The Neumann Problem on Product Domains in C n

Transcription

1 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-9 Wien, Austria The Neumann Problem on Product Domains in C n Dariush Ehsani Vienna, Preprint ESI 754 (5) December, 5 Supported by the Austrian Federal Ministry of Education, Science and Culture Available via

2 THE -NEUMANN PROBLEM ON PRODUCT DOMAINS IN C n DARIUSH EHSANI Abstract. Let Ω = Ω Ω n C n, where Ω j C is a bounded domain with smooth boundary. We study the solution operator to the -Neumann problem for (, )-forms on Ω. In particular, we construct singular functions which describe the singular behavior of the solution. As a corollary our results carry over to the -Neumann problem for (,q)-forms. Despite the singularities, we show that the canonical solution to the -equation, obtained from the Neumann operator, does not exhibit singularities when given smooth data.. Introduction The -Neumann problem was studied on product domains by the author in [4] and [5] in the setting of C. In this paper we consider the -Neumann problem on a product of n domains, each in C. Such domains are interesting because the solution operator, the Neumann operator, is seen to exhibit singularities, whereas an easy argument shows the related canonical solution operator to the -equation maps (, )-forms smooth up to the boundary to functions which are smooth up to the boundary. The singular behavior of the solution is expressed in terms of functions which are singular at the intersection of the boundaries of two or more domains. We obtain a description of the singularities in the solution to the -Neumann problem for (, )-forms by relating it to the solution to the Dirichlet problem. The idea of obtaining information for the Neumann operator from the solution operator to the Dirichlet problem on product domains is also used in a paper of Fu [6], in the setting of the bi-disk. Our description of the solution to the -Neumann problem is then obtained by inverting anti-holomorphic vector fields. In doing so, the main difficulties in extending our results from complex dimension to n are encountered. Whereas in dimension, each integration led to remainder terms which were functions of at most one distance to boundary function of one of the domains in the product, which were smooth, in dimension n a further study needs to be undertaken to examine the singular behavior of these remainder terms. Our paper is organized as follows. We set up the -Neumann problem in Section. Sections 3 and 4 are devoted to the study of the Dirichlet problem on product domains. In our study of the Dirichlet problem we obtain an asymptotic expansion of the solution as a sum of terms with increasing differentiability up to the boundary. We also construct singular functions which describe the singular behavior of the solution. The singular functions are then shown to describe the singular behavior to the solution of the -Neumann problem as well in Section 5. We end the paper with a theorem in Section 6 describing the phenomenon mentioned above that the Mathematics Subject Classification. Primary 3W5; Secondary 35B65. Partially supported by the Max-Planck-Institute for Mathematics in Bonn.

3 DARIUSH EHSANI singularities cancel when one constructs the solution to the -equation from the Neumann operator. Part of this paper was completed while the author was visiting the Math Institute at the University of Bonn and the Max-Planck-Institute for Mathematics in Bonn, and we wish to extend special thanks to Professor Ingo Lieb for his hospitality. Results of this paper were presented at the 5 Workshop on Complex Analysis and PDE s at the Erwin Schrödinger Institute in Vienna, and the author gratefully acknowledges support from the Schrödinger Institute to attend the workshop.. Preliminaries Let Ω j C, for j n, be bounded smooth domains, and set Ω = Ω Ω n. We set up the -Neumann problem on a product of half-planes in C n with a choice of metric which is related to Ω. Let H = {z C n Iz j > for j n}. Let z j = ϕ j (w j ) be a biholomorphic mapping from Ω j to {Iz j > }. We choose the metric on H to be n (.) ds = ϕ j (w j ) dz jd z j j= and an orthonormal basis for Λ, (H) to be ω j = dz j ϕ j (w j). The vector fields dual to ω,..., ω n are given by Thus if α is a smooth function L j = ϕ j(w j ) z j. α = L (α) ω + + L n (α) ω n, and if β = β ω + + β n ω n is a (, )-form, β = j<k ( L j β k L k β j ) ωj ω k. We will also need to know how, the formal adjoint of, acts on (, )-forms; with β as above β = L β L n β n. (.) The -Neumann problem for (, )-forms on H equipped with the metric ds is u = f on H u j = on Iz j = L k u j = k j on Iz k =, where is the operator +. The -Neumann problem on H with the metric ds is equivalent to the -Neumann problem on Ω with the Euclidean metric. Existence and uniqueness of an L solution to the -Neumann problem on Ω given the Euclidean metric is known from a theorem of Hörmander [7] (see also Theorem 4.4. in []).

