Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter October 11, 2016

Mixed type boundary value problems for polymetaharmonic equations

  • George Chkadua EMAIL logo

Abstract

In the paper we consider three-dimensional Riquier-type and classical mixed boundary value problems for the polymetaharmonic equation (Δ+k12)(Δ+k22)u=0. We investigate these problems by means of the potential method and the theory of pseudodifferential equations. We prove the existence and uniqueness theorems in Sobolev–Slobodetskii spaces, analyse the asymptotic properties of solutions and establish the best Hölder smoothness results for solutions.

A Properties of potentials

For the readers’ convenience, we collect here some results describing the properties of the layer potentials. For the potentials associated with the Helmholtz equation the following theorems hold; see [1, 6, 12, 11, 26, 21, 23].

Theorem A.1

Let sR. Then the single and double layer potentials can be extended to the following continuous operators:

Vj:Hs(S)Hs+3/2(Ω+),Vj:Hs(S)Hlocs+3/2(Ω-),
Wj:Hs(S)Hs+1/2(Ω+),Wj:Hs(S)Hlocs+1/2(Ω-).

Theorem A.2

Let sR. Then the following operators are continuous:

j:Hs(S)Hs+1(S),𝒦j,𝒦j*:Hs(S)Hs+1(S),j:Hs(S)Hs-1(S).
Theorem A.3

Let sR and kj a complex wave number, Imkj0. Then the following operators are invertible:

j:Hs(S)Hs+1(S),j:Hs(S)Hs-1(S).

Proof.

First we note that the operator j:H-1/2(S)H1/2(S) is a pseudodifferential operator of order -1; see Theorem A.2. Now we show that the operator j:H-1/2(S)H1/2(S) is invertible. Indeed, the principal homogeneous symbol of the operator j is 𝔖j(ξ)=-12|ξ|, where ξ2{0}. Hence the operator j is a strongly elliptic pseudodifferential operator. Therefore from the theory of pseudodifferential operators on the manifold S without boundary (see [14, 16]) it follows that the operator j:H-1/2(S)H1/2(S) is Fredholm with zero index.

Let us show that the operator j:H-1/2(S)H1/2(S) is injective. Indeed, let φH-1/2(S) be a solution of the homogeneous equation

jφ=0on S.

Write Green’s formulae in the domains Ω± as

(A.1)Ω±(Δ+kj2)uju¯j𝑑x+Ω±|uj|2𝑑x-kj2Ω±|uj|2𝑑x=±{nuj}±,{u¯j}±S,

where ujH1(Ω±,Δ). By summing Green’s formulae (A.1) and substituting uj=Vjφ, we obtain

(A.2)Ω+|(Vjφ)|2𝑑x+Ω-|(Vjφ)|2𝑑x-kj2Ω+|Vjφ|2𝑑x-kj2Ω-|Vjφ|2𝑑x=-φ,j¯(φ)S.

Since Imkj0, from (A.2) we have

Ω+|Vjφ|2𝑑x+Ω-|Vjφ|2𝑑x=0,

whence Vjφ=0 in 3 follows. Using the jump relation in Theorem 3.1, we have

{nVj(φ)}--{nVj(φ)}+=φ=0on S.

Therefore the pseudodifferential operator j:Hs(S)Hs+1(S) is invertible for all s; see [2, Chapter 3, Proposition 10.6].

Analogously, we can prove the invertibility of the operator j. ∎

Theorem A.4

The operator

-12I+𝒦j*+μj:Hs(S)Hs(S)

is invertible for all sR provided Imμ0.

For the potentials of the polymetaharmonic equation (2.8) the following theorems hold.

Theorem A.5

Let sR. Then the layer potentials V(k), k=0,1,2,3, can be extended to the following continuous operators:

𝐕(k):Hs(S)Hs-k+7/2(Ω+),𝐕(k):Hs(S)Hlocs-k+7/2(Ω-),k=0,1,2,3.
Theorem A.6

Let fH3/2(S), ψH-1/2(S). Then the following relations hold on the manifold S:

(A.3){𝐁1𝐕(3)(f)}+={𝐁1𝐕(3)(f)}-on S,
(A.4){𝐁3𝐕(1)(ψ)}+={𝐁3𝐕(1)(ψ)}-on S.

Proof.

