Polyharmonic polynomials and mixed boundary value problems in the Heisenberg group H n
aa r X i v : . [ m a t h . A P ] J un POLYHARMONIC NEUMANN AND MIXED BOUNDARYVALUE PROBLEMS IN THE HEISENBERG GROUP H n SHIVANI DUBEY, AJAY KUMAR* AND MUKUND MADHAV MISHRA
Abstract.
We study the polyharmonic Neumann and mixed bound-ary value problems on the Kor´anyi ball in the Heisenberg group H n .Necessary and sufficient solvability conditions are obtained for the non-homogeneous polyharmonic Neumann problem and Neumann-Dirichletboundary value problems. Introduction
As we know, by convolution, the expressions of iterated polyharmonicGreen and polyharmonic Neumann functions can not be easily constructedin explicit form because of complicated computation. Rewriting the higherorder Poisson equation ∆ m u = f in a plane domain as a system of Pois-son equations it is immediately clear what boundary conditions may beprescribed in order to get (unique) solutions. Neumann conditions for thePoisson equation lead to higher-order Neumann (Neumann-m) problems for∆ m u = f . Extending the concept of Neumann functions for the Lapla-cian to Neumann functions for powers of the Laplacian leads to an explicitrepresentation of the solution to the Neumann-m problem for ∆ m u = f .The representation formula provides the tool to treat more general partialdifferential equations with leading term ∆ m u in reducing them into somesingular integral equations. Integral representations for solutions to higherorder differential equations can be obtained by iterating those representa-tions formulas for first order equations.In case of the unit disk D of the complex plane C the higher order Greenfunction is explicitly known, see e.g. [1]. Although this is not true forthe higher-order Neumann function nevertheless it can be constructed itera-tively. A large number of investigations on various boundary value problemsin the Heisenberg group have widely been published [2, 9, 12]. However, thepolyharmonic Dirichlet problem on the Heisenberg group appeared [14]. Af-ter discussing the Neumann problem for Kohn-Laplacian on the HeisenbergGroup H n in [5], we consider here the polyharmonic Neumann problem withcircular data.In the next section of this article, we study the basic terminologies whichhave laid the foundation for harmonic analysis on the Heisenberg group. In Mathematics Subject Classification.
Key words and phrases.
Heisenberg group; sub-Laplacian; Kelvin transform; Kor´anyiball; Green’s function.*Corresponding author. Email: [email protected]. section 3, we discuss the solvability of the polyharmonic Neumann prob-lem and a representation formula for a solution of the polyharmonic Neu-mann problem on the Kor´anyi ball in H n is given. Section 4 deals with thepolyharmonic mixed boundary value problems like Neumann-Dirichlet andDirichlet-Neumann on the Kor´anyi ball in H n .2. The Heisenberg group H n and Horizontal normal vectors H n and the Kohn-Laplacian.The underlying set of the Heisenberg group is C n × R with