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## On the solution of a class ofpartial differential equations

1,2

### and Aygun T. Orujova

B2

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 domainGRn satisfying the flexibleλ-horn condition.

Keywords:flexibleλ-horn, generalized derivatives,generalized solution, smoothness of solution.

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

### 1Introduction

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

n

\

i=0

L<pli>

ii (G), (1.1)

1 ≤ pi∞, 1θi∞, (i = 0, 1, . . . ,n), li = (l1i, . . . ,lni), l0j ≥ 0, lii > 0, lij ≥ 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<pli>

ii,a,κ(G,λ), (1.2)

with the finite norm

kfkTn i=0L<li>

pi,θi,a,κ(G,λ) =

### ∑

n i=0

kfk

L<li>

pi,θi,a,κ(G,λ), (1.3)

kfk

L<li>

pi,θi,a,κ(G,λ) =





Z h0

0

mi hλ;G,λ Dkif

pi,a,κ

h(λ,liki)

θi

dh h





1 θi

, (1.4)

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

Najafov).

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kfkpi,a,κ,τ;G=kfkL

pi,a,κ(G)=sup

xG

( Z

0

"

[t]

(κ,a) pi

1 kfkpi,Gtκ(x)

#τ

dt t

)τ1

, (1.5)

where 1 ≤ pi < , 1 ≤ θi, miNn, kiNn0, a ∈ [0, 1], κ ∈ (0,∞)n, τ ∈ [1,∞], li = (l1i, . . . ,lin), l0j ≥ 0, lii > 0, lij ≥ 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 l0 = (0, . . . , 0), li = (0, . . . ,li, . . . , 0), pi = p,θi =θ space (1.1) coincides with the spaceBlp,θ(G)studied in [2], the space (1.2) coincides with the space Blp,θ,a,κ(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 spacesBlp,θ(G)andBlp,θ,a,

κ(G). Example 1.1. Let us consider an equation of the form

u(x52)y3+u(x42)y2+u(x32)y1 +u(xy2)+u(x1)+u= f(x), (1.6) in our case the solution of this equation is sought in the spaceL(20,0)TL(23,1)TL(21,4). One can look for the solution of equation (1.6) in the space W2(6,3) (B(2,26,3)), 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| = li, (i=1, 2, . . . ,n) fα belongs to a wider class. Furthermore, as in the papers [5,6,8,10–12,14,15]

here the coefficients do not require smoothness.

### 2Main results

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

Theorem 2.1 ([13]). Let the open set G ⊂ Rn satisfy the flexible λ-horn condition [2], λ = (λ1, . . . ,λn), λj > 0, (j = 1, 2, . . . ,n), 1 ≤ pi ≤ p ≤ ∞, 1 ≤ θi∞, (i=0, 1, 2, . . . ,n); ν= (ν1,ν2, . . . ,νn), νj0be entire(j=1, 2, . . . ,n);

1) νj ≥l0j (j=1, 2, . . . ,n);

2) νj ≥lij (j6=i,j=1, 2, . . . ,n), νi <lii (j=i, i=1, 2, . . . ,n); 1≤τ1τ2∞;κ= cκ, 1c =maxj=1,...,nκj

λj; f ∈ Tni=0L<li>

piia,κ(G,λ)and µi =

### ∑

n j=1

lijλjνjλjλj−κjaj 1

pi1 p

>0 (i=1, . . . ,n). Then Dν : Tni=0L<li>

pii,a,κ1(G,λ) ,→ Lp,b,κ2(G). Precisely, for f ∈ Tni=0L<pili>ia,

κ1(G,λ)in the domain G there exists the generalized derivative Dνf , for which the the following inequalities are valid:

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kDνfkp,G≤C1

### ∑

n i=0

Tµikfk

L<pi,θi a,li>

κ1

(G,λ), (2.1)

and

kDνfkp,b,κ

2;G ≤C2kfkn

T i=0

L<pi,θi a,li>

κ1

(G,λ) (pi ≤ p< ). (2.2) In particular,µi,0 =nj=1lijλjνjλjλj−κjaj 1

pi

>0,(i=1, . . . ,n), then Dνf is continuous on G and

sup

xG

|Dνf(x)| ≤C1

### ∑

n i=0

Tµi,0kfk

L<li>

pi,θi a,κ1

(G,λ) (2.3)

where T is an arbitrary number from(0, min(1,T0)], b= (b1,b2, . . . ,bn), bjany numbers and satisfy the conditions

0≤bj ≤1, forµi,0>0, 0≤bj <1, forµi,0=0, 0≤bj <aj+ µ

ip 1−aj

n λj−κjaj, forµi,0<0,

(2.4)

but withκ replaced byκ, C1and C2 are constants independent of f , moreover C1 is independent also of T.

Letγbe ann-dimensional vector.