Substituting the fundamental solution (2.14) of the polymetaharmonic equation (2.8) in Green’s formula (5.1) instead of v, we can derive a representation formula for a function uH2(Ω+),

(A.5)u=𝐏(F)-𝐕(0){𝐁3u}+-𝐕(1){𝐁2u}++𝐕(2){𝐁1u}++𝐕(3){𝐁0u}+in Ω+,

and

(A.6)0=𝐏(F)-𝐕(0){𝐁3u}+-𝐕(1){𝐁2u}++𝐕(2){𝐁1u}++𝐕(3){𝐁0u}+in Ω-,

where 𝐏(F) is the volume potential

𝐏(F)(x)=Ω+H(x-y)F(y)𝑑y,x3,F:=(Δ+k12)(Δ+k22)u.

From the asymptotic properties of the fundamental solution H near the origin it follows that, for FL2(Ω+),

(A.7){𝐁i𝐏(F)}+={𝐁i𝐏(F)}-on S,i=0,1,2,3.

Let uH2(Ω+) be a solution of the following Dirichlet problem for the polyharmonic equation (see [29]):

Δ2u=0in Ω+,
{𝐁0u}+=f,{𝐁1u}+=gon S,

where fH3/2(S), gH1/2(S).

Denote

χ:={𝐁2u}+H-1/2(S),φ:={𝐁3u}+H-3/2(S),G:=(k12+k22)Δu+k12k22uL2(Ω+).

Then we can rewrite formulae (A.5) and (A.6) as

𝐕(3)(f)=-𝐏(G)+u+𝐕(0)(φ)+𝐕(1)(χ)-𝐕(2)(g)in Ω+,
𝐕(3)(f)=-𝐏(G)+𝐕(0)(φ)+𝐕(1)(χ)-𝐕(2)(g)in Ω-.

Therefore using equalities (3.2), (3.3) and (A.7), we derive the following relations:

{𝐁1𝐕(3)(f)}+=-{𝐁1𝐏(G)}++{𝐁1u}++{𝐁1𝐕(0)(φ)}++{𝐁1𝐕(1)(χ)}+-{𝐁1𝐕(2)(g)}+
=-{𝐁1𝐏(G)}++g+{𝐁1𝐕(0)(φ)}++{𝐁1𝐕(1)(χ)}+-12g-𝒦1,2(g)
=-{𝐁1𝐏(G)}-+{𝐁1𝐕(0)(φ)}-+{𝐁1𝐕(1)(χ)}-+12g-𝒦1,2(g)on S,
{𝐁1𝐕(3)(f)}-=-{𝐁1𝐏(G)}-+{𝐁1𝐕(0)(φ)}-+{𝐁1𝐕(1)(χ)}-+12g-𝒦1,2(g)on S.

Hence relation (A.3) holds.

Let us introduce the notations

𝐊3,1±={𝐁3V(1)(ψ)}±,𝐊1,3±={𝐁1V(3)(f)}±,𝐊¯1,3±={𝐁1V¯(3)(f)}±on S,

where fH3/2(S), ψH1/2(S), and

V¯(3)(f)(x)=S𝐁3(y)H(x-y)¯ψ(y)dyS,xS,
H(x)¯=-1k22-k12e-ik1|x|4π|x|-1k12-k22e-ik2|x|4π|x|.

Note that 𝐊3,1±, 𝐊¯1,3± are pseudodifferential operators of order 1, implying that the operators

𝐊3,1±:Hs(S)Hs-1(S),𝐊¯1,3±:Hs(S)Hs-1(S)

are bounded for all s. It is easy to see that the pseudodifferential operators

𝐊3,1±:H1/2(S)H-1/2(S),𝐊¯1,3±:H1/2(S)H-1/2(S)

are mutually conjugate, i.e. 𝐊3,1±=(𝐊¯1,3±)*. Hence from the equalities 𝐊1,3+=𝐊1,3- and (𝐊¯1,3+)*=(𝐊¯1,3-)* it follows that 𝐊3,1+=𝐊3,1-. Therefore relation (A.4) holds. ∎

Theorem A.7

Let fH3/2(S), gH1/2(S). Then the following relations hold on the manifold S:

(A.8){𝐁2𝐕(3)(f)}++{𝐁2𝐕(2)(g)}+={𝐁2𝐕(3)(f)}-+{𝐁2𝐕(2)(g)}-,
(A.9){𝐁3𝐕(3)(f)}++{𝐁3𝐕(2)(g)}+={𝐁3𝐕(3)(f)}-+{𝐁3𝐕(2)(g)}-.

