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INTRODUCTION One of the most important example of a singular differential operator (in mathematical physics) is the Schr¨odinger operator S(ψ

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TYPE EQUATIONS WITH BOUNDARY CONDITIONS

CHIA-CHI TUNG

This paper is concerned with the weak solvability of the boundary value problems for Schr¨odinger type equations. For such equations with a real (admissible) spectral parameter µ, the nonhomogeneous Dirichlet, respectively, homogeneous Dirichlet-Neumann problem on a weak Stokes domain lying in a Riemann domain is shown to admit a unique weak solution by making use of aSchr¨odinger product.

If the parameter (denoted by λ) is complex-valued, the weak solvability (in a suitable sense) of a homogeneous Dirichlet-Neumann boundary value problem for a generalized Schr¨odinger equation withL2-density is proved in the case where λ does not belong to R.

AMS 2010 Subject Classification: Primary 35J10, 35J25; Secondary 32C30, 35A01.

Key words: generalized Schr¨odinger product, Schr¨odinger functional, weak Green’s operator.

1. INTRODUCTION

One of the most important example of a singular differential operator (in mathematical physics) is the Schr¨odinger operator

S(ψ) :=− 4ψ+ (µ−V)ψ

in L2(Rn), where µ is a real constant (usually called the spectral parameter) and V a real-valued L2-function (the potential). For wider applications, one may wish to take µ and/or V to be complex-valued. In the following, let Y denote a complex space of pure positive dimension m and DbY a rela- tively compact open subset, and ˜D⊆Y an open neighborhood of D. Assume throughout this paper that ( ˜D, p) is a Riemann subdomain of Y, meaning that ˜D (as a subspace of Y) admits a holomorphic map p : ˜D → Cm with discrete fibers. Denote by 4p the pull-back under p of the Laplace operator of the Euclidean metric on Cm to (the largest) open dense subset D ⊆ D where p is locally biholomorphic. Observe that if ψ∈C2(D), theSchr¨odinger operator S˜p,λ,V[ψ] :=−4˜p[ψ] + (λ−V) [ψ] associated with a constant λ∈C and V ∈ L1loc(D), regarded as acting on Cc2(D), is induced on D by the function

MATH. REPORTS15(65),4(2013), 497–510

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(1.1) Sp,λ,V(ψ) :=− 4pψ+ (λ−V)ψ.

If D is a weak Stokes domain ([6], p. 568) lying in a Riemann domain, for the Schr¨odinger equation

(1.2) Sp,λ,V(Ψ) =− 4pΨ + (λ−V) Ψ = g

in D with a real spectral parameter λ and density g ∈ L2(D) (or more generally, g ∈ H−1(D)), sufficient conditions can be given for the solvability, in a weak sense, of the equation (1.2) subject to a boundary condition (Theo- rem 4.1). Similarly, one may consider, for a generalized Schr¨odinger equation, a mixed homogeneous Dirichlet-Neumann problem relative to a partition

(1.3) ∂D\A= (∂D)D∪(∂D)N,

where A is a thin analytic subset of X (see Theorems 4.2 and 4.3). This is done by making use of a Schr¨odinger product Sλ,h defined on a (suitable) Sobolev space Hλ,h1 (D).

For the metaharmonic operator Mp,λ := Sp,λ,0 with a real spectral pa- rameter λ, it is possible to construct a Green’s operator (with explicitly de- termined kernel) (at least) on suitable pseudoballs in a complex space. The general question remains as to for what parameters λ the equation (1.2) ad- mits a classical or a weak solution (in a suitable sense). If D is a domain in the complex plane, Garnir [4] proved by an ingenious method that the equation

”Mp,λ(Ψ) =g” admits a distributional solution, provided λ belongs to C\R. This method can be extended to the case of ageneralized Schr¨odinger operator Pλ,hw, defined for complex parameters λand elements w of a linear subspace of L1loc(D) (see (3.6)), by making use of a generalized Poincar´e’s inequality (Lemma 3.1) and introducing a Schr¨odinger map and itsweak Green’s opera- torfor a real (admissible) spectral parameter µ(compare [4]). The solvability of a homogeneous Dirichlet-Neumann problem (see (4.9)) for an (admissible) L2-density g can then be ascertained, in the case λ∈C\R, in a weak sense, namely, there exists an element w (in a Sobolev space) satisfying suitable boundary conditions such that the associated Schr¨odinger functional satisfies the equation

(1.4) hPλ,hw, viD = h[g], viD,

where v ranges over a (suitable) test space (Theorem 5.1). For Neumann prob- lems for related Schr¨odinger type equations in a (possibly) irregular domain in Rn, see [1, 2]. The author is indebted to the referee for suggestions which led to improvements of this paper.