4 THE -NEUMANN PROBLEM ON PRODUCT DOMAINS IN C n 3 In terms of the vector fields L j, u = f is written n L j L j u i = f i i =,..., n. j= We have the following result concerning the location of the singularities of the solution; the proof follows as in the case of two dimensions. Proposition.. Let u L (,)(Ω) be the unique solution to the -Neumann problem on Ω. Let V Ω such that V Ω j for at most one j. Then u C(,) (V ). Proof. The proof can be divided into three cases depending on which boundary the neighborhood intersects and on which coefficient of the (, )-form u we are considering. If V lies in the interior of Ω, the result follows from interior ellipticity of the Laplacian. If V intersects a boundary and a coefficient of u satisfies a Dirichlet condition there, then regularity at the boundary of the Dirichlet problem gives smoothness up to the boundary. Lastly, in the case that V intersects a boundary and a Neumann type condition, given by the vanishing of a derivative, L j, holds, the difference between the coefficient of u in question and a Cauchy transform applied to a function which satisfies a Dirichlet condition is shown to be smooth up to the boundary, in the same manner as in the case of two dimensions (see [5]). 3. The Dirichlet problem Define v = L L n u and g = L L n f. Then v satisfies the Dirichlet problem n (3.) L j L j v = g on H j= v = on H. Let ρ j (z j ) denote a signed geodesic distance in the metric ϕ j (w j ) dz j d z j on {Iz j > } from z j to the boundary Iz j =. We will work in a small neighborhood of the origin intersected with H in which the ρ j and x j = Rz j form a coordinate system. In this neighborhood, we write L j = ϕ j(w j ) ρ j + κ j (x j, ρ j ), z j ρ j x j where the κ j are in C (H). Then in these coordinates, (3.) is ( n ϕ j(w j ) ρ j ) ( ) z j + a j (x ) j, ρ j, + b j x j, ρ j, v = g, x j ρ j x j j= ρ j where a j and b j are tangential derivatives of order and respectively. Since ρ j measures the distance in the metric ϕ j (w j) dz j d z j to the boundary Iz j = we can write ( ) n (3.) + a j + b j v = g. ρ j j= ρ j

5 4 DARIUSH EHSANI Let χ C (H) be such that χ in a neighborhood of the origin and equivalently outside some compact set. Multiplying (3.) by χ and commuting χ with the derivatives yields ( ) n (3.3) + a j + b j χv = χg + h, ρ j j= ρ j where h in a neighborhood of the origin. We utilize the Dirichlet boundary conditions by constructing extensions of the solution to all of C n. Extend each map ϕ j : {Iz j > } Ω j to all of C by defining ϕ j ( z j ) = ϕ j (z j ) for Iz j >. Furthermore, define a metric on all of C n which matches ds on H by n d s = ϕ j ( ϕ j (z j )) dz j d z j. j= Thus, the geodesics in {Iz j < } are just reflections of the geodesics in the upper half-plane about the real axis. With the convention that distances to Iz j = in the lower half-plane are negative, we extend v in an odd manner with respect to each ρ j by v( z j ) = v(z j ). We shall denote this extended function by v o. In what follows, the superscript o will denote an extension to C n by odd reflections, and the superscript e will denote even extensions. We extend (3.3) to all of C n using odd reflections about each Iz j =, and relate derivatives under Fourier transform to multiplication by a Fourier variable. ξ j will be the transform variable corresponding to x j, and η j the transform variable corresponding to ρ j. In this way, (3.3) becomes R n j= n ( η j + iη j a o j (x j, ρ j, iξ j ) + b e j (x j, ρ j, iξ j ) ) χv o e ix ξ e iρ η dξdη = ( χg + h) o e ix ξ e iρ η dξdη. R n A solution to the Dirichlet problem is thus obtained by solving ( (3.4) η j + iη j a o j + ) be j χv o = χg o + ĥo. Without loss of generality we will perform our calculations with the right hand side of (3.4) replaced with just the term χg o, and our results will be valid modulo terms which are C smooth up to the boundary in the neighborhood of the origin, arising from the inversion of the elliptic operator on the left hand side applied to ĥ o which is C smooth in a neighborhood of the origin in C n. Let K be the operator defined by n ( K ˆφ j= iηj a o j = + ) βe j n j= (η j + ξ j ) ˆφ, where βj e = be j ξ j. We write (3.4) as (3.5) (I K) χv o = ˆΦ, where χg ˆΦ o = n j= (η j + ξ j ).

