Elliptic model problems including mixed boundary conditions and material heterogeneities

8/10/2019 Elliptic model problems including mixed boundary conditions and material heterogeneities

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J. Math. Pures Appl. 89 (2008) 2548

www.elsevier.com/locate/matpur

Elliptic model problems including mixed boundary conditionsand material heterogeneities

Robert Haller-Dintelmann , Hans-Christoph Kaiser, Joachim Rehberg

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

Received 5 July 2007

Available online 25 September 2007

Abstract

We present model problems in three dimensions, where the operator maps the Sobolev space W1,p () isomorphicallyontoW

1,p

() for ap > 3. The emphasis is here on the case where different boundary conditions meet material heterogeneities.

2007 Elsevier Masson SAS. All rights reserved.

Rsum

Cet article prsente des situations modles, en trois dimensions, dans lesquelles loprateur est un isomorphisme deW

1,p

() sur W1,p

() pour un p >3. On sintresse notamment au cas o des conditions au bord mixtes Dirichlet/Neumann

sont combines avec des sauts du coefficient .

2007 Elsevier Masson SAS. All rights reserved.

MSC:35B65; 35J25; 35R05

Keywords: Elliptic transmission problems; Mixed boundary problems; W1,p regularity

1. Introduction

Many elliptic problems originating from science, engineering, and technology exhibit mixed boundary conditions

and non-smooth material parameters, see [1,41] and the references cited there. For instance, in the simulation of

operation and fabrication of semiconductor devices one is regularly confronted with heterogeneous materials in the

volume and on the boundary (contacts), see [46], while dealing with elliptic and parabolic equations as mathematicalmodels, see [17]. However, not much is known concerning maximal regularity for elliptic operators which include

mixed boundary conditions. Moreover, most of this is restricted to Hilbert space scales, see e.g. [44,43,25,12,8,7].

Unfortunately, the Hilbert spaceH3/2 is a principle threshold for mixed elliptic, second order problems at least in the

case when the Dirichlet and Neumann boundary part meet on smooth parts of the boundary, see [48] and also [44].

Thus, within this scale one cannot expect an embedding of the domains of these operators inL (or even in C ) in

* Corresponding author.

E-mail addresses:[email protected] (R. Haller-Dintelmann), [email protected] (H.-C. Kaiser),

[email protected] (J. Rehberg).

0021-7824/$ see front matter

2007 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.matpur.2007.09.001


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26 R. Haller-Dintelmann et al. / J. Math. Pures Appl. 89 (2008) 2548

case of three or more space dimensions. But exactly this is desirable in view of nonlinear, in particular quasilinear

problems, see [42,26,39].

Concerning optimal regularity in non-Hilbert spaces there are the results of [47,48,5,22,11,20,21]; for the pure

Dirichlet or pure Neumann case see [30] and [55], respectively. Grger proved in [22] that under only L(and ellip-ticity) assumptions on the coefficient function, the Lipschitz property of the domain and very weak assumptions

on the Dirichlet boundary part \ the operator, : W1,p () W1,p (), (1)

is a topological isomorphism for a certain p >2 (W1,p () denoting the subspace of W

1,p() including a trace

zero condition on the Dirichlet boundary part \ , andW1,p () the dual ofW1,p

()). This result has found

numerous applications within the treatment of applied problems. Nevertheless, it is well known that under thesegeneral assumptions one can only expect that p exceeds 2 arbitrarily little. This is the reason why the applications

of [22] remained restricted to two-dimensional problems. Because the demand for three-dimensional modelling and

simulation steadily increases, the question arises under which assumptions the isomorphism property of (1) can be

obtained for a p >3 and, in particular, whether this is true with mixed boundary conditions. Dauge proved in [11]

that if the domain is a convex polyhedron and the border between Dirichlet and Neumann boundary part consists

of (finitely many) line segments, then the Laplacian provides a topological isomorphism between W1,p andW1,pfor some p >3. In this paper we generalise this to prototypical situations where mixed boundary conditions and

heterogeneous, anisotropic coefficient functions occur simultaneously. Thus, this calculus allows for jumps in the

conormal derivative of solutions across internal interfaces. This means, e.g. in electrostatics, that the jump in the

normal component of the displacement + across a prescribed interface equals the surface chargedensity on the interface, and this surface charge density is represented by a distribution on the underlying domain .