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2.PRELIMINARIES

In what follows, every complex space is assumed to be reduced and has a countable topology. For the definition and basic properties of differential forms on a complex space, see ([5],§4.1). In particular, the exterior differentiationd, the operators ∂, ∂¯ and dc:= (1/4πi)(∂−∂) are well-defined ([5], Chap. 4).¯ Denote by kzk the Euclidean norm of z = (z1,· · · , zm) ∈ Cm, where zj = xj +i yj. Let the space Cm be oriented so that the form υm := (ddckzk2)m is positive. If p:Y →Cm is a holomorphic map, set a0 :=p(a), p[a]:=p−a0 and ra:=kp[a]k, for a∈Y. Clearly the form

υp :=ddcra2 = ( i

2π)∂∂ r¯ a2 is nonnegative and independent of a.

For an open subset G of Y, denote by Ckν(G) the set of C-valued k- forms of classCν on G, Ck,cν (G) the subspace of compactly supportedk-forms (dropping the degree if k= 0), with ν =β to mean locally bounded, ν =m measurable, and ν = λ locally Lipschitzian ([5], §4). Similarly, for Ckλ(G).

A measurable function ψ on Y is said to be locally integrable (f ∈L1loc(Y)) if so is every 2m-form ψχ with χ ∈ C2m0 (Y). Denote by dυ the Euclidean volume element of Cm and define the semivolume form d˜υ := p(dυ) on Y.

If f, ψ∈L1loc(D), set

(f, ψ)D :=

Z

D

f ψd˜υ,

provided the integral exists. Each element f ∈L1loc(D) gives rise naturally to a top-dimensional current, T = [f] :χ7→R

f∧χ, with induced functional h[f], φi:= (−1)m(m−1)2 πm

m!

Z

T ∧φ υmp = (f, φ)D, ∀φ∈Cc1(D).

3. THE SCHR ¨ODINGER PRODUCT

In ([5],§3) theDirichlet productof locally Lipschitz functions ψ, η:D→ C is defined:

[ψ, η][D] :=

Z

D

dψ∧dcη¯∧υm−1p ,

provided the integral exists. It is shown in ([7], (3.6)) that the Dirichlet product is Hermitian symmetric. The Hilbert scalar product on L2(D) is defined by the pairing (f, g)7→(f,g)¯ D (with induced norm k kL2(D)). A generalization of both the Hilbert and Dirichlet product is considered below.

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If ψ = v ∈ C1(D), denote by ∂νψ,1 ≤ ν ≤ 2m, the ν-th first order derivative ∂t∂ψ

ν on D, namely,

∂ψ

∂tν := ∂v

∂x˜k, ifν = 2k−1, ∂ψ

∂tν := ∂v

∂y˜k, if ν= 2k.

Define C1,1(D) := {φ ∈ C1(D)|φxj˜ and φyj˜ ∈ Cλ(D),1 ≤ j ≤ m}. If ψ∈L1loc(D), denote by (the same) ∂νψ theν-th distributionalweak derivative of ψ on D, provided it exists as an element of L1loc(D). Let MD be the subspace of L1loc(D) consisting of all elements w such that: (i) the first order weak derivatives ∂νw =φν, ν = 1,· · ·,2m, exist (as an element of L1loc(D)), namely,

h∂˜t

ν[w], vi=−(w, ∂v

∂˜tν

)D = (φν, v)D, ∀v∈Cc1(D);

(ii) each weak derivative ∂νw ∈ Cβ(D). In the following, let h denote an (2m+ 1)-tuple (h0,· · · , h2m), where each component hj is a locally integrable function on D, with hj ≥ 0 for 1 ≤ j ≤ 2m, and set V := −h0. Then it is easy to show that if w, v ∈ MD, then each of the forms hj(∂jw)(∂j¯v)d˜υ is integrable on D. It follows that for any given complex constant λ, the sesqilinear form

(3.1) Sλ,h(w, v) := ((λ−V)w,¯v)D+

m

X

j=1

Z

D

h2j−1

∂w

∂x˜j

∂¯v

∂x˜j

+h2j

∂w

∂y˜j

∂¯v

∂˜yj

d˜υ

is well-defined on MD × MD. Since an element of C1(D) gives rise to a member of MD, the Schr¨odinger form Sλ,h : C1(D)×C1(D) → C is well- defined by (3.1). In the following, the reference to ”µ,h” will be abbreviated simply as ”µ, V” whenever h=h{1} := (−V,1,· · ·,1). Thus,

(3.2) Sλ,V(w, v) =S

λ,h{1}(w, v) = ((λ−V)w, ¯v)D +S0,(0,h

1,···,hm)(w, v).