6 THE -NEUMANN PROBLEM ON PRODUCT DOMAINS IN C n 5 We define cutoff functions, χ j C (R n ) for j =,,... by letting χ = χ above and χ j = on suppχ j. We also define operators T = I and T j φ = ( K(χj T j φˆ) ˇ) for φ L for j =,,.... Our idea is to use the T j to replace the powers of K in the infinite sum j Kj in inverting (I K) in (3.5); the use of the cutoffs χ j ensures we stay in L spaces. After restricting the supports of the χ j if necessary, we easily verify that the following relations are possible T j φ < φ φ L and j N; T j v o as j. We obtain the following solution of the Dirichlet problem. Theorem 3.. Let the domain H be equipped with the metric ds defined above. Let v be the solution to the inhomogeneous Dirichlet problem on H with data g C (H), χ a smooth cutoff function with compact support such that χ near. With the operators T j and Φ defined as above (3.6) χv L = Proof. From (3.5) χ j+ T j Φ. j= (3.7) χ j+ T j v χ j+ T j+ v = χ j+ T j Φ + s j ( where s j = (χ j+ χ j+ )T j+ v +χ j+ Kŝj ˇ) and s = (χ χ )Φ. Equation 3.6 gives terms of a telescoping series which converges in L since χ j+ T j+ v as j. For any ǫ > we may also choose the χ j so that s j < ǫ and j+ j= s j < ǫ. Hence, we conclude (3.6). As a corollary, we can, through a change of variables, also read off the solution to the Dirichlet problem on Ω equipped with the Euclidean metric. The existence of a solution to the inhomogeneous Dirichlet problem on product domains is already common knowledge, and in fact the solution is in H 3, the Sobolev- 3 space, in the setting of domains with Lipschitz boundaries (see [8]). The usefulness of (3.6) comes from the observation, which we explore below, that it is actually an asymptotic expansion, whose successive terms exhibit increasing orders of differentiability up to the boundary of H. 4. Asymptotic expansion of the solution to the Dirichlet problem In this section we relate terms in Fourier space to corresponding singular functions on H. As a first step we show we can multiply χv by cutoff functions each of which is in a neighborhood of some η i =, without changing the singular behavior we wish to analyze. Define χ η (η) C (R n ) so that χ η for η < and χ η for η >. Also, define χ η = χ η. χv multiplied by any compactly supported function in Fourier space is the transform of a smooth function, by the Sobolev Embedding Theorem, hence the difference of χv and χ η χv is the transform of a function, which when restricted to H is in C (H). Let χ ηi (η i ) C (R) be a cutoff function in Fourier space with the property that χ ηi ( η i ) = χ ηi (η i ), χ ηi for η i < and χ ηi for η i >. Also, define χ η i = χ ηi and