In view of an adequate localisation principle, see [22], the geometric constellations we investigate may be viewedas local constituents of rather complex global settings.

Since the knowledge of the singularity of solutions is crucial for the efficiency of numerical methods, there exist of

course several numerical approaches to determine singular exponents of concrete anisotropic problems, see [35,10,51]

and the references therein. For a more general numerical approach to heterogeneous elliptic problems see for instance

[2,27,9,53] and the references cited there.In detail, our results are as follows:

Theorem 1. Let R2 be an open triangle, let further P be the center of one of its sides and the open legbetweenP and one of its neighbouring vertices. Define

def= ]1, 1[and the boundary part as ]1, 1[.Suppose to be a plane withinR3 that intersects {P}]1, 1[ in exactly one point. Assume that the elliptic coefficient

function takes its values in the set of real, symmetric, positive definite 33 matrices and is constant on bothcomponents of\ . Then there is a p > 3such that

: W1,p () W1,p () (2)

is a topological isomorphism.

Theorem 2. Let R2 be an open triangle, be one of its open sides or \one of its closed sides. Define

def= ]1, 1[ and the boundary partas ]1, 1[. Let furtherbe a plane the intersection of which withthe boundary ofconsists of exactly two points. Assume that the elliptic coefficient function takes its values in the

set of real, symmetric, positive definite 33matrices and is constant on both components of\ . Then there is ap > 3such that(2)is a topological isomorphism.

Corollary 3.Let, andbe as in Theorem2. Letbe ]1, 1[combined with the ground plate or/and theupper plate. Let furtherbe a plane as in Theorem2which does neither touch the upper/lower plate and let be as

in Theorem2. Then the conclusion of Theorem2also holds.

Remark 4. The supposition that the plane has only a finite intersection with edges where the Dirichlet boundary part

meets with the complementing boundary part is crucial. If this is not the case, a bimaterial outer edge (see Definition 10


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R. Haller-Dintelmann et al. / J. Math. Pures Appl. 89 (2008) 2548 27

Fig. 1. The model domains under consideration in Theorem 1 (left) and Theorem 2 (right) with generating triangle , Neumann boundary

(hatched area) and material interface(shaded area) cutting the domain.

below) with mixed boundary conditions occurs, for which the appearing singularities may be arbitrarily large, seeRemark 28 in Appendix A.

Remark 5. Let us further mention that in Theorem 1 can be taken as Grgers third model set, see [22],and thus Theorem 1 can be viewed as a regularity assertion for Grgers third model constellation if the coefficient

function has a discontinuity along a plane.

Operators of type (1)which may be seen as the principal part of the (Dirichlet)-homogenization of an ellip-tic operatorare of fundamental significance in many application areas. This is the case not only in mechanics

(see [35, Ch. IV/V]), thermodynamics (see [50]), and electrodynamics (see [49]) of heterogeneous media, but also in

mining, multiphase flow, mathematical biology (see [16,6]), and semiconductor device simulation (see [46,17,19]), in

particular quantum electronics (see [54,4,33,52,53,36]).The nonhomogeneous coefficient function represents varying material properties as the context requires. It may

be thermal conductivity in a heat equation (see [50, 21]), or dielectric permittivity in a Poisson equation, or diffusivityin a transport equation (see for instance [46, 2.2] for carrier continuity equations), or effective electron mass in a

Schrdinger equation (see [33]).

Let us emphasise that the matrices which constitute the coefficient function may be not diagonal and, in par-ticular, not multiples of the identity, see [1] and [35, Ch. IV /V]. This is motivated by the applications, for instance

in heat conduction, see [50, 21.B]. On the other hand anisotropic coefficients are absolutely necessary in view of

(local) deformation and transformation of the domain in the localisation procedure, see Proposition 16. It should benoted that in case of an essentially anisotropic coefficient matrix the generic properties of the elliptic operator differ

dramatically from the case of a scalar coefficient, see [13, Remark 5.1], [14, 4], and [45, Ch. 5].The outline of the paper is as follows: in the next section we will introduce some notation. In Section 3 the strategy

of proof is explained. Section 4 contains some preliminaries which establish the connection between the regularity

of the solution and the edge singularities. In Section 5 we collect some auxiliary results which justify at the end the

transformation of the problem to a Dirichlet one. Section 6 is devoted to the core of the proof of Theorem 1, essentiallybased on the discussion of the edge singularities. In Section 7 we give the proofs of Theorem 2 and of Corollary 3.