Green’s first identity ([6], (5.9)) yields a relation between the operator S˜p,λ,V and the form Sλ,h:

(3.3) hS˜p,λ,V [w], vi = Sλ,V(w,¯v), ∀(w, v)∈C1(D)×Cc2(D).

In the following, let µ denote a real constant. Then Sµ,h defines a Hermi- tian symmetric, continuous form on C1(D) (with induced seminorm denoted accordingly by k kµ,h). Let ∆ be the branch locus of p and set ∆0 =p(∆).

If a ∈ D0 := D\p−1(∆0), a connected neighborhood U b D can be chosen so that p|U is a covering projection onto a ball U0 bCm[1] with sheets Uν. In terms of such U =Uα, a C-partition of unity {(Uνα, ραν)}, each Uνα be- ing a sheet of Uα, is said to be distinguished for D0, if, for any real-valued

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φ∈C1(D), setting φαν :=ρανφ, one has Z

D0

2m

X

j=1

(X

α6=β

hj∂φαµ

∂tj

∂φβν

∂tj

) d˜υ≥0.1

Definition 1. A pair (µ,h) (as above) is said to beadmissible for D if:

(i) hD := min{ess infDhj|j= 1,· · ·,2m}>0; (ii) D0 admits a distinguished partition of unity; and (iii) denoting by P

Cm[1] the Poincar´e’s constant for the Euclidean unit ball,

µ−ess supDV >− hD (PCm[1])2.

A pair (µ, V) is said to beadmissible for D if so is (µ,h{1}).

If the map p exhibits D as a covering space of an open all in Cm, then the condition (ii) above holds trivially; in which case, every pair (µ,0) with µ≥0 is admissible for D.

Lemma 3.1 (Generalized Poincar´e’s inequality). If (µ,h) is admissible for D, then there exists a positive constantcD,µ,h (depending on a distinguished covering of D) such that

(3.4) kφkL2(D) ≤ cD,µ,hkφkµ,h, ∀φ∈C1(D).

Proof. It suffices to consider a real-valued φ ∈ C1(D). In terms of a distinguished covering for D0, by the Poincar´e’s inequality for the Euclidean unit ball, one has

hD (PCm[1])2

Z

D

kφk2d˜υ ≤ hD (PCm[1])2

X

α,ν

Z

(Uα)0

k(φαν)0k2

≤hDX

α,ν

Z

(Uα)0

k∇(φαν)0k2dυ≤ Z

D0

X

α,ν 2m

X

j=1

hjk∂(φαν)0

∂tj k2d˜υ

≤ Z

D0

2m

X

j=1

hj(X

α, β

∂(φαµ)0

∂tj

∂φ(φβν)0

∂tj

) d˜υ=S0,(0,h

1,···,hm)(φ, φ), where (φαν)0 : (Uα)0 →C is induced by the restriction of φ to a sheet of Uα. Therefore, one has

kφk2

µ,h ≥ µ−ess supDV + hD (PCm[1])2

kφk2

L2(D) =:κD,µ,hkφk2

L2(D).

1This condition is likely to be superfluous by making judicious choice of the partition functions.

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In the following, assume (unless otherwise mentioned) that (µ,h) is ad- missible for D. Then the Hermitian form Sµ,h is elliptic, hence defines a scalar product on C1(D). With respect to the induced norm k kµ,h the completion of C1(D) (respectively, Cc(D)) gives rise to theSobolev space Hµ:=Hµ,h1 (D) (respectively, Hµ,h,c1 (D)) equipped with the (naturally extended) scalar prod- uct Sµ,h given by the right-hand side expression of (3.1). It follows that, if φ∈Hµ, then the generalized Poincar´e’s inequality (3.4) holds for all φ∈Hµ. Clearly, this inequality implies that Hµ⊆L2(D). Taking (µ, V) = (1,0), the Sobolev spaces H1(D) and Hc1(D) are defined. Moreover, the Schr¨odinger form (3.2) extends to a scalar product [ , ]1,0,D =S1,(0,1,···,1) on H1(D) with associated energy norm denoted by k k1,0,D.