Proof.

Let uH2(Ω+) be a solution of the Dirichlet problem for the polyharmonic equation

Δ2u=0in Ω+,
{𝐁0u}+=f,{𝐁1u}+=gon S.

Denote

ψ:={𝐁2u}+H-1/2(S),φ:={𝐁3u}+H-3/2(S),G:=(k12+k22)Δu+k12k22uL2(Ω+).

Then we can rewrite formulae (A.5) and (A.6) as

-𝐕(2)(g)-𝐕(3)(f)=𝐏(G)-u-𝐕(0)(φ)-𝐕(1)(ψ)in Ω+,
-𝐕(2)(g)-𝐕(3)(f)=𝐏(G)-𝐕(0)(φ)-𝐕(1)(ψ)in Ω-.

Therefore using equalities (3.1), (3.3) and (A.7), we derive the relations

-{𝐁2𝐕(3)(f)}+-{𝐁2𝐕(2)(g)}+={𝐁2𝐏(G)}+-{𝐁2u}+-{𝐁2𝐕(0)(φ)}+-{𝐁2𝐕(1)(ψ)}+
={𝐁2𝐏(G)}--12ψ-𝒦2,1(ψ)-{𝐁2𝐕(0)(φ)}-,
-{𝐁2𝐕(3)(f)}--{𝐁2𝐕(2)(g)}-={𝐁2𝐏(G)}--12ψ-𝒦2,1(ψ)-{𝐁2𝐕(0)(φ)}-.

Hence relation (A.8) follows.

Further, using equalities (3.1), (3.3), (A.4) and (A.7), we derive the relations

-{𝐁3𝐕(3)(f)}+-{𝐁3𝐕(2)(g)}+={𝐁3𝐏(G)}+-{𝐁3u}+-{𝐁3𝐕(0)(φ)}+-{𝐁3𝐕(1)(ψ)}+
={𝐁3𝐏(G)}--12φ-𝒦3,0(φ)-{𝐁3𝐕(1)(ψ)}-,
-{𝐁3𝐕(3)(f)}--{𝐁3𝐕(2)(g)}-={𝐁3𝐏(G)}--12φ-𝒦3,0(φ)-{𝐁3𝐕(1)(ψ)}-.

Therefore (A.9) holds. ∎

The following assertions hold.

Theorem A.8

The operators Ki,j, i+j=3, i,j=0,1,2,3, are pseudodifferential operators of order -1, i.e., the operators

𝒦i,j:Hs(S)Hs+1(S)

are continuous for all sR.

Theorem A.9

The operators

𝐊i,j:Hs(S)Hs+3-(i+j)(S),i+j<3,i,j=0,1,2 and i+j=4,i,j=1,3,
𝐊i,j±:Hs(S)Hs+3-(i+j)(S),i,j=2,3,

are continuous for all sR.

B Auxiliary propositions

The fundamental solution of the differential operator (Δ-k12)(Δ-k22) for all k1,k2(0,+) has the form

H~(x)=1k22-k12(e-k1|x|4π|x|-e-k2|x|4π|x|).

The difference H-H~ has the following behaviour in a neighbourhood of the origin:

(B.1)H(x)-H~(x)=i-14π(k1+k2)+O(|x|2),
(B.2)xα[H(x)-H~(x)]=O(|x|2-|α|),α=(α1,α2,α3),|α|>0.

Let us introduce the layer potentials

𝐕~(j)(h)(x):=S𝐁~j(y)H~(x-y)h(y)dyS,j=0,1,2,3,

where 𝐁~j:=𝐁j for j=0,1,2, and 𝐁~3:=2nΔ-n3-(k12+k22)n. Denote

𝐊~i,j(hj):={𝐁~i𝐕~(j)(hj)}±,i+j<3,i,j=0,1,2 and i=1,j=3 or j=1,i=3,
𝐊~i,j±(hj):={𝐁~i𝐕~(j)(hj)}±,i,j=2,3.