multiplicationgiven by[ z, t ] . [ ζ, s ] = [ z + ζ, t + s + 2 ℑ ( z. ¯ ζ )] , ∀ [ z, t ] , [ ζ, s ] ∈ C n × R . With this operation and the usual C ∞ structure on C n × R (= R n × R ) , itbecomes a Lie group denoted by H n . A basis of left invariant vector fieldson H n is given by { Z j , ¯ Z j , T : 1 ≤ j ≤ n } where, Z j = ∂ z j + i ¯ z j ∂ t ;¯ Z j = ∂ ¯ z j − iz j ∂ t ; T = ∂ t . If we write z j = x j + iy j and define, X j = ∂ x j + 2 y j ∂ t ,Y j = ∂ y j − x j ∂ t , then, Z j = 12 ( X j − iY j ) , and { X j , Y j , T } is a basis.Given a differential operator D on H n , we say that it is left invariant if itcommutes with left translation L g for all g ∈ H n and rotation invariant if itcommutes with rotations R θ for all θ ∈ U ( H n ) , the unitary group H n . It issaid to be homogeneous of degree α if D ( f ( δ r g )) = r α Df ( δ r g ) ,δ r being the Heisenberg dilation defined by δ r [ z, t ] = [ rz, r t ] . It can beshown that up to a constant multiple, there is a unique left invariant, rota-tion invariant differential operator that is homogeneous of degree 2 [3, 15].This unique operator is called the sub-Laplacian or Kohn-laplacian on theHeisenberg group. The sublaplacian L on H n is explicitly given by L = − n X j =1 (cid:0) X j + Y j (cid:1) . Let L denote the slightly modified subelliptic operator − L .The fundamental solution for L on H n with pole at identity is given in [4]as g e ( ξ ) = g e ([ z, t ]) = a ( | z | + t ) − n , where, a = 2 n − (Γ( n )) π n +1 , is the normalization constant and ξ = [ z, t ]. The fundamental solution of L with pole at η is given by OLYHARMONIC NEUMANN PROBLEM 3 g η ( ξ ) = g e ( η − ξ ) . For a function f on H n , the Kelvin transform is defined as in [10] by Kf = N − n f oh, where h is the inversion defined as h ([ z, t ]) = (cid:20) − z | z | − it , − t | z | + t (cid:21) , for [ z, t ] ∈ H n \ { e } . From [13, (3.3)] we have, for η = e ∈ H n K ( g η ) = N ( η ) − n g η ∗ , where we wrote η ∗ for h ( η ).An integrable function f on H n is called circular if it is invariant under circleaction i.e., f = ¯ f = 12 π Z π f ([ e iθ z, t ]) dθ. M ) as follows. On the linear span of X j , Y j (1 ≤ j ≤ n )we define an inner product h ., . i by the condition that the vectors X j , Y j form an orthonormal system. For vectors not in this span we say that theyhave infinite length. A vector is said to be horizontal if it has finite length.The horizontal gradient ∇ F of a function F on H n is defined as the uniquehorizontal vector such that h∇ F, V i = V.F, where
V.F is action of V on F , for all horizontal vectors V . ∇ F can beexplicitly written as ∇ F = X j { ( X j F ) X j + ( Y j F ) Y j } . If a hypersurface in H n is given as the level set of a smooth function F , atany regular point i.e. a point at which ∇ F = 0 the horizontal normal unitvector will be defined by ∂∂n := 1 ||∇ F || ∇ F. More precisely, this is the horizontal normal pointing outwards for the do-main { F < } . A point ζ on the boundary surface is termed “characteristic”if ∇ F ( ζ ) = 0 . For smooth F , the set of characteristic points form a lowerdimensional (and hence of measure zero) subset of the boundary.From here onwards the operators ∂∂n and L , whenever applied to a functionwill be with respect to the variable ξ only.3. Polyharmonic Neumann Boundary Value Problem