Theorem 2.2([13]). Let the conditions of Theorem2.1be fulfilled. Then for µi >0(i= 1, 2, . . . ,n) the derivative Dνf satisfies on G the Hölder condition in the metrics Lp with the exponent σ, more exactly,

k(γ,G)Dνfkp,G≤CkfkTn i=0L<li>

pi,θi,a,κ(G,λ)|γ|σ, (2.5) hereσis any number satisfying the inequalities:

0≤σ≤1, for µ0 λ0 >1, 0≤σ<1, for µ0

λ0 =1, 0≤σµ0

λ0, for µ0 λ0 <1,

(2.6)

whereµ0 = minµi (i= 1, 2, . . . ,n), λ0 = maxλj (j=1, 2, . . . ,n), while C is a constant indepen- dent of f and|γ|.

In particular, if µi,0>0(i=1, 2, . . . ,n), then sup

xG

|(γ,G)Dνf(x)| ≤CkfkTn i=0L<li>

pi,θi,a,κ(G,λ)|γ|σ0, (2.7) σ0satisfies the same conditions thatσsatisfies, but withµi replaced byµi,0.

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

consider a problem of the form

|α|≤|

### ∑

li|,

|β|≤|li| i=1,2,...,n

Dα(aαβ(x)Dβu(x)) =

### ∑

|α|≤|li|, i=1,2,...,n

Dαfα(x), (2.8)

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Dνu|∂G= ϕν|∂G, (2.9) where it is assumed that G is a bounded n-dimensional domain with piecewise-smooth boundary ∂G, ν = (ν1, . . . ,νn), where |ν| < |li|, i = 1, 2, . . . ,n while α = (α1, . . . ,αn),

|α|=α1+α2+. . .+αn,β = (β1, . . . ,βn);αj,βj ≥0 are integer (j=1,n). We assume that the coefficientsaαβ(x)are bounded measurable functions in the domain G, aαβ(x) ≡ aβα(x)and forξ ∈ Rn

|α|≤|

### ∑

li|,

|β|≤|li| i=1,2,...,n

(−1)|α|aα,β(x)ξαξβ ≥C0

### ∑

|α|≤|li| i=1,2,...,n

|ξα|2, C0=const>0, (2.10)

we also assume that fα∈ L2(G), for allα= (α1, . . . ,αn).

The function u ∈ Tni=0L<2li>(G)is called a generalized solution of problem (2.8)–(2.9) in the domainG, if Dνu−ϕνTni=0L˚<2mi>(G), when|mi+ν|<|li|,(i=1, 2, . . . ,n)and for any functionϑ(x)∈ Tni=0L˚<2li>(G)the following integral identity is valid:

|α|≤|

### ∑

li|,

|β|≤|li|, i=1,2,...,n

Z

Gaαβ(x)Dβu(x)Dαϑ(x)dx =

### ∑

|α|≤|li|

(−1)|α|

Z

G fαDαϑ(x)dx. (2.11)

The space Tni=0<2li>(G) is completion of C0(G) in the metric Tni=0L<2li>(G). Prove that there exists a unique generalized solution of problems (2.8) and (2.9). Consider for φ,ψ

Tn

i=0L<2li>(G)the bilinear functional F(φ,ψ) =

### ∑

|α|≤|li|,

|β|≤|li|, i=1,2,...,n

(−1)|α|

Z

G

aαβ(x)Dβφ(x)Dαψ(x)dx

|α|≤|li|

(−1)|α|

Z

G

fαDαψ(x)dx−

### ∑

|β|≤|li|

(−1)|β|

Z

G

fβDβφ(x)dx

= I(φ,ψ)−(fα,ψ)−(fβ,φ), where

I(φ,φ) = I(φ)≥ kφk2Tn

i=0L<2li>(G).

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

Tn

i=0L<2li>(G), l = (l1, . . . ,ln), lj > 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)−(fα+ fβ,φ)

≥ I(φ)−

|α|≤|li|

(−1)|α| Z

G

|fα|2dx+

Z

G

|Dαφ(x)|2dx

### ∑

|α|≤|li|

(−1)|β| Z

G

|fβ|2dx+

Z

G

|Dβφ(x)|2dx

≥ kφkTn

i=0L<2li>(G)−2kφkTn

i=0L<2li>(G)−d=−d1.