In what follows, let λ denote a complex (possibly real) constant, and (λ,h) a pair with h = {−V, h} where each hj ∈ Cλ(D) (unless otherwise mentioned). It can be shown that, if w∈ MD, the integral

(3.5) h∂k[hjjw],φi¯ =−(hjjw, ∂kφ)¯ D

exists for all φ∈Cc1(D) and j, k = 1,· · · ,2m ([8], §3, Remark 1). Therefore, each such element w gives rise to aSchr¨odinger functional Pλ,hw:Cc1(D)→C defined by

(3.6) hPλ,hw, viD := Sλ,h(w, v), ∀v∈Cc1(D).

Note also, that the right-hand side of (3.6) serves to extend (i) the action of the functional Pµ,hw to Hµ,h1 (D) and (ii) its definition to w∈Hµ,h1 (D). If w∈C1,1(D), the functional

(3.7) hSλ,h(w), φiD := ((λ−V)w,φ)¯ D−X

j

Z

D

j(hjjw) ¯φd˜υ

is well-defined for all elements φ∈Cc1( ˜D). Consequently, the functional ”φ7→

hSµ,h(w), φiD” extends naturally to an operator (denoted by the same) on Hµ,h1 (D). In the following, assume that D⊆Y is a weak Stokes domain with nonempty (maximal) boundary manifold dD of Dreg (oriented to the exterior of the manifold Dreg of simple points of D) ([5], p. 218).

Proposition 3.1. Assume that w∈C1,1(D). Then for all φ∈C1,1(D), (3.8) hSλ,h(w), φiD +

2m

X

j=1

Z

dD

φ¯nj(hjjw) dσ =Sλ,h(w, φ),

where dσ denotes the surface element of dD, n the unit outward normal to dD. If further λ = µ ∈ R, the above formula remains valid for all φ ∈ H1

µ,h,c(D).

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Proof. For all w, φ ∈ C1,1(D), the formula (3.8) is a consequence of the Green’s formula. Assume now that λ = µ ∈ R. An arbitrary element φ∈Hµ,h,c1 (D) is approximable in the L2-norm on L2(D) by a sequence φn∈ Cc(D). Then

Sλ,h(w, φn)= ((µ−V)w,φ¯n)D+

2m

X

j=1

Z

dD

φ¯nnj(hjjw) dσ− Z

D

φ¯nj(hjjw) d˜υ

=hSµ,h(w), φni

D+

2m

X

j=1

Z

dD

φ¯nnj(hjjw) dσ

→ hSµ,h(w), φiD +

2m

X

j=1

Z

dD

φ¯nj(hjjw)dσ, as n→ ∞,

thus, proving formula (3.8).

4. WEAK SOLVABILITY OF SCHR ¨ODINGER TYPE EQUATIONS

A pair (µ,h) (with h as in (3.1)) is said to bestrictly admissible for D, if

(4.1) kφk1,0,D ≤Const.kφkµ,h, ∀φ∈Cc( ˜D),

(noting that Cc( ˜D)⊂Cc1(D)). To each such pair (µ,h) the associated ”Sµ,h” is positive definite, hence defines a scalar product on Hµ,h1 (D).

It is shown in ([8], §3) that the k-th partial derivatives ∂x˜kT and ∂y˜kT of a regular current T = [ψ] (where ψ∈L1loc(D)) can be defined as (2m−1)- dimensional quasicurrents on D (which are currents if D is nonsingular). As a consequence, given g= (g0, . . . , g2m) with gj ∈L2(D), j= 0,1,· · · ,2m, by extending each gj to ˜D by setting ˜gj = 0 on ˜D\D, there is an associated quasicurrent

(4.2) bgc := ˜g0+

m

X

j=1

˜xk[˜gj] +∂˜yk[˜gj]

on ˜D, and an induced functional (denoted by the same) acting on Cc( ˜D):

(4.3) hbgc, φiD := (g0,φ)¯ D

m

X

j=1

(gj, ∂φ¯

∂x˜j

)D + (gj, ∂φ¯

∂y˜j

)D

.

Let H−1(D) denote the linear space of all such 2m-tuples g. It follows from (4.3) and H¨older’s inequality that, if (µ,h) is strictly admissible for D,

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then

(4.4) |hbgc, φiD| ≤Const. max

0≤j≤2mkgjkL2(D)kφkµ,h, ∀φ∈Cc( ˜D).

