(x0)u^{2}dx
^{1}_{2}

dη
*η* ≤C

Z _{1}

0

db

b^{1}^{−}^{1}^{2}* ^{σ}* <

_{∞.}

From 0 < *ξ* = (κ^{,}^{a}) < 1, it follows that u(x) ∈ L_{2,a,}_{κ}_{,1}(G_{d}) ⊂ L_{2,a,}_{κ}_{,τ}(G_{d})and from the
condition of Theorem2.3it follows that *µ*_{i} > 0,*µ*_{i,0}> 0(i=1, 2, . . . ,n), i.e. the conditions of
Theorems2.1and2.2are fulfilled. Thus, by Theorem2.1,u(x)is continuous, by Theorem2.2,
u(x)satisfies the Hölder condition onG_{d}.

Let a* _{αβ}* = 0, except a

*, for which |*

_{αβ}*α*| = |

*β*| = |l

^{i}|, and the right hand sides of equa- tion (2.8) be nonzero. Let u

_{b,x}

_{0}be a generalized solution of this equation in Πb(x

_{0}) from T

_{n}

i=0L˚^{<}_{2}^{l}^{i}^{>}(_{Π}_{b}(x0)). Existence of such a solution is proved by the functional method. Put in
(2.11)*ϑ*≡u_{b,x}_{0} then from (2.10) we get

Z

Πb(x0)

### ∑

|*α*|=|l^{i}|,
i=1,2,...,n

(D* ^{α}*u

_{b,x}

_{0})

^{2}dx≤

### ∑

|*α*|=|l^{i}|,
i=1,2,...,n

b^{2}^{|}^{l}^{i}^{|−}^{2}^{|}^{α}^{|}
Z

Πb(x0) f_{α}^{2}dx+

### ∑

|*α*|=|l^{i}|,
i=1,2,...,n

Z

Πb(x0)f_{α}^{2}dx≤ C_{1}b^{r},
if

r = min

|*α*|=|l^{i}|,
i=1,2,...,n

n

2|l^{i}| −2|*α*|,(κ^{,}^{a})^{o}>0,
hereC_{1}andrare independent of uandx0.

Hence it follows that

A(u_{b,x}_{0},Πb(x0))≤C_{1}b^{r}. (2.16)
Asu = u−u_{b,x}_{0} is the solution of homogeneous equation (2.8) then the following inequality
is valid for it:

A(u,Πb(x_{0}))≤C_{2}
*δ*

b
*ξ*−*σ*

A(u,G). (2.17)

From inequalities (2.16) and (2.17) we get

A(u,Πb(x_{0}))≤C_{3}A(u,Πb(x_{0})) +C_{3}A(u_{b,x}_{0},Πb(x_{0}))

≤C_{4}
*δ*

b
*ξ*−*σ*

A(u,¯ G) +C_{5}b^{r}≤C_{6}
*δ*

b
*ξ*−*σ*

, and hence we get

Z _{1}

0

*η*^{−}^{ξ}

Z

Πb(x0)u^{2}dx
^{1}_{2}

dη
*η* ≤C

Z _{1}

0

db

b^{1}^{−}^{1}^{2}* ^{σ}* <

_{∞.}

Here using Theorems 2.1 and 2.2 we get that u(x) is continuous and satisfies the Hölder
condition onG_{d}.

Finally, we consider equations (2.8) where there are nonzero coefficients at minor deriva- tives of the solution. Then we take these terms to the right hand side of the equation and in this case we get the desired result.

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