Some concluding remarks are given in Section 8. Appendix A finishes the paper by establishing the required estimates

for the occurring singularities for geometric edges and bimaterial outer edges.

2. Notation

Throughout this paper

Rd always denotes a bounded Lipschitz domain (see [24] for the definition) and is an open part of its boundary. W1,p() denotes the (complex) Sobolev space on consisting of those


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Lp() functions whose first order distributional derivatives also belong to Lp() (see [24] or [37]). Note that

enjoys the extension property for W1,p() in view of being a bounded Lipschitz domain, see [18, Thm. 3.10] or [37,

Ch. 1.1.16]. Thus,W1,p() is identical with the completion of the set{v| : vC(R3)}with respect to the normvW1,p def= (

(|v|p + |v|p) dx)1/p . We use the symbol W1,p ()for the closure of,

v| : v C(R3), supp v (\ ) = ,inW1,p(). If= we write as usualW1,p0 ()instead ofW

1,p ().W

1,p ()denotes the dual to W

1,p ()and

W1,p()denotes the dual toW1,p0 (), when 1p+ 1

p=1 holds. If is understood, then we sometimes abbreviateW

1,p ,W

1,p0 andW

1,p , respectively.,X always indicates the duality between a Banach spaceX and its dual; incase ofX= Cd we mostly write,. If is a Lebesgue measurable, essentially bounded function on taking itsvalues in the set of real, symmetric d dmatrices, then we define : W1,2 () W1,2 ()by:

v, wW

1,2

def=

v, w dx; v, w W1,2 (). (3)

The maximal restriction of to any of the spaces W1,p

() (p >2) we will denote by the same symbol.Finally, we define for any two complex numbers ,:

def=exp log || + i arg , arg ], ]; (4)

and for, ], ]with < we define the sector,K

def= (rcos , r sin ): r > 0, ], [.3. Strategy of proof

Because the core of the proof of Theorem 1 is very technical, we will give here an exposition of the ideas

behind it for the convenience of the reader. Clearly, the problem is a mixed boundary value problem with discon-

tinuous coefficients on a convex polyhedron. One should expect that this couldin principlebe treated as in [11]

(see also [38]), where the following is shown for the mixed problem with Hlder continuous coefficients:

If for any edge point or vertex x a certain spectrum (x)satisfies,

p < infz (x),z ]0,1[

2

1 z for any edge point x, (5)

and, additionally,

p < infz (x),z ]0,1[

3

1 z for any vertex x, (6)

then the associated differential operator with Hlder continuous coefficients provides a topological isomorphism

betweenW1,p

andW1,p

. Here, the spectrum (x) is the spectrum of an associated (generalized) SturmLiouvilleoperator, if x is a point from an edge, see the next section for details, and in the case of a vertex it is the spectrum of

an associated LaplaceBeltrami operator, see [11] for details. The problem is that it is already difficult to determine

the spectrum of this LaplaceBeltrami operator if the coefficient function is constant, and we have no idea how to do

this in the case of heterogeneous materials.

Fortunately, there is a way out of this dilemma: for Dirichlet problems a deep idea of Mazya [39] permits to restrict

the investigation to the edge singularities as far as the integrability of the gradient of the solution up to an index p > 3

is concerned, see Proposition 11 below. (This heavily rests on the a priori known Hlder continuity of the solution,

see [34, Ch. III.14].) So we may circumvent the analysis of the vertex singularities, if we can transform the problem

to an equivalent one with Dirichlet boundary conditions. The strategy of proof is thus the following: We first deform

the problem via a bi-Lipschitz (in fact: piecewise linear) mapping, such that in the resulting polyhedron the Neumann

boundary part is a complete side of it, see Fig. 2. When doing so, we have to show that under this deformation

the occurring spaces W1,p and W1,p are suitably mapped on spaces of the same quality and that, additionally,


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Fig. 2. Model domain, original (left) and transformed (right), with Neumann boundary (hatched area) and material interface (shaded area). N.B.

the material interface is flexed after transformation, and there is an additional material interface framed by the dashed lines after transformation.(,, andkeep their names after transformation.)

the differential operator goes over into a similar one (see Proposition 16). In a second step we reflect the problemacross the Neumann boundary part (see Proposition 17) and identify the detailed structure of the resulting (Dirichlet)

problem. One especially obtains additional edges in the interior of the polyhedron coming from the transform and thereflection process (see also Fig. 2). The main part of the proof is then to show, that all edges fulfill the supposition of

Proposition 11, which stands in an obvious relation to (5). Regrettably, this latter is a touchy business, heavily restingon an adequate reformulation of the transmission conditions for the SturmLiouville problem (see Section 6.2) and,

finally, on the sophisticated estimates for the geometrical and bimaterial outer edges, which we present in Appendix A.