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Then means that F(φ) is lower bounded on Tin=0L2<li>(G). Show that there exists φ0

Tn

i=0L<2li>(G)such thatF(φ0) =min

φTni=0L<2li>(G)F(φ). Indeed, letk=minΦTn

i=0L<2li>(G)F(φ). Fix some sequence φmTni=0L<2li>(G) such that limmF(φm) = k and let ε > 0. Choose mε so that for m ≥ mε and µ = 0, 1, 2, . . . would hold F(φm+µ) < k+ε. Then noting

1

2(φm+µ+φm) ∈ Tni=0L<2li>(G), we have F φm+µ2+φm

≥ k. Further, by direct calculations we show that I φm+µ2φm

< 4ε. From the ellipticity condition (2.10) it follows that kφm+µφmkTn

i=0L<2li>(G) < 2q

ε

c0. It means that the sequence {φm} in fundamental in the space Tni=0L<2li>(G). Therefore, because of completeness in the space Tni=0L<2li>(G) there exists the function φ0Tni=0L<2li>(G)such that limmkφmφ0kTn

i=0L<2li>(G) =0. By [7, The- orem 1], it is proved that |F(φm)−F(φ0)| ≤Ckφmφ0kTn

i=0L<2li>(G), and hence it follows that k = limmF(φm) = F(φ0). Show that the function delivering minimum to the functions F(φ)in the spaceTni=0L<2li>(G)is unique and satisfies

Dνu|∂G= ϕν|∂G. Indeed,φTni=0L<2li>(G)andF(φ0) =k, we have:

0≤ I

φφ0 2

= 1

2F(φ) + 1

2F(φ0)−F

φ+φ0 2

k 2+ k

2−k=0, I(φφ0) =0,

then again by the ellipticity condition (2.10),kφmφ0kTn

i=0L<2li>(G) m0

−→0, hence it follows that φcoincides withφ0as an element ofTni=0L<2li>(G). By Theorem 1 in [7] we have:

kDν(φmφ0)|∂GkL

2(∂G) ≤ kφmφ0kTn

i=0L<2li>(G) →0, m→∞, |ν|<|li|(i=1, 2, . . . ,n), as

kDνφm|∂Gφν|∂GkL

2(G) →0, m→∞, |ν| ≤ |li|(i=1, 2, . . . ,n), therefore

kDνφ0|∂Gφν|∂GkL

2(G)=0,

|ν| ≤ |li|(i = 1, 2, . . . ,n). Taking into account the conditions d F(φ0+λψ)

λ=0 = 0, show that the function φ0Tni=0L2<li>(G), minimizing the integral F(φ), satisfies the following equation:

I(φ0,ψ)−(fα,ψ) =0. (2.12) Now prove that the function φ0Tni=0L<2li>(G), minimizing the integralF(φ)is the solution (generalized) of problem (2.8)–(2.9). For that we suppose thataα,β(x)are bounded in absolute value in the domain G together with its derivatives and the function fα has derivatives be- longing to the space L2(G). Denote byΘ(t) some monotonically decreasing function on the interval 12 ≤t ≤1, and possessing the following properties:

Θ 1

2+0

=1; Θ(1−0) =−1;

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Θ(s) 1

2 +0

=Θ(s)(10) =0 for anys>0.

The function

0(t), 12 ≤t≤1,

0, −<t ≤ 12, 1≤t <

is infinitely differentiable and finite over the whole axis. Letη>0 andGη ={y:ρ(y,Rn\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

h1

γ r

h2

.

For 0< h1 < h2 < η. Obviously,ψis infinitely differentiable finite function with support of annular domain h21 <r< h22. Therefore,ψTni=0L<2li>(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

Gω r

h1

φ(x)dx=

Z

Gω r

h2

φ(x)dx,

where ω

r hi

=Dβ

aα,β(x)Dαγ r

hi

−(−1)|α|fαDαγ r

hi

, (i=1, 2). The function ω hr

i

possesses all the properties of a kernel. Then for the function φ0 (the solution of the variational problem) we can construct the Sobolev averagingφ0,hi(x), i = 1, 2 over the ballhi, (i=1, 2)centered at the pointx:

φ0,hi(x) = 1 σnhni

Z

Rnω

|z−x| hi

φ0(z)dz, i=1, 2.

Then we can rewrite equality (2.12) in the form φ0,h1(x) =φ0,h2(x). Consequently, forh <η φ0,h(x) =φ0(x).

As the average function φ0,h(x)is continuous and has any order continuous derivatives, then φ0(x) also possesses these properties. Making integration in parts in the equality I(φ0,h,ψ)−(fα,ψ) =0, in the limiting case

Z

G

### ∑

|α|≤|li|,

|β|≤|li| i=1,2,...,n

ψ(x)hDα

aα,β(x)Dβφ0(x)−Dαfα(x)idx =0.

Hence, by arbitrariness of the functionψ(x)it follows

|α|≤|

### ∑

li|,

|β|≤|li| i=1,2,...,n

Dα

aα,β(x)Dβφ0(x)=

### ∑

|α|≤|li|, i=1,2,...,n

Dαfα(x).

Thus the solution of the variational problem from the classTni=0L<2li>(G)is also the solu- tion of problem (2.8)–(2.9) and this solution is unique.