In view of estimates (B.1) and (B.2) it follows that the principal parts of the operators 𝐊i,j± and 𝐊~i,j± (also for 𝐊i,j and 𝐊~i,j) are the same, therefore the operators

𝐊i,j±-𝐊~i,j±:Hs(S)Hs+3-(i+j)(S),𝐊i,j-𝐊~i,j:Hs(S)Hs+3-(i+j)(S)

are compact for all s. Hence the operator

𝒫μ-𝒫~:H3/2(S)×H1/2(S)H3/2(S)×H1/2(S)

is compact, where

𝒫~:=(-12I+𝒦~0,3𝐊~0,2𝐊~1,3-12I+𝒦~1,2),μ=μ1+iμ2,
𝒦~i,j(h)(z):=S𝐁~i(z)𝐁~j(y)H~(z-y)hj(y)dyS,zS,j=0,1,2,3,i+j=3.

Theorem B.1

The operator

𝒫~:H3/2(S)×H1/2(S)H3/2(S)×H1/2(S)

is invertible.

Proof.

Since 𝒫μ is an invertible operator, the operator 𝒫~ is Fredholm with zero index.

Let Φ=(φ,ψ)H3/2(S)×H1/2(S) be a solution of the homogeneous equation

𝒫~Φ=0.

Introduce the function

v=𝐕~(3)φ+𝐕~(2)ψ.

Then (cf. Theorems 3.2, A.6 and A.7)

{𝐃v}-=({𝐁0v}-,{𝐁1v}-)=𝒫~Φ=0on S.

Therefore v is a solution of the homogeneous Dirichlet problem

(B.3)(Δ-k12)(Δ-k22)v=0in Ω-,
(B.4){𝐁0v}-=0,{𝐁1v}-=0on S.

It follows from the uniqueness of the solution of the Dirichlet problem (B.3)–(B.4) that v=0 in Ω-. Due to the jump formulae on S (cf. again Theorems 3.2, A.6 and A.7),

{𝐁0v}+-{𝐁0v}-=φ,{𝐁1v}+-{𝐁1v}-=ψon S,
{𝐁2v}+-{𝐁2v}-=0,{𝐁~3v}+-{𝐁~3v}-=0on S,

we arrive at the equations

(B.5){𝐁0v}+=φ,{𝐁1v}+=ψon S,
(B.6){𝐁2v}+=0,{𝐁~3v}+=0on S.

Taking into account (B.5) and (B.6) in Green’s formula (2.5), we obtain the equality

-(k12+k22)Ω+|v|2𝑑x-k12k22Ω+|v|2𝑑x-Ω+|α|=22α!|Dαv|2dx=0.

Hence v=0 in Ω+, and it follows that φ=0 and ψ=0 on S. Consequently, the operator

𝒫~:H3/2(S)×H1/2(S)H3/2(S)×H1/2(S)

is invertible. ∎

Theorem B.2

The operator

𝒜~:=(𝐊~3,3-𝐊~3,2-𝐊~2,3-𝐊~2,2-)𝒫~-1:H3/2(S)×H1/2(S)H-3/2(S)×H-1/2(S)

is a positive-definite operator, i.e.

𝒜~Φ,Φ¯ScΦH3/2(S)×H1/2(S)2for all ΦH3/2(S)×H1/2(S),

where the symbol ,S denotes the duality between H-3/2(S)×H-1/2(S) and H3/2(S)×H1/2(S), and c is a positive constant.

Proof.

Let Φ=(φ,ψ)H3/2(S)×H1/2(S). Substituting the functions u=(𝐕~(3),𝐕~(2))𝒫~-1Φ and v=u¯ in Green’s formula for the differential operator (Δ-k12)(Δ-k22) (cf. (2.7)),

Ω-(Δ-k12)(Δ-k22)uv𝑑x-(k12+k22)Ω-uvdx-k12k22Ω-uv𝑑x-Ω-|α|=22α!DαuDαvdx
=-{𝐍~u}-,{𝐃v}-S,

where 𝐍~u:=(𝐁~3u,𝐁2u), we obtain the equality

(k12+k22)Ω-|u|2𝑑x+k12k22Ω-|u|2𝑑x+Ω-|α|=22α!|Dαu|2dx=𝒜~Φ,Φ¯S.

Therefore for some positive constants c and c we get

𝒜~Φ,Φ¯ScuH2(Ω-)2
c{(𝐁0u,𝐁1u)}-H3/2(S)×H1/2(S)2
=c{𝐃u}-H3/2(S)×H1/2(S)2
=c{𝐃(𝐕~(3),𝐕~(2))𝒫~-1Φ}-H3/2(S)×H1/2(S)2
=cΦH3/2(S)×H1/2(S)2for all ΦH3/2(S)×H1/2(S).