7 6 DARIUSH EHSANI χ η = χ η χ η n. χ η χ η can be written as a sum of terms each having support contained in a neighborhood of η i = for some i and is equivalently in a neighborhood of (4.) η + + η i + η i+ + + η n =. Write η = n ηj, j= and let η ī be the left hand side of (4.). Defining χ η ī as above but with respect to all η variables but η i, we consider χ η ī χ η i for φ L (R n ), arising in the expansion (3.6). We follow an argument below (see Equation 4.8) to conclude that infinite order of decay of terms in the η i variable can be translated to a transform of a sum of smooth terms each multiplied by (4.) χ η ī (η, ī )l for l some integer, whereby an induction argument on the dimension in transform space completes our assertion, the case of two dimensions being already worked out in [4]. That is, the error terms involved by analyzing χ η as opposed to χ η will involve explicit singular functions which we now explore. We see that up to multiplication by a constant the following relation holds for l < n where ρ = (ρ + + ρ n). Let (4.3) Φ n l = ˆφ η R n (η ) l eiρ η dη = ρ n l, ρ n l l < n. For l n and n even, we define Φn l to be the unique solution of the form p (ρ)log ρ + p (ρ) where p and p are polynomials of degree l n to the equation (4.4) Φ n l ρ = ρ Φ n l (see [3]). For n odd we define Φ n l for l n as in (4.3). In what follows, we let s denote any term which is the transform of a C function in some neighborhood of the origin in C n. Then the Φ n l, expressed in the ρ coordinates, have the property that (4.5) χ χφ η n l = χ η (η ) l + s up to multiplication by a constant. We can redefine Φ n l by multiplying by a constant so that (4.5) holds; we again use Φ n l to denote this term. Note that the term χ η is necessary when considering the Fourier transform of log( ρ ), and also of Φ n l

8 THE -NEUMANN PROBLEM ON PRODUCT DOMAINS IN C n 7 dependant on this transform determined by the recursion formula (4.4), in the case l = n for n even. For k = (k,..., k n ) we further define Φ n lk = ρn t kn ρ k t Φ n l dtk dtk k dt kn dtkn k n ; thus integration by parts shows Φ n lk has the property (4.6) χ χφ η n lk = χ η η k (η ) l + s, where we use the notation η k = η k ηkn n for k = (k,..., k n ). For a multi-index q = (q,..., q i ), we define Φ q lk for i n in the same fashion as we did Φ n lk but with respect to the i variables ρ q,..., ρ qi, in particular k is a multi-index of length i. We also use the notation Φ n lk = Φq lk when q = (,..., n). The multiplication by χ η is useful in the calculations as it avoids complications arising from the vanishing of the η i variables in denominators of the asymptotic expansion we write for the solution to the Dirichlet problem. Furthermore, there is no loss of information pertaining to the description of the singularities of the solution. From (4.6) and the construction of the Φ q lk we see that ( (η )η m ηn mn χ η( χφ q lk ) χφ ) q lk s for large enough m,..., m n. Thus, by the ellipticity of the operator whose symbol is η we obtain derivatives of the difference of the quantity in question is a C function in R n. We now show for all N N, each term in (3.6) can be written on H in a neighborhood of as a sum of terms of the form (4.7) T j Φ = n l(q)= N mq +(l n )+ k =n mα,l, k c klmq ρ m Φ q lk + s N, where l(q) is the number of terms of the multi-index, q, m q = (m q,..., m ql(q) ), k = (k..., k n ), m q = m q + +m ql(q), and k = k + +k n, the c klmq depend smoothly on x,..., x n, and on ρ qi for q i / q, and s N is used to denote any function which, when restricted to H is in C N (H). We prove (4.7) by induction, the first step verifiable by integrating by parts in the Fourier integral of χg o and expanding η +ξ in a geometric series in η, as was done in the case n = in [4]. The remainder terms in such an expansion must be shown to give rise to terms s N. We can, without loss of generality, assume that N + n is even. We integrate by parts in the Fourier integral of χg o so that the remainder terms have decay in one variable, which we suppose to be η j, to the order. We consider the term η N+n j χ η φ j ηj N+n resulting from integration by parts in the Fourier integral of χg o, where φ j = η η n N+n+ η ρ N+n+ j (χg) o cos(ρ j η j )dρ j,