Thus, the linear operator bgc is continuous on Cc(D) with respect to the norm k kµ,h (or k kµ,V, if h = h{1}). The subspace Cc(D) being dense in Hµ,{h},c1 (D), the operator bgc admits a unique continuous linear extension (denoted by the same) to the latter (relative to k kµ,h), hence also to Hµ,h1 (D), by the Hahn-Banach’s theorem. The operator bgc is said to be representableby an element η∈L2(D), if the action of bgc on Cc(D) agrees with the (conjugated) action of [η]. Note that if each gj has weak derivative

νgj belonging to L2(D) for ν= 1,· · · ,2m, then bgc is representable by an element of L2(D). The subspace of ”Schr¨odinger functions” of type (µ, V), namely,

Sµ,V(D) := {w∈H1

µ,V(D)|P

µ,h{1}w= 0},

is given precisely by the orthogonal complement of Hµ,V,c1 (D) (with respect to Sµ,V):

(4.5) H1

µ,V(D) =H1

µ,V,c(D)⊕ Sµ,V(D).

This is an easy consequence of the identity (3.3).

Definition 2. Let g ∈H−1(D) and f ∈C1,1(D). An element Ψ =w∈ Hµ,V1 (D) is called aweak solution of the Dirichlet problem

(4.6) Sp,µ,V(Ψ) =− 4pΨ + (µ−V) Ψ = bgc, Ψ = f on dD, provided that w≡f mod (Hµ,V,c1 (D)) and

Sµ,V(w, ρ) = ((µ−V)w, ρ)¯D +S0,0(w, ρ) = hbgc, ρiD, ∀ρ∈Hµ,V,c1 (D).

It is a consequence of Green’s first identity that, if g ∈ L2(D), every weak solution Ψ =ψ∈C1,1(D) of the Dirichlet problem

(4.7) Sp,µ,V (Ψ) = g in D, Ψ = 0 on dD, is a classical solution of the Schr¨odinger equation (1.2) in D.

Theorem 4.1. Assume that either (i) g ∈ L2(D) and (µ, V) is ad- missible for D or (ii) g ∈ H−1(D) such that bgc is representable by an element η ∈ L2(D) and (µ, V) is strictly admissible for D. Then for each f ∈C1,1(D)∩C2(D) with 4pf ∈L2(D), the Dirichlet problem (4.6)admits a unique weak solution in Hµ,V1 (D).

Proof. Let F := 4pf. The homogeneous Dirichlet problem (4.7) with

“g” replaced by “ ˆg:=η+F−(µ−V)f”, where η=g whenever g∈L2(D),

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admits a weak solution ψ=w0 ∈Hµ,V,c1 (D), by Riesz’s representation theorem (applied to the functional bˆgc). Then the function Ψ :=w0+f gives a weak solution of the Dirichlet problem (4.6). For the equation (3.3) (with µ = 0) and the Hermitian symmetry of the Dirichlet product imply that

S0,0(f, φ) = −(4pf, φ)¯ D, ∀φ∈Cc(D), hence, by virtue of definition (3.2), one has

Sµ,V(Ψ, φ) =Sµ,V(w0, φ) +Sµ,V(f, φ)

=hbˆgc, φiD+ ((µ−V)f, φ)¯ D+S0,0(f, φ) =hbgc, φiD, thus, proving the above claim.

The difference ψ= Ψ1−Ψ2 of two solutions of (4.6) belongs to Hµ,V,c1 (D) and the functional ”Sµ,V(ψ, )” annihilates Hµ,V,c1 (D). Hence, the decomposi- tion (4.5) implies that ψ= 0.

To consider a mixed Dirichlet-Neumann boundary value problem with a partition (1.3) of∂D\A, let N be an open neighborhood of (∂D)N contained in ˜D and set ˆD := D∪ N. Denote by R0,D the closure in Hµ,h1 (D) of the subspace consisting of φ∈ Cc( ˆD) that vanishes in the intersection with D of some neighborhood of (∂D)D (possibly depending on φ). Let Aµ,h be a maximal linear subspace of Hµ,h1 (D) admitting a (Schr¨odinger) continuous linear mapping sµ,h :Aµ,h →L2(D) satisfying the equation

(4.8) hPµ,hw, viD = (sµ,h(w),v)¯ D, ∀v∈Cc(D).

It can be shown that, for each w ∈ C1,1(D)∩ Aµ,h, the (conjugated) action of [sµ,h(w)] on R0,D agrees with the action of [Sµ,h(w)]. Define

N0,Dµ :={w∈ Aµ,h∩R0,D|Sµ,h(w, φ) = (sµ,h(w),φ)¯ D, ∀φ∈R0,D}.

Definition 3.A weak solution to the Dirichlet-Neumann problem (4.9) Sµ,h(w) = bgc, w∈ N0,Dµ ,

where g∈H−1(D), is an element w∈ N0,Dµ satisfying the equation (4.10) hPµ,hw, φiD = hbgc, φiD, ∀φ∈R0,D.