In this section we solve the higher-order Neumann problems for L m u = f to get an explicit representation of the solution. Let S denotes the set ofcharacteristic points of B = { [ z, t ] ∈ H n : ( | z | + t ) / ≤ } . S. DUBEY, A. KUMAR AND M. M. MISHRA
We define, for each positive integer k , the function N k ( η, ξ ) inductively. Let N ( η, ξ ) = N ( η, ξ ) , and when N k − ( η, ξ ) is defined, define N k ( η, ξ ) = Z B N k − ( η, ζ ) N ( ζ, ξ ) dζ, (3.1)where N ( η, ξ ) is the Neumann function for B given in [5] as N ( η, ξ ) = g η ( ξ ) + K ( g η )( ξ − ) + h ( η, ξ ) , (3.2) h ( η, ξ ) = ∞ X k =1 ∞ X m =1 n k X j =1 nm a m ; k C ( n + k, n + k ) ( m − k ) ( t + i | z | ) Y k ; j ( z ) C ( n + k, n + k ) m ( t ′ + i | z ′ | ) Y k ; j ( z ′ ) + b ,b is a constant, n k , is dimension of the space H k = H k,k where the space H k,l of complex (solid) spherical harmonics of bidegree ( k, l ) on C n consistsof all polynomials P in z , . . . z n , ¯ z , . . . ¯ z n , homogeneous of degree k in the z ′ j s and homogeneous of degree l in the ¯ z ′ j s satisfying n X j =1 ∂ ∂z j ∂ ¯ z j P = 0 . For each k, l the Y k,l ; j ′ s form a basis for H k,l , where Y k,l ; j ′ s have the form Y k,l ; j ( z ) = r X q =0 c q | z ∗ | q z k − q (¯ z ) l − q ,z = ( z , z ∗ ) ∈ C n , | z ∗ | = | z | + . . . + | z n | , where , r = min( k, l ) and c , . . . , c r are constants, c = 1 and c q ′ s are determined by the relation( k − q )( l − q ) c q + ( q + 1)( n + q − c q +1 = 0 , ≤ q < r, for detailed study of spherical harmonics refer to [8].The only possible points of singularities of integrands in (3.1) are at ζ = η and ζ = ξ . Estimating near the points of singularities we show that integralon the right-hand side of (3.1) is convergent and this kernel is well defined. Lemma 3.1.
The functions N k as defined in (3.1) are integrable as func-tions of ξ and hence are defined finitely a.e.Proof. We first show that N = N is integrable over B . Using polar co-ordinates in H and the fact that measure on B is translation invariant, itfollows that g η ( ξ ) is integrable. The second term in (3.2) is integrable as itis equal to N ( η ) − n g η ∗ ( ξ − ) and h ( η, ξ ) being continuous function over B so the function N ( η, ξ ) is integrable over B .Next we claim that N ( η, ξ ) is integrable by writing N ( η, ξ ) as sum of nine OLYHARMONIC NEUMANN PROBLEM 5 integrals I i (1 ≤ i ≤ I ( ζ ) = Z B g η ( ξ ) g ξ ( ζ ) dξ, I ( ζ ) = N ( η ) − n Z B g η ∗ ( ξ − ) g ξ ( ζ ) dξ,I ( ζ ) = Z B h ( η, ξ ) g ξ ( ζ ) dξ, I ( ζ ) = Z B g η ( ξ ) N ( ξ ) − n g ξ ∗ ( ζ − ) dξ,I ( ζ ) = N ( η ) − n Z B g η ∗ ( ξ − ) N ( ξ ) − n g ξ ∗ ( ζ − ) dξ,I ( ζ ) = Z B h ( η, ξ ) N ( ξ ) − n g ξ ∗ ( ζ − ) dξ,I ( ζ ) = Z B g η ( ξ ) h ( ξ, ζ ) dξ, I ( ζ ) = N ( η ) − n Z B g η ∗ ( ξ − ) h ( ξ, ζ ) dξ,I ( ζ ) = Z B h ( η, ξ ) h ( ξ, ζ ) dξ. (3.3) I ( ζ ) exists for almost all ζ as R B g ξ ( ζ ) is uniformly bounded. Similarly I ( ζ )and I ( ζ ) exist for almost all ζ . Also I ( ζ ) ≤ N ( η ) − n sup ξ ∈ B g η ∗ ( ξ ) Z B g ξ ( ζ ) dξ, so exists. Similarly I ( ζ ) exists for almost all ζ . Next, I ( ζ ) ≤ sup ξ ∈ B g ξ ∗ ( ζ ) Z B g η ( ξ ) N ( ξ ) − n dξ, and Z B g η ( ξ ) N ( ξ ) − n dξ ≤ " sup B \ ( B ∪ B ) g η ( ξ ) N ( ξ ) − n ν ( B \ ( B ∪ B ))+ sup B g η ( ξ ) Z B N ( ξ ) − n dξ + sup B N ( ξ ) − n Z B g η ( ξ ) dξ, where ν is the Haar measure on the Heisenberg group and B , B are disjointsmall neighbourhoods of e and η respectively. Applying dominated conver-gence theorem and Fubini’s theorem, it can be shown that sup ξ ∈ B g ξ ∗ ( ζ ) isintegrable function of ζ .For η = 0, g η ∗ ( ζ ) is a bounded function of ζ . So, I ( ζ ) ≤ N ( η ) − n sup ζ ∈ B g η ∗ ( ζ ) Z B N ( ξ ) − n g ξ ∗ ( ζ ) dξ ≤ K ′ sup ξ ∈ B g ξ ∗ ( ζ ) Z B N ( ξ ) − n dξ = K sup ξ ∈ B g ξ ∗ ( ζ ) ,K ′ and K being constants independent of ζ . Since sup ξ ∈ B g ξ ∗ ( ζ ) is inte-grable function of ζ , so is I . On same lines, it can be shown that I ( ζ )is integrable function of ζ . Since h ( η, ξ ) is continuous function on B so byFubini’s theorem I ( ζ ) exists for almost all ζ . Having proved integrability of I i (1 ≤ i ≤ N ( η, ζ )is integrable function of ζ . S. DUBEY, A. KUMAR AND M. M. MISHRA
We shall, by induction, prove that N k ( η, ζ ) exists for η = ζ and as a functionof ζ it is integrable over B . Assume, we have proved that N k exists and isintegrable over B . We have, N k +1 ( η, ζ ) = Z B N k ( η, ξ ) N ( ξ, ζ ) dξ = Z B N k ( η, ξ ) g ξ ( ζ ) dξ + N ( ζ ) − n Z B N k ( η, ξ ) g ξ ( − ζ ∗ ) dξ + Z B N k ( η, ξ ) h ( ξ, ζ ) dξ. (3.4)Consider Z B Z B N k ( η, ξ ) g ξ ( ζ ) dζdξ = Z B N k ( η, ξ ) (cid:18)Z B g ξ ( ζ ) dζ (cid:19) dξ = (cid:18)Z B g e ( ζ ) dζ (cid:19) (cid:18)Z B N k ( η, ξ ) dξ (cid:19) . So R B N k ( η, ξ ) g ξ ( ζ ) dξ exists a.e. and is integrable. Since for ζ ∈ B , ζ / ∈ B ,the second term in (3.4) can be estimated as N ( ζ ) − n Z B N k ( η, ξ ) g ξ ( − ζ ∗ ) dξ ≤ N ( ζ ) − n sup ξ ∈ B g ξ ( − ζ ∗ ) Z B N k ( η, ξ ) dξ. As N ( ζ ) − n sup ξ ∈ B g ξ ( − ζ ∗ ) is integrable function of ζ , we obtain the inte-grability of the second term in (3.4). Similarly we can estimate the thirdterm in (3.4). Finally, we obtain the integrability of N k +1 . (cid:3) Theorem 3.2.
For continuous circular functions f on B and g j ’s, ≤ j ≤ m − on ∂B , the polyharmonic Neumann BVP L m u = f in B∂∂n ( L j u ) = g j on ∂B \ S , ≤ j ≤ m − (3.5) is solvable if and only if Z B Z B N m − j − ( η, ξ ) f ( η ) dv ( η ) − m − X µ = j +1 Z ∂B \ S N µ − j ( η, ξ ) g µ ( η ) dσ ( η ) dv ( ξ )= Z ∂B \ S g j ( ξ ) dσ ( ξ ) , ≤ j ≤ m − , m ≥ , Z B f ( ξ ) dv ( ξ ) = Z ∂B \ S g m − ( ξ ) dσ ( ξ ) , (3.6) and the circular solution is given by u ( ξ ) = Z B N m ( η, ξ ) f ( η ) dv ( η ) − m − X µ =0 Z ∂B \ S N µ +1 ( η, ξ ) g µ ( η ) dσ ( η ) . (3.7) Proof.