9 8 DARIUSH EHSANI all other such remainder terms resulting from integration by parts, being handled in a similar manner. We use (4.8) χ φ j η η N+n j η = χ η = χ η φ j η N+n j η j φ j ηj N+n η j + η j η j η j η j + + ( ) N+n ( η j η j ) N+n + η j η j The first N+n terms in (4.8) are easily seen to be of the form of (4.7), made up of terms of the form Φ q lk in which l(q) = n. Furthermore, the last term in (4.8), (4.9) χ ηφ j N+n (η ) + η j. is in s N because χ ηφ j (η N+n ) + j is the transform of a function, odd in the variable ρ j, in C N (R n {y j > }) by the Sobolev Embedding Theorem. Thus (4.9) is the solution to a Dirichlet problem on the domain R n {y j > }, and as such, from the ellipticity of the Dirichlet problem on half-spaces, when viewed as a function on H, is in C N (H). The induction argument starts with the recursive formula (4.) T j Φ = ( K(χ j T j Φˆ)ˇ). From the definition of K we consider the terms iη j a o j + βe j η + ξ χ(ρm Φ q lk )o. Without affecting the singular behavior of the solution on H, we may consider either an even or odd extension in each variable of each particular term of T j Φ, and thus ignore the odd extension under the transform. The difference in using such an extension from the original (odd) terms in T j Φ will be terms smooth up to the boundary of H, due to the parametrix for the Laplacian,, and the fact that η +ξ T j Φ satisfies Dirichlet conditions along each ρ i =. Hence we must prove (4.) χ iη j a o j + βe j η χρ η + ξ m qφ q lk = iη j a o χ j + βe j mq q η χφ η + ξ η mq lk = iη ja o j + βe j η + ξ = iη ja o j + βe j η + ξ mq η χ χφ q mq η lk + s mq ( χ η mq η η k q η ki ) (ηq + + ηq i ) l + s q i is of the form (4.7) when inverted and restricted to H. This is accomplished by expanding η +ξ as a geometric series in, Taylor expanding the terms a j P qα q η qα + s

10 THE -NEUMANN PROBLEM ON PRODUCT DOMAINS IN C n 9 and β j on H in the ρ variables about ρ = = ρ n =, and using the relations (4.) Φ n lk ρ = Φn l(k,,...,k i,...,k n) i Φ n l = ρ i Φ n l ρ, i χ χ η j Φ n lk = χ η η k (η ) + s, l with the obvious analogies for the Φ q lk, to interpret resulting terms from taking derivatives with respect to η variables, as well as multiplication by η j. The remainder terms from the Taylor expansions of a j and β j, and the geometric series expansion are absorbed into the term s N given the expansion is carried out to sufficiently many terms, and it remains to examine s N terms in (4.7) under the operator K. K(ŝ N ) leads to terms which can be absorbed in s N again by the parametrix for the Laplacian, η +ξ, and the fact that T j Φ satisfies Dirichlet conditions along each ρ i =. These same induction steps show, modulo the s N terms, the T j are of increasing degree of differentiability at the origin. In particular, we have Theorem 4.. Let v L (Ω) be the unique solution to the inhomogeneous Dirichlet problem on Ω. Let ρ j (w j ) denote the distance to the boundary of Ω j. Then near the distinguished boundary, Ω Ω n, v is of the form (4.3) v = c klq Φ q lk k,l, l(q) n where c klq are dependent on the variables ρ j and tangential variables, and is C up to the boundary, and where Φ q lk are defined as above. Proof. The proof follows from (4.7), valid in a neighborhood of a point on the distinguished boundary, written in terms of the w variables on Ω and from Borel s theorem to give the coefficients c klq as C functions from the power series terms in (4.7). From Proposition (.), we know v is C in terms of the tangential variables, therefore we can patch together neighborhoods of points on Ω Ω n so that (4.3) is valid near the distinguished boundary. 5. Solution to the -Neumann problem In this section we obtain an asymptotic expansion for the solution to the - Neumann problem from our expansion of v = L L n u above. Theorem 5.. Let u L (,) (Ω) be the unique solution to the -Neumann problem on Ω with data f C(,) (Ω). Let ρ j(w j ) denote the distance to the boundary of Ω j. Then near the distinguished boundary, Ω Ω n, each component of u is of the form u i = c klq Φ q lk k,l, l(q) n where c klq are dependent on the variables ρ j and tangential variables, and is C up to the boundary, and where Φ q lk are defined as above.