4. Edge singularities

In this section we first recall the optimal regularity result from [39] for heterogeneous Dirichlet problems on

polyhedral domains and explain how to identify the occurring edge singularities.

Definition 6. Let numbers 0 < 1


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Definition 8. Let R3 be a polyhedron which, additionally, is a Lipschitz domain and{k}k a (finite, disjoint)polyhedral partition of . Let be a matrix function on which is constant on each k and takes real, symmetric,

positive definite 3 3 matrices as values. Take any edge E of any of the k s and consider an arbitrary inner point Pof this edge. Choose a new orthogonal coordinate system (x,y, z) with origin at the point P such that the direction

ofE coincides with the z-axis. We denote by OE the corresponding orthogonal transformation matrix and by E,P

the piecewise constant matrix function which coincides in a neighbourhood ofP with OE (O1E (x+P ))O1E andwhich satisfies:

E,P(tx,ty,z) = E,P(x,y, 0), for all(x,y, z) R3, t > 0. (9)ByE (,)we denote the upper left 2 2 block ofE,P(,, 0).

Remark 9.There exist angles 0< 1 0arbitrarily small), then there is ap > 3such that

: W1,p

0 () W1,p

() (10)is a topological isomorphism.

Remark 12.Unfortunately, there are some errors in the paper [39], cf. also [13, Remark 2.2]. First, the assertion of

[39, Thm. 2.3] that the exponent p can be taken from the interval[2, 2/(1 )[is erroneous, since the assumptionsof [39, Thm. 2.4] have to be taken into account. The correct formulation of the linear regularity result proved in [39]

is given in Proposition 11 above. Furthermore, the signs in formulas for the coefficients of certain generalized Sturm

Liouville equations are not correct, in detail: in [39, p. 240] there is a wrong sign in the formula for the Mellin

transformrr u = u, which has to be replaced byrr u = u. Therefore the formulas [39, (3.33)] for the sesquilinearforma(u, v; )and [39, (3.32)] for the corresponding differential problem differ in sign from the correct formulas (8)and (11), (12). The correctness of the other considerations given in [39] is not affected by this.

Thus the question arises how to find the parameters for which the operatorAhas only a trivial kernel. One proceeds

as follows: standard arguments show that any function u from the kernel of the operator A obeys the differential

equation,

(b2u)+ (b1u)+ b1u+ 2b0u =0, (11)

on each of the intervals]j, j+1[. Additionally, in every point {1, . . . , n1}the transmission conditions,[u]=0, [b2u+ b1u]=0, (12)

have to be satisfied. (As usual,[w]stands for limw() limw().) In order to find the critical parameters, one employs the elementary solutions of the differential equation (11) on each of the subintervals ]j, j+1[,

ei(e2i + 1), ei(e2i + 1),


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which were announced in the pioneering paper [10] (see also [39, Ch. 3.6] for further details). The complex number

= jis determined by the matrix:

m =

m11 m12m12 m22

def=

j

11 j

12

j

12 j

22

,

as

def= i(m22 D

1/2m ) m12

i(m22+ D1/2m ) + m12, (13)

whereDm denotes the determinant of the matrix m.

Remark 13.Becausem22is positive, necessarily satisfies 0 || < 1. Moreover, ifm11 m12m12 m22

=

m11 m12m12 m22

,

then

= .

Making on any interval]j, j+1[an Ansatz

uj( )def= cj,+ei(je2i + 1) + cj,ei(je2i + 1), (14)

these functions automatically satisfy (11), while the boundary conditions together with the transmission condi-

tions (12) for= j (j {1, . . . , n 1}) lead to a 2n 2nhomogeneous linear system for the coefficients cj,+, cj,.The usual criterion for the (nontrivial) solvability of this system gives the characteristic equation of the prob-

lem (11), (12) and allows (in principle) to determine the critical values or at least to give estimates for the real part

of them. In the next sections we will do this for all edges resulting from our problems.