It is easy to see that, using Lemma 3.1, Z B N k ( η, ξ ) f ( η ) dv ( η ) and Z ∂B \ S N k ( η, ξ ) g µ ( η ) dv ( η ) exist for all 1 ≤ k ≤ m . The proof follows by OLYHARMONIC NEUMANN PROBLEM 7 induction. Assume the result is true for m = k − m = k . Let L u = w in ¯ B . Then (3.5) reduces to L k − w = f in B∂∂n ( L j w ) = g j +1 , on ∂B, ≤ j ≤ k − . From induction hypothesis, the above BVP is solvable if and only if Z B Z B N k − j − ( η, ζ ) f ( η ) dv ( η ) − k − X µ = j +1 Z ∂B \ S N µ − j ( η, ζ ) g µ +1 ( η ) dσ ( η ) dv ( ζ )= Z ∂B \ S g j +1 ( ζ ) dσ ( ζ ) , ≤ j ≤ k − , k ≥ Z B f ( ζ ) dv ( ζ ) = Z ∂B \ S g k − ( ζ ) dσ ( ζ ) , (3.8)and the circular solution is given by w ( ζ ) = Z B N k − ( η, ζ ) f ( η ) dv ( η ) − k − X µ =0 Z ∂B \ S N µ +1 ( η, ζ ) g µ +1 ( η ) dσ ( η ) . (3.9)The solution of (3.5) with m = k is then the solution of L u = w in B∂∂n u = g on ∂B. Using ([5], Theorem 4.1) the above BVP is solvable if and only if Z B w ( ζ ) dv ( ζ ) = Z ∂B \ S g ( ζ ) dσ ( ζ ) , (3.10)and since w is circular using (3.9), the solution is given by u ( ξ ) = Z B N ( ζ, ξ ) w ( ζ ) dv ( ζ ) − Z ∂B \ S N ( ζ, ξ ) g ( ζ ) dσ ( ζ ) . (3.11)Combining the solvability conditions (3.8) with (3.10), substituting the valueof w from (3.9) into (3.11), using expression of N k ( η, ξ ) in (3.1) and Lemma3.1 we obtain the result to be true for m = k . Now, we verify that u ( ξ )given by (3.7) is the solution of the Neumann BVP (3.5) if conditions (3.6)are satisfied.By using definition of N m ( η, ξ ), we have L N m ( η, ξ ) = N m − ( η, ξ ) in B .Therefore, by induction L m N m ( η, ξ ) = δ η ( ξ ) in B .Now it is obvious that L m u ( ξ ) = f ( ξ ) in B with u as in (3.7). Next, for S. DUBEY, A. KUMAR AND M. M. MISHRA ≤ j ≤ m − , L j u ( ξ ) can be expressed as= Z B N m − j ( η, ξ ) f ( η ) dv ( η ) − m − X µ = j Z ∂B \ S N µ − j +1 ( η, ξ ) g µ ( η ) dσ ( η )= Z B (cid:18)Z B N m − j − ( η, ζ ) N ( ζ, ξ ) dv ( ζ ) (cid:19) f ( η ) dv ( η ) − m − X µ = j Z ∂B \ S (cid:18)Z B N µ − j ( η, ζ ) N ( ζ, ξ ) dv ( ζ ) (cid:19) g µ ( η ) dσ ( η ) , = Z B Z B N m − j − ( η, ζ ) f ( η ) dv ( η ) − m − X µ = j +1 Z ∂B \ S N µ − j ( η, ζ ) g µ ( η ) dσ ( η ) N ( ζ, ξ ) dv ( ζ ) . Therefore, by ([5], Theorem 4.1) ∂∂n ( L j u ( ξ )) = g j ( ξ ) on ∂B \ S , ≤ j ≤ m − , if first equality of (3.6) is satisfied.For j = m − ∂∂n ( L j u ( ξ )) = g m − ( ξ ) if R B f ( ξ ) dv ( ξ ) = R ∂B \ S g m − ( ξ ) dσ ( ξ ) . (cid:3) Polyharmonic Mixed Boundary Value Problems