11 DARIUSH EHSANI Proof. Define u i for i n by u i = L L ī L n u, where ī means the corresponding operator with index i is to be omitted. Hence, L i u i = v holds. We claim that there is no loss in the generality of our results if we assume that u i and v are supported near Ω Ω n. Granted this claim for the time being, we will obtain an expression for u and similar results will hold upon the inversion of the other vector fields. Define coordinates by the ρ j, j =,..., n and x j = ϕ j (Rz j ). ( α (x, ρ ) + κ (x, ρ ) ) u = v, ρ x where α (x, ρ ) = ϕ (w ) ρ z. Since α (x, ρ ) we can divide by α and relate derivatives with respect to x to multiplication in Fourier space: ( ρ + iγ ξ ) ũ = ṽ α, where ũ i is the partial Fourier transform in only the x i variable, γ = κ(x,ρ) α. (x,ρ ) Therefore, we solve e Γξ ũ = e ṽ Γξ, ρ α where Γ = i γ dρ, and we have the solution for ξ >, (5.) ũ = and, for ξ <, (5.) ũ = ρ ρ e (Γ(x,t) Γ(x,ρ))ξ ṽ α dt, e (Γ(x,t) Γ(x,ρ))ξ ṽ α dt + e Γξ ũ ρ=. Without loss of generality we assume the case Iγ >, and that RΓ < and is decreasing with respect to ρ. We want to show that for all N N u can be written as n N (5.3) d klm ρ m Φ q lk + s N. l(q)= mq +(l n )+ k =n mα,l, k In light of the expansion for v in (3.6) in terms of T j Φ, and the singular functions, Φ lk, which compose T j Φ as in (4.7), after Taylor expanding α it suffices to show u is of the desired form when ṽ α in the integrals of (5.) and (5.) is replaced with s N or c klm ρ m Φ q lk for any k, l, m, and q. That terms of the type s N again lead to terms of type s N after integration is obvious, and we look to solve ρ e Γξ ũ = e Γξ c klm χρ m Φ q lk,

12 THE -NEUMANN PROBLEM ON PRODUCT DOMAINS IN C n where χ is a cutoff function equivalently near the origin, q, / q being trivial, and which, for ξ >, has the solution (5.4) ũ = c klm ρ m where ρ m = ρ m ρm3 3 ρ mn n (5.5) ũ = c klm ρ m ρ ρ e (Γ(x,t) Γ(x,ρ))ξ χt m Φq lk dt,, and, for ξ <, has the solution e (Γ(x,t) Γ(x,ρ))ξ χt m Φq lk dt + e Γξ ũ ρ=. In (5.4) and (5.5) we expand the factor e (Γ(x,t) Γ(x,ρ))ξ as a series in t about ρ, obtaining powers of ξ in the process. We first invert the integral terms with respect to ξ, and obtain C coefficients in the x variables, since our data function g is C up to the boundary. We check that (5.6) ρ (t ρ ) q χt m Φq lk dt, arising in the expansion of e (Γ(x,t) Γ(x,ρ))ξ, is of the form (5.3). This follows from integration by parts and the first two relations in (4.). The remainder term from the Taylor series of e (Γ(x,t) Γ(x,ρ))ξ, t ( ) N+n (5.7) (t s ) N+n+ e (Γ(x,s) Γ(x,ρ))ξ ds ρ s leads to terms which can be absorbed into s N when inserted into the integrals in (5.4) and (5.5). Inserting (5.7) into (5.4) and performing a change of order of integration leads us to examine (t s ) N+n+ χt m Φq lk dt. s Similar arguments applied to the integral in (5.6) show that the remainder terms are absorbed in s N. The case in which (5.7) is inserted into (5.5) is handled in the same manner. The last step in showing u has the desired form is in handling the boundary term in (5.5). Let f = L 3 L n f. The relation between u and f is given by n L j Lj u = f j= with Dirichlet conditions holding along ρ i = for i and L u = along ρ =. Thus applying the Laplacian to (5.8) ũ = ρ and setting ρ = yields e (Γ(x,t) Γ(x,ρ))ξ ṽ α dt + e Γξ ũ ρ= σ(ξ ) ũ + n ρ= L j L j ũ = f ρ= ρ= + Φ, ρ= j where σ(ξ ) comes from differentiating the last term in (5.8) with respect to ρ, and Φ ṽ are terms of the type (5.3) coming from the differentiation of α. Applying the same Fourier analysis as in Section 4 in n variables shows u ρ= is composed