5. Auxiliary results

Lemma 14. In the terminology from above let with ]0, 1[ be a number such that there exists a (nontrivial)function vH from the kernel ofA, see Definition6. Let13, 23 and33 be real valued, bounded, measurablefunctions onK

n0

and define the coefficient function on Kn0

def= K n0 R by:

(x,y,z)def= j11 j12 13(x,y)j12 j22 23(x,y)

13(x, y ) 23(x , y) 33(x,y)

, if(x,y) Kj+1j . (15)Then there is a compactly supported elementf W1,6(Kn0 )such that thealso compactly supportedvariationalsolution

W

1,2

0 (K

n

0)of

=

f on Kn

0does not belong to W

1, 210

(Kn

0).

Proof. It is not hard to calculate that the function0 given by,

0(x,y) = (x2 + y2)/2v

arg(x+ iy), (16)belongs to W

1,ploc (K

n0

) if p [2, 21 [ but not to W1, 21loc (K

n0

). (Recall that v does not vanish identically on]0, n[.) By construction ofA, the function0satisfies,

0=0, (17)

in the distributional sense, see [39]. We define now the function by=(x,y, z)def= 0(x,y)and notice that belongs to W

1,p

loc (K

n

0) for p

[2, 2

1 [, but not to W

1, 21loc

(Kn

0). Suppose

=

1

2with

1C

0 (K

n

0) and

2 C0 (R), then


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32 R. Haller-Dintelmann et al. / J. Math. Pures Appl. 89 (2008) 2548K

n0

, R3dx dy dz =R

K

n0

0, 1R2dx dy 2(z) dz

+K n0

130

x+ 23

0

y 1 dx dy

R2

zdz. (18)

The first addend vanishes by (17) and the second by 2C0 (R). The set of s with the above tensor productstructure is total in C0 (K

n0

), therefore (18) is also zero for any from this latter space. Let be a function from

C0 (R3)which equals 1 in a neighbourhood of 0R3 and which vanishes outside a ballB . Then one calculates for

any C0 (Kn0 ):K

n0

(), dx=

Kn0

, dx+K

n0

, dx+K

n0

, ()dx. (19)

Kn0 , () dx vanishes because (18) always is zero ifC0 (Kn0 ). On the other hand, it is not hard tosee that the other two addends on the right-hand side definein their dependence on continuous linear forms

on W1,6/50 (K

n0

), namely: the property W1,2loc(Kn0 ) and the compact support property of imply, L2(K

n0

). Combining this with the embedding W1,6/50 (K

n0

) L2(Kn0 ), the claim becomes clear for the first addendfrom the right-hand side of (19). Concerning the second addend, one easily estimates:

K

n0

, dx L(Kn0 )L6(BKn0 )W1,6/50 (Kn0 )

L(Kn0 )

W1,2(BKn0 )

W1,6/50 (K

n0

).

Thus, setting def= , one obtains the assertion.

Remark 15.Ifn= 0+ 2 , thenK n0 =R2,Kn0

=R3; hence,W1,p0 (Kn0 ) = W1,p(R3).

Proposition 16.LetRd be a bounded Lipschitz domain andbe an open subset of its boundary. Assume thatis a mapping from a neighbourhood of into Rd which is bi-Lipschitz. Let us denote ()= and ( )=.Then

(i) For anyp ]1, [ induces a linear, topological isomorphism,p: W

1,p

()

W

1,p (),

which is given by(pf)(x) = f((x)) = (f )(x);(ii)

p is a linear, topological isomorphism between W1,p ()andW

1,p ();

(iii) If is a bounded measurable function on , taking its values in the set ofddmatrices, thenp p= , (20)

with

(y) = (D)

1(y)

1(y)

(D)T

1(y) 1| det(D)(1y)| . (21)

(D denotes the Jacobian of anddet(D)the corresponding determinant.)

If, in particular,

: W

1,p ()

W

1,p () is a topological isomorphism, then

: W

1,p

(

)

W1,p ()also is(and vice versa).