In this section we consider the mixed BVP arising out of m -Neumann and n -Dirichlet conditions.We define, for each positive integer k , the functions G k ( η, ξ ) and P k ( η, ξ )inductively.Let G ( η, ξ ) = G ( η, ξ ) and P ( η, ξ ) = P ( η, ξ ) and when G k − ( η, ξ ) and P k − ( η, ξ ) are defined, define G k ( η, ξ ) = Z B G k − ( η, ζ ) G ( ζ, ξ ) dv ( ζ ) (4.1) P k ( η, ξ ) = Z B P k − ( η, ζ ) G ( ζ, ξ ) dv ( ζ ) , (4.2)where G ( η, ξ ) is the Green’s function for B given by G ( η, ξ ) = g η ( ξ ) − K ( g η )( ξ − ) and P ( η, ξ ) is the corresponding Poisson kernel for B given by P ( η, ξ ) = − ∂∂n G ( η, ξ ) as in [13]. Similar computations as in Lemma 3.1,show that G k ( η, ξ ) are well defined.The details of the following lemma can be worked out by showing for anycontinuous φ defined on ∂B , R ∂B P k ( η, ξ ) φ ( ξ ) dσ ( ξ ) exists. Lemma 4.1.
The functions P k ( η, ξ ) as defined in (4.2) are integrable over ∂B for k ∈ N , and hence exists finitely a.e. For 0 ≤ k ≤ m − ≤ j ≤ n −
1, we define M k,j ( η, ξ ) = Z B N k ( η, ζ ) G j ( ζ, ξ ) dv ( ζ ) , (4.3) S k,j ( η, ξ ) = Z B N k ( η, ζ ) P j ( ζ, ξ ) dv ( ζ ) , (4.4) OLYHARMONIC NEUMANN PROBLEM 9 where N k ( η, ξ ) is defined in (3.1). The only possible points of singularitiesof integrands in (4.3) and (4.4) are at ζ = η and ζ = ξ . Writing M k,j as combination of six integrals I i , ≤ i ≤ M k,j are well defined. Furtherin a similar way one can show that S k,j is convergent and the kernel is welldefined.4.1. Polyharmonic Neumann-Dirichlet problem.Theorem 4.2.
For m, n ≥ and continuous circular functions f definedon B , g r ’s, ≤ r ≤ m − and h s ’s, ≤ s ≤ n − defined on ∂B , the mixedBVP L m + n u = f in B∂∂n ( L r u ) = g r on ∂B \ S , ≤ r ≤ m − L m + s u = h s on ∂B, ≤ s ≤ n − (4.5) is solvable if and only if Z B Z B M m − r − ,n ( η, ξ ) f ( η ) dv ( η ) + n − X s =0 Z ∂B S m − r − ,s +1 ( η, ξ ) h n − s − ( η ) dσ ( η ) − m − X µ = r +1 Z ∂B \ S N µ − r ( η, ξ ) g µ ( η ) dσ ( η ) dv ( ξ )= Z ∂B \ S g r ( ξ ) dσ ( ξ ) , ≤ r ≤ m − , Z B Z B G n ( η, ξ ) f ( η ) dv ( η ) + n − X s =0 Z ∂B P s +1 ( ζ, η ) h n − s − ( η ) dσ ( η ) ! dv ( ξ )= Z ∂B \ S g m − ( ξ ) dσ ( ξ ) . (4.6) and the circular solution is given by u ( ξ ) = Z B M n,m ( η, ξ ) f ( η ) dv ( η ) + n − X s =0 Z ∂B S m,s +1 ( η, ξ ) h n − s − ( η ) dσ ( η ) − m − X µ =0 Z ∂B \ S N µ +1 ( η, ξ ) g µ ( η ) dσ ( η ) . (4.7) Proof.