Corollary B.3

The operator

𝒜:=rS2𝒜~:H~3/2(S2)×H~1/2(S2)H-3/2(S2)×H-1/2(S2)

is a positive-definite operator, i.e.

𝒜Ψ,Ψ¯S2cΨH~3/2(S2)×H~1/2(S2)2for all ΨH~3/2(S2)×H~1/2(S2),

where the symbol ,S2 denotes the duality between H-3/2(S)×H-1/2(S) and H~3/2(S)×H~1/2(S), and c is a positive constant.

References

[1] Abboud T. and Starling F., Scattering of an electromagnetic wave by a screen, Boundary Value Problems and Integral Equations in Nonsmooth Domains, Lect. Notes Pure Appl. Math. 167, Marcel Decker, New York (1993), 1–17. Search in Google Scholar

[2] Agranovich M., Elliptic singular integro-differential operators (in Russian), Uspekhi Mat. Nauk 20 (1965), no. 5(125), 3–120. 10.1070/RM1965v020n05ABEH001190Search in Google Scholar

[3] Begehr H., Vu T. N. H. and Zhang Z.-X., Polyharmonic Dirichlet problems (in Russian), Tr. Mat. Inst. Steklova 255 (2006), 19–40; translation in Proc. Steklov Inst. Math. 255 (2006), 13–34. 10.1134/S0081543806040031Search in Google Scholar

[4] Brakhage H. and Werner P., Über das Dirichletsche Aussenraumproblem für die Helmholtzsche Schwingungsgleichung, Arch. Math. 16 (1965), 325–329. 10.1007/BF01220037Search in Google Scholar

[5] Brenner A. V. and Shargorodsky E. M., Boundary value problems for elliptic pseudodifferential operators, Partial Differential Equations. IX: Elliptic Boundary Value Problems, Encyclopaedia Math. Sci. 79, Springer, Berlin (1997), 145–215. 10.1007/978-3-662-06721-5_2Search in Google Scholar

[6] Buchukuri T., Chkadua O., Duduchava R. and Natroshvili D., Interface crack problems for metallic-piezoelectric composite structures, Mem. Differ. Equ. Math. Phys. 55 (2012), 1–150. Search in Google Scholar

[7] Chkadua G., Screen-type boundary value problems for polymetaharmonic equations, Math. Methods Appl. Sci. 36 (2013), no. 3, 358–372. 10.1002/mma.2602Search in Google Scholar

[8] Chkadua O. and Duduchava R., Asymptotics of functions represented by potentials, Russ. J. Math. Phys. 7 (2000), no. 1, 15–47. Search in Google Scholar

[9] Chkadua O. and Duduchava R., Pseudodifferential equations on manifolds with boundary: Fredholm property and asymptotic, Math. Nachr. 222 (2001), 79–139. 10.1002/1522-2616(200102)222:1<79::AID-MANA79>3.0.CO;2-3Search in Google Scholar

[10] Chow S.-N. and Dunninger D. R., A maximum principle for n-metaharmonic functions, Proc. Amer. Math. Soc. 43 (1974), 79–83. Search in Google Scholar

[11] Colton D. and Kress R., Integral Equation Methods in Scattering Theory, Pure Appl. Math. (New York), John Wiley & Sons, New York, 1983. Search in Google Scholar

[12] Colton D. and Kress R., Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Appl. Math. Sci. 93, Springer, Berlin, 1998. 10.1007/978-3-662-03537-5Search in Google Scholar

[13] Costabel M., Dauge M. and Duduchava R., Asymptotics without logarithmic terms for crack problems, Comm. Partial Differential Equations 28 (2003), no. 5–6, 869–926. 10.1081/PDE-120021180Search in Google Scholar

[14] Eskin G. I., Boundary Value Problems for Elliptic Pseudodifferential Equations, Transl. Math. Monogr. 52, American Mathematical Society, Providence, 1981. Search in Google Scholar

[15] Gunther N. M., La Théorie du Potentiel et ses Applications aux Problemes Foundamentaux de la Physique Mathématique, Gauthier-Villars, Paris, 1934. Search in Google Scholar

[16] Hörmander L., The Analysis of Linear Partial Differential Operators. III. Pseudodifferential Operators, Grundlehren Math. Wiss. 274, Springer, Berlin, 1985. Search in Google Scholar