13 DARIUSH EHSANI of the terms Φ q lk for l(q) n, and hence that u is composed of terms of the form Φ q lk for l(q) n. The steps to invert the other L derivatives to finally obtain an expression for u are accomplished in a similar manner as the inversion of L, only the step in analyzing boundary value terms as in (5.8) differing slightly. We present here the argument for u 3 = L 4 L n u. For ξ 3 < u 3 satisfies (5.9) ũ 3 = ρ3 (Γ3(x3,t3) Γ3(x3,ρ3))ξ3 ũ e dt 3 + e Γ3ξ3 ũ 3 α, ρ3= 3 ũ 3 referring to the partial transform with respect to x 3 variable, the Γ 3 and α 3 functions being defined in analogous fashions to Γ and α, respectively, above. Similarly u 3 satisfies (5.) ũ 3 = ρ Setting ρ 3 = in (5.) gives ρ (5.) ũ 3 = ρ3= e (Γ(x,t) Γ(x,ρ))ξ ũ3 e dt + e Γξ ũ 3 α. ρ= (Γ(x,t) Γ(x,ρ))ξ ũ3 α dt + e Γξ ũ 3. ρ=ρ 3= ρ3= Upon entering (5.) in (5.9), we see we only need to find the singular functions composing u 3. ρ=ρ 3= Applying the Laplacian to u 3 and setting ρ = ρ 3 = yields (5.) L j Lj u 3 + (L ρ=ρ 3= u 3 ) + (L ρ=ρ 3= 3u ) = f ρ=ρ 3= 3, ρ=ρ 3= j,3 where f 3 = L 4 L n f. We have used the fact that L 3 u 3 = u and L u 3 = u 3. We may thus rely on our results for u and u 3 from above and our Fourier analysis developed in Section 4 to conclude that u 3 ρ=ρ 3= is composed of terms of the form Φ q lk, hence, by (5.9) and (5.), so is u 3. The proof will therefore be completed once we verify our claim that we may assume support near the distinguished boundary. Let φ j C (R) such that φ j near ρ j =, and define ϕ = φ φ n C (Rn ). Then L (ϕu ) = ( L ϕ ) u + ϕ L u (5.3) = ( L ϕ ) u + ϕv. As in (5.4) and (5.5) above, after using partial Fourier transforms with respect to x we may write the left hand side as an integral solution of the right hand side. For ξ >, the solution to (5.3) involves the term and, for ξ <, the term (5.4) If we set ρ = in ϕu = e (Γ(x,t) Γ(x,ρ))ξ ρ ρ e (Γ(x,t) Γ(x,ρ))ξ ( L ϕ ) u dt, α ( L ϕ ) u dt. α e ( (Γ(x,t) Γ(x,ρ))ξ ( L ϕ ) ) u ρ α + ϕv dt,

14 THE -NEUMANN PROBLEM ON PRODUCT DOMAINS IN C n 3 we can solve for e Γ(x,t)ξ ( L ϕ ) u dt, α and this is enough to determine ϕu since v is known from Theorem 4. and ϕu ρ= is the solution to a Dirichlet problem in Rn and hence can be determined by the methods of Sections 3 and 4, and lastly, because near ρ = we have ϕu = e Γ(x,ρ)ξ e Γ(x,t)ξ ( L ϕ ) u dt + α ρ e (Γ(x,t) Γ(x,ρ))ξ ϕvdt by definition of ϕ. Similarly, it is enough to solve for ϕu 3 in order to determine ϕu ρ= 3. By setting ρ = in the equation ϕu 3 = e ( ( L (Γ(x,t) Γ(x,ρ))ξ ϕ ) ) u ρ α 3 + ϕu 3 dt, we see that an expression for ϕu 3 ρ= would give us e Γ(x,t)ξ ( L ϕ ) u 3 dt, α from which ϕu 3 is determined, since u 3, and therefore u 3 ρ=, may be obtained as was u above. To obtain ϕu 3 we set ρ ρ= = in (5.5) ϕu 3 = e ( ( L (Γ3(x3,t3) Γ3(x3,ρ3))ξ3 3 ϕ ) ) u ρ 3 α 3 + ϕu dt 3, 3 and by doing so, we see we yet need e Γ3(x3,t3)ξ3 ( L 3 ϕ ) u 3 ρ= dt 3, α 3 which is easily given in terms of ϕu 3 ρ=ρ 3=, which is the solution to a Dirichlet problem in R n, and ϕu by setting ρ ρ= = ρ 3 = in (5.5). The process is continued until we obtain u. As a corollary to the proof of Theorem 5. we can also state properties of the solution for (, Q)-forms. Let u = u J ω J J =Q f = J =Q f J ω J, in which the sums are taken over strictly increasing sequences.