As in Lemma 3.1, we can show that Z B M k,l ( η, ξ ) f ( η ) dv ( η ); Z ∂B S k,l ( η, ξ ) h r ( η ) dσ ( η ) and Z ∂B \ S N k ( η, ξ ) g µ ( η ) dσ ( η ) exist and are definedfinitely a.e. We decompose the above mixed boundary value problems into m -Neumann and n -Dirichlet problems. Let L m u = w in B . Then above problem reduces into the following system of equations: L m u = w in B∂∂n ( L r u ) = g r on ∂B \ S , ≤ r ≤ m − w instead of f and the solution is given by u ( ξ ) = Z B N m ( ξ, ζ ) w ( ζ ) dv ( ζ ) − m − X µ =0 Z ∂B \ S N µ +1 ( ξ, ζ ) g µ ( ζ ) dσ ( ζ ) . Next consider L n w = f in BL s w = h s on ∂B, ≤ s ≤ n − . The solution of above polyharmonic Dirichlet problem is given in [14] as w ( ζ ) = Z B G n ( ζ, η ) f ( η ) dv ( η ) + n − X s =0 Z ∂B P s +1 ( ζ, η ) h n − s − ( η ) dσ ( η ) . Hence, by substituting the value of w ( ζ ) in u ( ξ ) and in the solvability condi-tion, combining the integrals using Fubini’s theorem and (4.3), (4.4), we getthe required solvability conditions (4.6) and the circular solution is given by(4.7).As we have verified in the last section, one can easily verify that u ( ξ ) givenby (4.7) is the solution of the BVP (4.5) if the solvability condition (4.6) issatisfied. (cid:3) Polyharmonic Dirichlet-Neumann problem.Theorem 4.3.
For m, n ≥ and continuous circular functions f definedon B , g r ’s, ≤ r ≤ n − and h s ’s, ≤ s ≤ m − defined on ∂B , the mixedBVP L m + n u = f in BL r u = g r on ∂B, ≤ r ≤ n − ∂ ⊥ ( L n + s u ) = h s on ∂B, ≤ s ≤ m − (4.8) is solvable if and only if Z B Z B N m − s − ( η, ξ ) f ( η ) dv ( η ) − m − X µ = s +1 Z ∂B N µ − s ( η, ξ ) h µ ( η ) dσ ( η ) dv ( ξ )= Z ∂B h s ( ξ ) dσ ( ξ ) , ≤ s ≤ m − , m ≥ , Z B f ( ξ ) dv ( ξ ) = Z ∂B h m − ( ξ ) dσ ( ξ ) , (4.9) OLYHARMONIC NEUMANN PROBLEM 11 and the circular solution is given by u ( ξ ) = Z B M m,n ( η, ξ ) f ( η ) dv ( η ) − m − X µ =0 Z ∂B M µ +1 ,n ( η, ξ ) h µ ( η ) dσ ( η )+ n − X r =0 Z ∂B P r +1 ( η, ξ ) h n − r − ( η ) dσ ( η ) . (4.10) Proof.