[17] Hsiao G. C. and Wendland W. L., Boundary Integral Equations, Appl. Math. Sci. 164, Springer, Berlin, 2008. 10.1007/978-3-540-68545-6Search in Google Scholar

[18] Jentsch L., Natroshvili D. and Wendland W. L., General transmission problems in the theory of elastic oscillations of anisotropic bodies (basic interface problems), J. Math. Anal. Appl. 220 (1998), no. 2, 397–433. 10.1006/jmaa.1997.5764Search in Google Scholar

[19] Jentsch L., Natroshvili D. and Wendland W. L., General transmission problems in the theory of elastic oscillations of anisotropic bodies (mixed interface problems), J. Math. Anal. Appl. 235 (1999), no. 2, 418–434. 10.1006/jmaa.1999.6360Search in Google Scholar

[20] Lions J.-L. and Magenes E., Non-Homogeneous Boundary Value Problems and Applications. Vol. 1, Grundlehren Math. Wiss. 181, Springer, New York, 1972. 10.1007/978-3-642-65217-2_1Search in Google Scholar

[21] McLean W., Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. Search in Google Scholar

[22] Natroshvili D., Boundary integral equation method in the steady state oscillation problems for anisotropic bodies, Math. Methods Appl. Sci. 20 (1997), no. 2, 95–119. 10.1002/(SICI)1099-1476(19970125)20:2<95::AID-MMA839>3.0.CO;2-RSearch in Google Scholar

[23] Natroshvili D., Chkadua O. and Shargorodsky E., Mixed problems for homogeneous anisotropic elastic media (in Russian), Tr. Inst. Prikl. Mat. Im. I. N. Vekua 39 (1990), 133–181. Search in Google Scholar

[24] Natroshvili D. G., Dzhagmaidze A. Y. and Svanadze M. Z., Some Problems in the Linear Theory of Elastic Mixtures (in Russian), Tbilisi University Press, Tbilisi, 1986. Search in Google Scholar

[25] Nedelec J. C., Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems, Appl. Math. Sci. 144, Springer, New York, 2001. 10.1007/978-1-4757-4393-7Search in Google Scholar

[26] Paneah B., On existence and uniqueness of the solution of n-metaharmonic equation in unbounded space (in Russian), Mosc. State Univ. Vestnic Math. Mech. 5 (1959), 123–135. Search in Google Scholar

[27] Riquier C., Sur quelques problèmes relatits a l’équation aux dérivées parteilles (x2+y2)nu=0, J. Math. Pures Appl. (9) 5 (1926), 297–393. Search in Google Scholar

[28] Shargorodsky E., An Lp-analogue of the Vishik–Eskin theory, Mem. Differ. Equ. Math. Phys. 2 (1994), 41–146. Search in Google Scholar

[29] Sobolev S. L., On a boundary value problem for polyharmonic equations (in Russian), Mat. Sb. N. S. 2(44) (1937), 465–499; translation in Amer. Math. Soc. Transl. (2) 33 (1963), 1–40. 10.1090/trans2/033/01Search in Google Scholar

[30] Sobolev S. L., Cubature Formulas and Modern Analysis. An Introduction, Gordon and Breach Science Publishers, Montreux, 1992. Search in Google Scholar

[31] Stephan E. P., Boundary integral equations for mixed boundary value problems in 𝐑3, Math. Nachr. 134 (1987), 21–53. 10.1002/mana.19871340103Search in Google Scholar

[32] Tavkhelidze I., On some properties of solutions of polyharmonic equation in polyhedral angles, Georgian Math. J. 14 (2007), no. 3, 565–580. 10.1515/GMJ.2007.565Search in Google Scholar

[33] Triebel H., Interpolation Theory, Function Spaces, Differential Operators, 2nd ed., Johann Ambrosius Barth, Heidelberg, 1995. Search in Google Scholar

[34] Vekua I., On metaharmonic functions (in Russian), Tr. Math. Inst. Razmadze 12 (1943), 105–174. Search in Google Scholar

[35] Vu T. N. H., Helmholtz operator in quaternionic analysis, Ph.D. thesis, FU Berlin, 2005. Search in Google Scholar

Received: 2015-9-16
Accepted: 2016-3-11
Published Online: 2016-10-11
Published in Print: 2016-12-1

© 2016 by De Gruyter

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.1515/gmj-2016-0042/html
Scroll to top button