15 4 DARIUSH EHSANI The analogue of (.) for the -Neumann problem for (, Q)-forms on H equipped with the metric ds given by (.) is We state u = f on H u J = on Iz j = when j J L k u J = on Iz k = when k / J. Theorem 5.. Let Q n, f C(,Q) (Ω), and let u L (,Q)(Ω) be the unique solution to the -Neumann problem on Ω with data f. Then in a small neighborhood of some point on the distinguished boundary, Ω Ω n, each component of u is of the form u J = c klq Φ q lk k,l, l(q) n where c klq are dependent on the variables ρ j and tangential variables, and is C up to the boundary, and where Φ q lk are defined as above. Proof. In the manner in which we obtained our solution to the -Neumann problem for (, ) forms, we relate the problem for (, Q)-forms to the Dirichlet problem and invert L derivatives. In the case of (, Q)-forms n Q vector fields are inverted with the methods of Theorem 5.. Remark 5.3. The expression of the solution to the -Neumann problem in terms of singular functions cannot be avoided. That there do exist forms f such that N f is not in C(,Q) (Ω) can be seen by the simple case of each component of f being equivalently equal to near the distinguished boundary. 6. Canonical solution to the -equation While singularities exist in the solution to the -Neumann problem, a simple argument shows that the canonical solution (the solution orthogonal to holomorphic functions) to the -equation, given in terms of the Neumann operator, is a smooth function up to the boundary. Theorem 6.. Let f C (,) (Ω). Define u = Nf to be the canonical solution to the equation u = f. Then u C (Ω). Proof. We shall work in the case of the poly-disk in C n, the general case holding by biholomorphic equivalence. We look to solve (6.) u = f, in which f C (,) (Ω) for Ω = Dn = { z j < ; j =,..., n}, and f =. It is known [] that there is a solution, not necessarily the canonical solution, to (6.) which is in C (Ω). Let g C (Ω) such that g = f. Then using Kohn s formula relating the Bergman projection on functions, P, to the Neumann operator we see that P = I N (,), (6.) Pg = g N (,) f.

16 THE -NEUMANN PROBLEM ON PRODUCT DOMAINS IN C n 5 Furthermore, Pg is in C (Ω) because each rotational derivative, θ j with the Bergman projection. Since the derivatives z j write normal derivatives, r j in terms z j commutes of Pg are, and we can and θ j, estimates of the form Pg k g k follow for all k N. Thus, from (6.) N (,) f C (Ω), as claimed. References [] J. Bertrams. Randregularität von Lösungen der -Gleichung auf dem Polyzylinder und zweidimensionalen analytischen Polyedern. Bonner Mathematische Schriften, 76: 64, 986. [] S. Chen and M. Shaw. Partial Differential Equations in Several Complex Variables. AMS/IP Studies in Advanced Mathematics. American Mathematical Society and International Press,. [3] D. Ehsani. The solution of the -Neumann problem on non-smooth model domains. PhD thesis, University of Michigan,. [4] D. Ehsani. Solution of the -Neumann problem on a bi-disc. Math. Res. Letters, (4):, 3. [5] D. Ehsani. Solution of the -Neumann problem on a non-smooth domain. Indiana Univ. Math. J., 5(3):69 666, 3. [6] S. Fu. Spectrum of the -Neumann Laplacian on polydiscs. preprint. [7] L. Hörmander. L estimates and existence theorems for the operator. Acta. Math., 3:89 5, 965. [8] D. Jerison and E. Kenig. The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal., 3():6 9, 995. Department of Mathematics, Penn State - Lehigh Valley, Fogelsville, PA 85 address: ehsani@psu.edu