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Assume also that fα ∈ L2,α,κ(G)for|α|= |li|(i= 1, 2, . . . ,n), 0<d <1, d= const,b≤d, x0 ∈Gd;Gdis a subdomain of the domain Gsuch that

Gd={x00 :|x00−x0j|>dλj, x0∂G, j=1, 2, . . . ,n}, and

Πb(x0) ={x :|xj−xj,0|>bλj, j=1, 2, . . . ,n}.

Theorem 2.3. If |λ2| ≤ (λ,li) (i= 1, 2, . . . ,n), then any generalized solution of equation(2.8) from Tn

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

Proof. Let at first all aαβ(x) ≡ 0, except the ones for which |α| = |β| = |li| (i = 1, 2, . . . ,n) and the left hand side equals zero. For any Θ(x)∈ Πb(x0), such thatΘ≡1 in the vicinity of

Πb(x0)any polynomial

P(x) =

### ∑

|α|=|li|, i=1,2,...,n

Cαxα

and for arbitrary solutionu(x)from the variational principle it follows that Z

Πb(x0)

### ∑

|α|=|β|=|li|, i=1,2,...,n

(−1)|α|aαβ(x)Dβ(Θ(x)(u(x)−P(x)))Dα(Θ(x)(u(x)−P(x)))dx

Z

Πb(x0)

### ∑

|α|=|β|=|li|, i=1,2,...,n

(−1)|α|aαβ(x)Dβ((u(x)−P(x)))Dα((u(x)−P(x)))dx

= A(u(x)−P(x),Πb(x0)), (2.13)

moreover,θ(x) = 1−ni=1ωj xx0

bλj

,x ∈ G, whereωj(t)∈ C(R)is such thatωj(t)≡ 1 for

|t| <2λj,ωj(t)≡ 0 for|t|> 1, 0≤ωj(t)≤ 1. It is seen thatθ(x)≡ 0 inΠb

2 (x0),θ(x)≡1 in the vicinity of Πb(x0), and the coefficientsP(x)are chosen so that

Z

(Πb(x0))\ Πb

2

(x0)(u−p(x))xαdx=0.

By means of (2.1) and (2.2) we get A(u(x)−P(x),Πb(x0))

≤ A

u(x)−P(x);Πb(x0)\Πb

2 (x0)

+

Z

(Πb(x0))\ Πb

2

(x0)

### ∑

|α|<|li|

b2|α|−2|li|Dα(u(x)−P(x))2dx

≤qA

u(x)−P(x), Πb(x0)\Πb

2 (x0). (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),Πb

2k (x0)

1− 1 q

k

A(u(x),Πb(x0)).

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Let 0<δ < b

2k,ζ =1−1q, then A(u(x),Πδ(x0))≤

δ b

|lnln 2ζ||lnb/δlnζ |

A(u(x),G) = δ

b ξσ

A(u(x),G) (2.15) for anyδ≤ b, and consequently,

Z 1

0

ηξ

Z

Πη(x0)u2dx 12

η ≤C

Z 1

0

db

b112σ <∞.

From 0 < ξ = (κ,a) < 1, it follows that u(x) ∈ L2,a,κ,1(Gd) ⊂ L2,a,κ(Gd)and from the condition of Theorem2.3it follows that µi > 0,µi,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 onGd.

Let aαβ = 0, except aαβ, for which |α| = |β| = |li|, and the right hand sides of equa- tion (2.8) be nonzero. Let ub,x0 be a generalized solution of this equation in Πb(x0) from Tn

i=0<2li>(Πb(x0)). Existence of such a solution is proved by the functional method. Put in (2.11)ϑ≡ub,x0 then from (2.10) we get

Z

Πb(x0)

### ∑

|α|=|li|, i=1,2,...,n

(Dαub,x0)2dx≤

### ∑

|α|=|li|, i=1,2,...,n

b2|li|−2|α| Z

Πb(x0) fα2dx+

### ∑

|α|=|li|, i=1,2,...,n

Z

Πb(x0)fα2dx≤ C1br, if

r = min

|α|=|li|, i=1,2,...,n

n

2|li| −2|α|,(κ,a)o>0, hereC1andrare independent of uandx0.

Hence it follows that

A(ub,x0b(x0))≤C1br. (2.16) Asu = u−ub,x0 is the solution of homogeneous equation (2.8) then the following inequality is valid for it:

A(u,Πb(x0))≤C2 δ

b ξσ

A(u,G). (2.17)

From inequalities (2.16) and (2.17) we get

A(u,Πb(x0))≤C3A(u,Πb(x0)) +C3A(ub,x0b(x0))

≤C4 δ

b ξσ

A(u,¯ G) +C5br≤C6 δ

b ξσ

, and hence we get

Z 1

0

ηξ

Z

Πb(x0)u2dx 12

η ≤C

Z 1

0

db

b112σ <∞.

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

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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