Theorem4.2. Assume that either(i) g∈L2(D) and (µ,h) is admissible for D or (ii) g ∈ H−1(D) such that bgc is representable by an element of L2(D) and (µ,h) is strictly admissible for D. Then the Dirichlet-Neumann problem (4.9)admits a unique weak solution.

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Proof. Let Bµ,h be the set of all elements w∈Hµ,h1 (D) such that the anti- linear form v 7→ hPµ,hw, viD is continuous on Hµ,h,c1 (D) with respect to the L2-norm on L2(D). Bµ,h is clearly a linear subspace of Hµ,h1 (D). According to the Hahn-Banach’s theorem, the said anti-linear form is extendable to a unique continuous anti-linear form on L2(D), and this extension is given by taking the scalar product of a unique element of L2(D), to be denoted by sµ,h(w), with v. Thus the equation (4.8) holds for all v∈Hµ,h1 (D). With g given as either in (i) or, in (ii), the functional φ 7→ hbgc, φiD is well-defined and bounded on R0,D (by the inequality (4.4)). Hence, the Riesz’s representation theorem implies that there is a unique element w0 ∈R0,D satisfying the equation (4.11) hbgc, φiD =Sµ,h(w0, φ), ∀φ∈R0,D

with

(4.12) kw0kµ,h ≤ Const. max

0≤j≤2mkgjkL2(D).

If bgc is representable (on Cc(D)) by an element η ∈ L2(D) (or, if g ∈ L2(D)), then by the formula (3.6), the equation (4.8) holds with sµ,h(w0) =η(or g), for all v∈Cc(D). One has, for such v, |hPµ,hw0, viD| ≤ kηkL2(D)kvkL2(D), thereby showing that w0 ∈ Bµ,h. Moreover, by the equation (4.11), this element w0 belongs to N0,Dµ , thus giving a weak solution to the problem (4.9).

The difference ψ = w1−w2 of two weak solutions of (4.9) satisfies the conditions ψ∈ N0,Dµ ⊂R0,D and hPµ,hψ, φiD = 0 on R0,D. Hence, it follows from the Poincar´e’s inequality (3.4) (or (4.1)) that kψkL2(D) = 0, and thus ψ= 0.

Theorem 4.3. If (µ,h) is admissible for D, then the Schr¨odinger map sµ,h : Aµ,h → L2(D) is invertible with inverse given by a continuous, self- adjoint and strictly positive isomorphism Gµ,h : L2(D) → N0,Dµ (called the weak Green’s operator for (4.9)); in particular, for g ∈ L2(D), the element w = Gµ,hg gives the unique weak solution to the Dirichlet-Neumann problem (4.9). Moreover, for admissible (µj,h), j= 1,2, the operators Gµj,h are com- mutative.

Proof. Given g ∈ L2(D), the association g 7→ Gµ,hg := w, the unique weak solution to Dirichlet-Neumann problem (4.9) is well-defined by Theo- rem 4.2. The equations (4.11)–(4.12) imply that

(4.13) Sµ,h(Gµ,hg, φ) = (g,φ)¯ D, ∀φ∈R0,D, with

(4.14) kGµ,hgkµ,h ≤Const.kgkL2(D).

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Furthermore, the induced mapping Gµ,h : L2(D) → N0,Dµ is a bounded linear operator giving the left-inverse of sµ,h|N0,Dµ . Setting φ = Gµ,hf with f ∈L2(D) in (4.13) yields

(4.15) Sµ,h(Gµ,hg,Gµ,hf) = (g,Gµ,hf)D. Also, by (4.13) one has

Sµ,h(Gµ,hf,Gµ,hg) = (f,Gµ,hg)D. Consequently,

(4.16) (g,Gµ,hf)D = (f,Gµ,hg)D = (Gµ,hg,f)¯D,

thus proving that Gµ,h is a self-adjoint endomorphism of L2(D). Formula (4.15) shows that the integral (Gµ,hg,g)¯ D is real-valued, and thus, by virtue of (3.4), one has, by relations (4.15) and (4.16),

(Gµ,hg,g)¯ D ≥(cD,µ,h)−2kGµ,hgk2

L2(D).

