On the solution of a class of partial differential equations

On the solution of a class of partial differential equations

Alik M. Najafov

and Aygun T. Orujova

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⊂Rⁿ satisfying the flexibleλ-horn condition.

Keywords:flexibleλ-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^li>

i,θi (G), (1.1)

1 ≤ pⁱ ≤ _{∞, 1} ≤ θⁱ ≤ _∞, (i = 0, 1, . . . ,n), lⁱ = (l₁ⁱ, . . . ,l_nⁱ), l⁰_j ≥ 0, l_iⁱ > 0, lⁱ_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^li>

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) =

∑

kfk

L^<li>

pi,θi,a,κ,τ(G,λ), (1.3)

kfk

L^<li>

pi,θi,a,κ,τ(G,λ) =









 Z _h0

0







∆^mi h^λ;G,λ D^kif

pⁱ,a,κ,τ

h^(λ,li−ki)







θⁱ

dh h











1 θi

, (1.4)

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

Najafov).

kfk_pi,a,κ,τ;G=kfk_L

pi,a,κ,τ(G)=sup

x∈G

( Z _∞

0

"

[t]⁻

(κ,a) pi

1 kfk_pi,G_tκ(x)

#τ

dt t

)_τ¹

, (1.5)

where 1 ≤ pⁱ < _∞, 1 ≤ θⁱ ≤ _∞, mⁱ ∈ _Nⁿ, kⁱ ∈ _Nⁿ₀, a ∈ [0, 1], κ ∈ (0,∞)ⁿ, τ ∈ [1,∞], lⁱ = (l₁ⁱ, . . . ,lⁱ_n), l⁰_j ≥ 0, lⁱ_i > 0, lⁱ_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 l⁰ = (0, . . . , 0)_, lⁱ = (0, . . . ,l_i, . . . , 0), pⁱ = p,θⁱ =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)_andB^l_p,θ,a,

κ,τ(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⁽₂^0,0)TL⁽₂^3,1)TL⁽₂^1,4). One can look for the solution of equation (1.6) in the space W₂^(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| = _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.

2 Main 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, . . . ,λ_n), λ_j > 0, (j = 1, 2, . . . ,n), 1 ≤ pⁱ ≤ p ≤ _∞, 1 ≤ θⁱ ≤ _∞, (i=0, 1, 2, . . . ,n); ν= (ν₁,ν₂, . . . ,ν_n)_, ν_j ≥₀be entire(j=1, 2, . . . ,n)_;

1) ν_j ≥l⁰_j (j=1, 2, . . . ,n);

2) ν_j ≥lⁱ_j (j6=i,j=1, 2, . . . ,n)_, ν_i <l_iⁱ (j=i, i=1, 2, . . . ,n); 1≤τ₁ ≤τ₂ ≤_∞;κ= cκ^{, 1}_c =max_j=1,...,nκj

λj; f ∈ ^Tn_i=0L^<li>

pⁱ,θⁱa,κ,τ(G,λ)and µⁱ =

∑

lⁱ_jλ_j−ν_jλ_j− λ_j−κja_j 1

pⁱ − ¹ p

>0 (i=1, . . . ,n). Then D^ν : ^Tn_i=0L^<li>

pⁱ,θⁱ,a,κ,τ1(G,λ) ,→ L_p,b,κ,τ2(G). Precisely, for f ∈ ^Tn_i=0L^<_pi^l_,θ^i>_ia,

κ,τ1(G,λ)in the domain G there exists the generalized derivative D^νf , for which the the following inequalities are valid:

kD^νfk_p,G≤C₁

∑

T^µikfk

L^<_{pi,θi a,}^li>

κ,τ1

(G,λ), (2.1)

and

kD^νfk_p,b,κ,τ

2;G ≤C₂kfkn

T i=0

L^<_{pi,θi a,}^li>

κ,τ1

(G,λ) (pⁱ ≤ p< _∞). (2.2) In particular,µ^i,0 =_∑ⁿ_j=1lⁱ_jλ_j−ν_jλ_j− λ_j−κja_{j 1}

pⁱ

>_0,(_i=_{1, . . . ,n})_{, then D}^νf is continuous on G and

sup

x∈G

|D^νf(x)| ≤C₁

∑

T^µi,0kfk

L^<li>

pi,θi a,κ,τ1

(G,λ) (2.3)

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

0≤b_j ≤1, forµ^i,0>0, 0≤b_j <1, forµ^i,0=0, 0≤b_j <a_j+ ^µ

ip 1−a_j

n λ_j−κja_j, forµ^i,0<_0,

(2.4)

but withκ replaced byκ^{, C}1and 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 D^νf satisfies on G the Hölder condition in the metrics L_p with the exponent σ, more exactly,

k_∆(γ,G)D^νfk_p,G≤CkfkTn i=₀L^<li>

pi,θi,a,κ,τ(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≤σ≤ ^µ0

λ₀, for µ₀ λ₀ <1,

(2.6)

whereµ₀ = minµⁱ (i= 1, 2, . . . ,n), λ₀ = maxλ_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^νf(x)| ≤CkfkT_n i=0L^<li>