We decompose the above mixed boundary value problems into m -Neumann and n -Dirichlet problems. Let L n u = w in B . Then above prob-lem reduces into the following system of equations: L n u = w in BL r u = g r on ∂B, ≤ r ≤ n − (cid:27) (4.11)and L m w = f in B∂ ⊥ ( L s w ) = h s on ∂B, ≤ s ≤ m − . ) (4.12)The solution of polyharmonic Dirichlet problem (4.11) as given in [14] is u ( ξ ) = Z B G n ( ξ, ζ ) w ( ζ ) dv ( ζ ) + n − X r =0 Z ∂B P r +1 ( ξ, ζ ) g n − r − ( ζ ) dσ ( ζ ) . By Theorem 3.2, the polyharmonic Neumann problem (4.12) is solvable ifand only if (4.9) is satisfied and the solution is given by w ( ζ ) = Z B N m ( ζ, η ) f ( η ) dv ( η ) − m − X µ =0 Z ∂B N µ +1 ( ζ, η ) h µ ( η ) dσ ( η ) . Hence, by substituting the value of w ( ζ ) in u ( ξ ), combining the integralsusing Fubini’s theorem and (4.3), we get the circular solution as given by(4.10). As we have verified in the last section, one can easily show that u ( ξ )given by (4.10) is the solution of the BVP (4.8) if the solvability condition(4.9) is satisfied. (cid:3) acknowledgements The first author is supported by the Senior Research Fellowship of Councilof Scientific and Industrial Research, India (Grant no. 09/045(1152)/2012-EMR-I) and the second author is supported by R & D grant from Universityof Delhi, Delhi, India.
References [1] H. Begehr,
Orthogonal decompositions of the function space L ( ¯ D ; C ),J. reine angew.Math. 549 (2002), pp. 191–219.[2] A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie Groups and Potential Theoryfor their sub-Laplacians , Springer Monograph in Mathematics, Springer-Verlag Berlin-Heidelberg 2007.[3] G. B. Folland and J. J. Kohn,
The Neumann Problem for the Cauchy Riemann Com-plex , Princeton University Press, Princeton, NJ, 1972.[4] G. B. Folland,
A Fundamental Solution for a subelliptic operator , Bull. Am. Math.Soc.79 (1973), pp. 373–376. [5] S. Dubey, A. Kumar, M. M. Mishra,
The Neumann problem for the Kohn-Laplacianon the Heisenberg Group H n , Potential Anal. (2016), 1–15, doi:10.1007/s11118-016-9538-1.[6] B. Gaveau, Principe de moindre action, propagation da la chaleur et estim´ees sous-elliptiques sur certaiins groupes nilpotents , Acta Math, 139 (1977), pp. 95–153.[7] B. Gaveau,
Syst ` e mes dynamiques associ ´ e s ` a certains op ´ e rateurs hypoelliptiques , Bull.Sc. Math., Paris, 2 c s´ e rie, T. 102, 1978, pp. 203–229.[8] P. C. Greiner and T. H. Koornwinder, Variations on the Heisenberg spherical harmon-ics , Report ZW 186/83 Mathematisch Centrum, Amsterdam, 1983.[9] David S. Jerison,
The Dirichlet problem for the Kohn Laplacian on the Heisenberggroup , I, J. Func. Anal. 43 (1981), pp. 97–142.[10] A. Kor´anyi,
Kelvin transforms and harmonic polynomials on the Heisenberg group ,J. Funct. Anal. 49 (1982), pp. 177–185.[11] A. Kor´anyi,
Geometric aspects of analysis on the Heisenberg group , in “Topics inmodern harmonic analysis”, Instituto nazionale di Alta mathematica, Roma, 1983,pp. 209–258.[12] A. Kor´anyi,
Poisson formulas for circular functions and some groups of type H , Sci.China Ser. A: Math. 49 (2006), pp. 1683–1695.[13] A. Kor´anyi and H. M. Riemann,
Horizontal normal vectors and conformal capacityof spehrical rings in the Heisenberg group , Bull. Sci. Math. Ser. 2 111 (1987), pp. 3–21.[14] A. Kumar and M. M. Mishra,
Polyharmonic Dirichlet problem on the Heisenberggroup , Complex Var. Elliptic Equ. 53 (2008), no. 12, pp. 1103–1110.[15] S. Thangavelu,
Harmonic Analysis on the Heisenberg Group , Birkh¨auser, Boston,Basel, Berlin, 1998.
Department of Mathematics, University of Delhi, Delhi, India
E-mail address : [email protected] Department of Mathematics, University of Delhi, Delhi, India
E-mail address : [email protected] Department of Mathematics, Hans Raj College, University of Delhi, Delhi,India
E-mail address ::