On the other hand, the H¨older’s and the generalized Poincar´e inequalities imply that

(Gµ,hg,g)¯ D ≤Const.kGµ,hgkµ,hkgkL2(D). Therefore, by the inequality (4.14),

kGµ,hgk2

L2(D) ≤Const.kgk2

L2(D),

so that Gµ,h, is a continuous endomorphism of L2(D). Furthermore, Gµ,h is strictly positive, since in fact, the condition Gµ,hg= 0 gives g=s{µ,h}(Gµ,hg) = 0, so that, if g ∈ L2(D)\{0}, then (Gµ,hg, ¯g)D > 0. Finally, for admissible (µj,h), j= 1,2, one has

Gµ1,hg=Gµ1,h(sµ

0,h(Gµ0,hg)) =Gµ1,h([sµ

1,h−(µ1−µ0)]Gµ0,hg)

=Gµ1,h(sµ

1,h(Gµ0,hg))−(µ1−µ0)Gµ1,h(Gµ0,hg)), and thus, the identity

(4.17) Gµ1,h− Gµ0,h=−(µ1−µ0)Gµ1,hGµ0,h

holds. In particular, the commutativity of the Gµj,h follows.

5. THE GENERALIZED GARNIR’S THEOREM

In the rest of this paper, assume that h = (−V, h1,· · ·, h2m) is given (as in (3.1)) with hD > 0. The purpose of this section is to consider, for a given g∈L2(D), the solvability of the operator equation (1.4) with v ranging

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over R0,D. If λ∈C\R, a positive number µ0 is called an admissible value for (V, D, λ), provided (µ0, V) is admissible for D and λ−µ0 ∈ C\R. The method of Garnir [4] (compare [3], 5.13.8) can be extended to yield the following:

Theorem 5.1 (Generalized Garnir’s Theorem). Let λ∈C\R and µ0

be an admissible value for (V, D, λ). Then there exists a continuous endomor- phism Gλ,h of L2(D) (independent of the choice of µ0) taking values in N0,Dµ0 , such that the element w:=Gλ,hg gives a weak solution, in the sense of (4.10), of the Dirichlet-Neumann Problem (4.9) (relative to a partition (1.3)), for each g∈L2(D).

Proof. The existence of the endomorphism Gλ,h, for each λ∈C\R, will be shown in Lemma 6.1 (of the Appendix). Observe that for such λ one has

hPλ,hGλ,hg, viD =λ(Gλ,hg,v)¯ D +hP0,hGλ,hg, viD

=λ(Gλ,hg,v)¯ D + (s0,h(Gλ,hg),¯v)D.

for all g, v ∈L2(D). As will be shown in Lemma 6.2, the terms on the right- hand side of this equation are holomorphic in C\R (as a function of λ). The equation (1.4) itself is known to hold for all v ∈ R0,D, in case λ = µ is a real ”admissible” spectral parameter, namely, µ≥ ess supDV. Therefore, the principle of analytic continuation ensures its validity for all λ∈C\R.

6.APPENDIX: GREEN OPERATORS WITH COMPLEX SPECTRAL PARAMETERS

Lemma 6.1. Let λ∈C\R. Then for each admissible µ0 for (V, D, λ), the mapping

(6.1) Gλ,h :=Gµ

0,h[I+ (λ−µ0)Gµ

0,h]−1

is a continuous endomorphism of L2(D) (independent of the choice of µ0), taking values in N0,Dµ0 .

Proof. Since Gµ0,h is a continuous, positive, self-adjoint endomorphism of L2(D), the inverse map [I+ (λ−µ0)Gµ0,h]−1 exists as a continuous endomor- phism of L2(D) (by [3], Lemma, p. 359). To show that the operator Gλ,h is well-defined by (6.1), choose admissible µj >0, j = 0,1 for (V, D, λ). Then, equivalent to the identity (4.17), one has

(6.2) [I+ (λ−µ0)Gµ0,h]Gµ1,h = [I+ (λ−µ1)Gµ1,h]Gµ0,h, One the other hand one has, clearly,

Gµ1,h[I+ (λ−µ0)Gµ0,h] = [I + (λ−µ0)Gµ0,h]Gµ1,h

= [I + (λ−µ1)Gµ1,h]Gµ0,h. (6.3)

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Multiplying (6.3) on the right by [I+ (λ−µ0)Gµ0,h]−1 and on the left by [I+ (λ−µ1)Gµ1,h]−1 yields

[I+ (λ−µ1)Gµ1,h]−1Gµ1,h = Gµ0,h[I+ (λ−µ0)Gµ0,h]−1. It will be shown below that the following identity holds:

(6.4) [I+ (λ−µ1)Gµ1,h]−1Gµ1,h = [I+ (λ−µ0)Gµ0,h]−1Gµ0,h. It follows from this that

Gµ0,h[I+ (λ−µ0)Gµ0,h]−1 = [I+ (λ−µ0)Gµ0,h]−1Gµ0,h,

and the same identity holds with µ0 replaced by µ1. Consequently from this and (6.4) one concludes, as desired, that

Gµ0,h[I+ (λ−µ0)Gµ0,h]−1 = Gµ1,h[I+ (λ−µ1)Gµ1,h]−1.