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

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

consider a problem of the form

|α|≤|

∑

|β|≤|lⁱ| i=1,2,...,n

D^α(a_αβ(x)D^βu(x)) =

∑

|α|≤|lⁱ|, i=1,2,...,n

D^αf_α(x), (2.8)

D^νu|_∂G= ϕ_ν|_∂G, (2.9) where it is assumed that G is a bounded n-dimensional domain with piecewise-smooth boundary ∂G, ν = (ν₁, . . . ,ν_n), where |ν| < |lⁱ|, i = 1, 2, . . . ,n while α = (α₁, . . . ,α_n),

|α|=α₁+α₂+. . .+α_n,β = (β₁, . . . ,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_αβ(x) ≡ a_βα(x)and forξ ∈ Rⁿ

|α|≤|

∑

|β|≤|lⁱ| i=1,2,...,n

(−1)^|α|a_α,β(x)ξ_αξ_β ≥C₀

∑

|α|≤|lⁱ| i=1,2,...,n

|ξ_α|², C₀=const>0, (2.10)

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

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

|α|≤|

∑

|β|≤|lⁱ|, i=1,2,...,n

Z

Ga_αβ(x)D^βu(x)D^αϑ(x)dx =

∑

|α|≤|lⁱ|

(−1)^|α|

Z

G f_αD^αϑ(x)dx. (2.11)

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

T_n

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

∑

|α|≤|lⁱ|,

|β|≤|lⁱ|, i=1,2,...,n

(−₁)^|α|

Z

G

a_αβ(x)D^βφ(x)D^αψ(x)dx

−

∑

|α|≤|lⁱ|

(−1)^|α|

Z

G

f_αD^αψ(x)dx−

∑

|β|≤|lⁱ|

(−1)^|β|

Z

G

f_βD^βφ(x)dx

= I(φ,ψ)−(f_α,ψ)−(f_β,φ), where

I(φ,φ) = I(φ)≥ kφk²_Tn

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^li>(G), l = (l₁, . . . ,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|≤|lⁱ|

(−1)^|α| _Z

G

|f_α|²dx+

Z

G

|D^αφ(x)|²dx

−

∑

|α|≤|lⁱ|

(−1)^|β| _Z

G

|f_β|²dx+

Z

G

|D^βφ(x)|²dx

≥ kφkT_n

i=0L^<₂^li>(G)−2kφkT_n

i=0L^<₂^li>(G)−d=−d₁.

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

T_n

i=0L^<₂^li>(G)such thatF(φ₀) =min

φ∈^Tn_i=0L^<₂^li>(G)F(φ). Indeed, letk=min_Φ∈Tn

i=0L^<₂^li>(G)F(φ). Fix some sequence φ_m ∈ ^Tn_i=0L^<₂^li>(G) such that lim_m→_∞F(φ_m) = 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(φ_m+µ+φ_m) ∈ ^Tn_i=0L^<₂^li>(G), we have F ^φm+µ₂^+φm

≥ k. Further, by direct calculations we show that I ^φm+µ₂^−φm

< 4ε. From the ellipticity condition (2.10) it follows that kφ_m+µ−φ_mkT_n

i=0L^<₂^li>(G) < 2q

ε

c₀. It means that the sequence {φ_m} in fundamental in the space ^Tn_i=0L^<₂^li>(G). Therefore, because of completeness in the space ^Tn_i=0L^<₂^li>(G) there exists the function φ₀∈^Tn_i=0L^<₂^li>(G)such that lim_m→_∞kφ_m−φ₀kTn

i=0L^<₂^li>(G) =0. By [7, The- orem 1], it is proved that |F(φ_m)−F(φ₀)| ≤Ckφ_m−φ₀kT_n

i=0L^<₂^li>(G), and hence it follows that k = _limm→∞F(φ_m) = F(φ₀). Show that the function delivering minimum to the functions F(φ)in the space^Tn_i=0L^<₂^li>(G)is unique and satisfies

D^νu|_∂G= ϕ_ν|_∂G. Indeed,φ∈^Tn_i=0L^<₂^li>(G)andF(φ₀) =k, we have:

0≤ I

φ−φ₀ 2

= ¹

2F(φ) + ¹

2F(φ₀)−F

φ+φ₀ 2

≤ ^k 2+ ^k

2−k=_0, I(φ−φ₀) =_0,

then again by the ellipticity condition (2.10),kφ_m−φ₀kTn

i=0L^<₂^li>(G) m→0

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

kD^ν(φ_m−φ₀)|_∂Gk_L

2(_∂G) ≤ kφ_m−φ₀kTn

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

kD^νφ_m|_∂G−φν|_∂Gk_L

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

kD^νφ₀|_∂G−φ_ν|_∂Gk_L

2(G)=0,

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

λ=0 = 0, show that the function φ₀ ∈ ^Tn_i=0L₂^<li>(G), minimizing the integral F(φ), satisfies the following equation:

I(φ₀,ψ)−(f_α,ψ) =0. (2.12) Now prove that the function φ₀ ∈^Tn_i=0L^<₂^li>(G), minimizing the integralF(φ)is the solution (generalized) of problem (2.8)–(2.9). For that we suppose thata_α,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;

Θ^(s) 1

2 +₀

=_Θ^(s)(₁−₀) =_{0 for any}s>_0.