To prove the identity (6.4), observe that the operators I + (µ−µ1)Gµ1 and I+ (µ−µ0)Gµ0 commute; hence, setting w:= [I+ (µ−µ1)Gµ1]−1v, one has

[I+ (µ−µ1)Gµ1][I+ (µ−µ0)Gµ0]w= [I+ (µ−µ0)Gµ0]v.

Then,

[I+ (µ−µ0)Gµ0][I+ (µ−µ1)Gµ1]−1v= [I+ (µ−µ1)Gµ1]−1[I+ (µ−µ0)Gµ0]v.

Therefore,

[I+ (µ−µ0)Gµ0][I+ (µ−µ1)Gµ1]−1 = [I+ (µ−µ1)Gµ1]−1[I+ (µ−µ0)Gµ0].

Consequently, the identity (6.4) will result from the relation Gµ0,h = [I+ (λ−µ0)Gµ0,h][I+ (λ−µ1)Gµ1,h]−1Gµ1,h

= [I+ (λ−µ1)Gµ1,h]−1[I+ (λ−µ0)Gµ0,h]Gµ1,h, namely, the above relation (6.2).

Lemma 6.2. Given a pair (g, ρ)∈L2(D)×L2(D), the function gh(λ) :=

(s0,h(Gλ,hg),ρ)¯D is holomorphic in C\R. The same is true for the function ˆ

gh(λ) := (Gλ,hg,ρ)¯D.

Proof. Let λ0 ∈ C\R be given and choose µ0 for (V, D, λ0) as in Lemma 6.1. Write λ00+ε. Taking ε with sufficiently small modulus, the same µ0 can be used in defining Gλ0,h. With K:= [I+ (λ0−µ0)Gµ0,h]−1, the definition (6.1) implies that

gh0)−gh0) = (s0,h(Gµ0,h(K−1+εGµ0,h)−1g),ρ)¯ D−(s0,h(Gµ0,hKg),ρ)¯ D

= (s0,h(Gµ0,h(I +εKGµ0,h)−1Kg),ρ)¯D −(s0,h(Gµ0,hKg),ρ)¯D.

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If |ε| is so small that

|ε| kGµ0,hk

L2(D) ≤(kKk

L2(D))−1, then the series P

n≥0(−1)nεn(KGµ0,h)n) converges (relative to the L2 operator norm) and its sum is none other than (I+εKGµ0,h)−1. Consequently, one has

s0,h(gh0))−s0,h(gh0)) =X

n≥1

(−1)nεn(s0,h◦ Gµ0,h) (KGµ0,h)nKg),ρ)¯D, the series on the right being necessarily convergent for ε with sufficiently small modulus. The holomorphy of gh(λ) in a neighborhood of λ0 is thereby established.

REFERENCES

[1] Angelo Alvino, Andrea Cianchi, Vladimir Maz’ya and Anna Mercaldo,Well-posed elliptic Neumann problems involving irregular data and domains. Ann. Inst. H. Poincar´e Anal.

Non Lin´eaire27(2010),4, 1017–1054.

[2] Andrea Cianchi and Vladimir Maz’ya,Boundedness of solutions to the Schr¨odinger equa- tion under Neumann boundary conditions. J. Math. Pures Appl. (9) 98 (2012), 6, 654–688.

[3] R.E. Edwards, Functional Analysis. Holt, Rinehart and Winston, Chicago, Toronto, London, 1965.

[4] H.G. Garnir,Les Probl´emes aux Limites de la Physique Math´ematique.Birkh¨auser, Basel- Stuttgart, 1958.

[5] C. Tung, The first main theorem of value distribution on complex spaces. Memoire dell’Accademia Nazionale dei Lincei, Serie VIII,XV(1979),1, 91–263.

[6] C. Tung, Semi-harmonicity, integral means and Euler type vector fields. Adv. Appl.

Clifford Algebr.17(2007), 555–573.

[7] C. Tung, Integral products, Bochner-Martinelli transforms and applications. Taiwanese J. Math. 13(2009),5, 1583–1608.

[8] C. Tung,On Wirtinger derivations of currents and conditions for weak holomorphy and harmonicity. Preprint (2013) 22 pp.

Received 1 August 2013 Minnesota State University, Department of Mathematics and Statistics,

Mankato, MN 56001 chia.tung@mnsu.edu

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