The function

(Θ⁰(t), ¹₂ ≤t≤1,

0, −_∞<t ≤ ¹₂, 1≤t <_∞

is infinitely differentiable and finite over the whole axis. Letη>0 andG_η ={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,ψ∈^Tn_i=0L^<₂^li>(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 h_i

=D^β

a_α,β(x)D^αγ r

h_i

−(−1)^|α|fαD^αγ r

h_i

, (i=1, 2). The function ω _h^r

i

possesses all the properties of a kernel. Then for the function φ₀ (the solution of the variational problem) we can construct the Sobolev averagingφ_0,hi(x), i = 1, 2 over the ballh_i, (i=1, 2)centered at the pointx:

φ_0,hi(x) = ¹ σ_nhⁿ_i

Z

Rⁿω

|z−x| h_i

φ₀(z)dz, i=1, 2.

Then we can rewrite equality (2.12) in the form φ_0,h1(x) =φ_0,h2(x). Consequently, forh <η φ_0,h(x) =φ₀(x).

As the average function φ_0,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(φ_0,h,ψ)−(f_α,ψ) =0, in the limiting case

Z

G

∑

|α|≤|lⁱ|,

|β|≤|lⁱ| i=1,2,...,n

ψ(x)^hD^α

a_α,β(x)D^βφ₀(x)−D^αfα(x)ⁱdx =0.

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

|α|≤|

∑

|β|≤|lⁱ| i=1,2,...,n

D^α

a_α,β(x)D^βφ₀(x)=

∑

|α|≤|lⁱ|, i=1,2,...,n

D^αf_α(x).

Thus the solution of the variational problem from the class^Tn_i=0L^<₂^li>(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|= |lⁱ|(i= 1, 2, . . . ,n), 0<d <1, d= const,b≤d, x0 ∈G_d;G_dis a subdomain of the domain Gsuch that

G_d={x⁰⁰ :|x⁰⁰−x⁰_j|>d^λj, x⁰ ∈ ∂G, j=1, 2, . . . ,n}, and

Πb(x₀) ={x :|x_j−x_j,0|>b^λj, j=1, 2, . . . ,n}.

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

i=0L^<₂^li>(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| = |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

C_αx^α

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

Πb(x0)

∑

|α|=|β|=|lⁱ|, 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(_x0)

∑

|α|=|β|=|lⁱ|, i=1,2,...,n

(−₁)^|α|a_αβ(x)D^β((u(x)−P(x)))D^α((u(x)−P(x)))dx

= A(u(x)−P(x),Πb(x₀)), (2.13)

moreover,θ(x) = 1−_∏ⁿ_i=1ω_j ^x−x0

b^λj

,x ∈ G, whereω_j(t)∈ C^∞(_R)is such thatω_j(t)≡ 1 for

|t| <2^−λj,ω_j(t)≡ 0 for|t|> 1, 0≤ω_j(t)≤ 1. It is seen thatθ(x)≡ 0 inΠ^b

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))x^αdx=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|

b^{2|α|−2|li|}D^α(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),Π^b

2k (x₀)

≤

1− ¹ q

k

A(u(x),Πb(x₀)).

Let 0<δ < ^b

2^k,ζ =1−¹_q, then A(u(x),Πδ(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 anyδ≤ b, and consequently,

Z ₁

0

η^−ξ

Z

Πη(x0)u²dx ¹₂

dη η ≤C

Z ₁

0

db

b^1−12σ <_∞.

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

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

Z

Πb(x0)

∑

|α|=|lⁱ|, i=1,2,...,n

(D^αu_b,x0)²dx≤

∑

|α|=|lⁱ|, i=1,2,...,n

b^{2|li|−2|α|} Z

Πb(x0) f_α²dx+

∑

|α|=|lⁱ|, i=1,2,...,n

Z

Πb(x0)f_α²dx≤ C₁b^r, 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(u_b,x0,Πb(x0))≤C₁b^r. (2.16) Asu = u−u_b,x0 is the solution of homogeneous equation (2.8) then the following inequality is valid for it:

A(u,Πb(x₀))≤C₂ δ

b ξ−σ

A(u,G). (2.17)

From inequalities (2.16) and (2.17) we get

A(u,Πb(x₀))≤C₃A(u,Πb(x₀)) +C₃A(u_b,x0,Πb(x₀))

≤C₄ δ

b ξ−σ

A(u,¯ G) +C₅b^r≤C₆ δ

b ξ−σ

, and hence we get

Z ₁

0

η^−ξ

Z

Πb(x0)u²dx ¹₂

dη η ≤C

Z ₁

0

db

b^1−12σ <_∞.

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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