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Proof. The proof of (i) is contained in [23, Thm. 2.10]. (ii) follows from (i) by duality. We prove (iii): For

f W1,p (),g W1,p ()we get by the change of variables formula:

p (pf ) , g

W

1,p ()

=

(pf ) , pg

W

1,p ()

=

(f ),g

W

1,p ( )

=

(x)(f )(x), (g )(x)dx=

(x)(D)T(x)(f )(x),(D)T(x)(g)(x)dx=

(D)(x)(x)(D)T(x)(f )(x), (g)(x) | det(D)(x)|| det(D)(x)|dx

=

(D)

1(y)

1(y) (D)T(1(y))| det(D)(1y)|f (y), g(y)

dy

=

(D)

1()

1()

(D)T(1())| det(D)(1())|f

, g

W

1,p ()

.

The essential point is thatas a Lipschitz continuous functionis differentiable almost everywhere and its (weak)derivative is essentially bounded (see [15, Ch. 4.2.3]). The last assertion follows from (i), (ii) and (20).

Proposition 17. Let R3 be a bounded, convex, polygonal domain and be an open subset of such that {(x, 0, z): x, zR} =. Let for anyx= (x,y,z)the symbolx denote the element(x , y,z)and define asthe interior of

{x: x } .If is a bounded, measurable function on taking its values in the set of real, symmetric 33 matrices, then wedefine:

(x)def=

(x

), ifx

, 11(x) 12(x) 13(x)12(x) 22(x) 23(x)13(x) 23(x) 33(x)

, ifx . (22)(i) If W1,2 () satisfies the equation = f W1,2 (), then the equation =

f W1,2()holds for with

(x) =

(x), ifx ,(x), ifx ,

and

fdefined by

f ,

W1,2(

)

def

= 1

2

f,

|

+

|

W

1,2

( ). The function

is defined by

(x)

def

= (x

)for

W1,1().(ii) Moreover, if f W1,p (), then f W1,p(); and if : W1,p0 ()W1,p() is a topological

isomorphism, then : W1,p () W1,p ()also is.

Proof. (i) It is known that belongs to W1,p0 (), see [18, Lemma 3.4]. Thus, (i) is obtained by the definitions of , f, , and straightforward calculations, based on Proposition 16 when applied to the transforma-tion xx.

(ii) The operatorffis the adjoint to 12 (|+ | ). The latter maps eachW1,p0 (

)continuously intoW

1,p ()for anyp ]1, [. The last statement is then implied by the preceding ones and the definition of .

Remark 18.The proposition is mutatis mutandis true for the reflection at other planes.


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Remark 19. In fact it can be shown that : W1,p ()W1,p () is a topological isomorphism, if and

only if is a topological isomorphism between the symmetric part ofW1,p0 () and the symmetric part ofW1,p(). The point is that this is not of use to us here because the reduction procedure of [39] applies to the wholespace and not only to the symmetric parts of the spaces (see also Remark 23).

In the sequel we will transform our model problems which include mixed boundary conditions to the case of Dirichletconditionswhich are imposed in Proposition 11. In essence, this happens via a linear transformation leading to apeculiar triangle, a bi-Lipschitz transformation and a reflection argument. All of this is carried out in the next section.

6. Proof of Theorem 1

6.1. Transformation of the problem

Proposition 16 allows us in a first step to reduce the case of an arbitrary triangle to that one where is the

triangle with the vertices (1,1), (1, 1), (1/2, 1/2) and, additionally, is the line segment between (0, 0) and(1, 1), see Figs. 1 and 3. Namely, first one shifts the triangle such that Pbecomes the origin. Let P1 denote thevertex where (the shifted)ends andP2the vertex which does not touch. We now transform

R2

under the linearmapping which assignsP1to (1,1)andP2to (1/2, 1/2). Extending this mapping to R3 by letting thez-componentinvariant, one obtains the special geometric constellation of Fig. 3 stated above. Clearly, the transformed plane maintains the properties demanded in the suppositions of Theorem 1. In particular, we denote the point, where the

(transformed) plane intersects the z-axis, by P. In a natural sense we may speak of an upper half space Gu and alower half space Gl (each on one side of the intersecting plane), where the coefficient function takes the values:

+= a11 a12 a13a12 a22 a23

a13 a23 a33

on Gu, = b11 b12 b13b12 b22 b23

b13 b23 b33

on Gl . (23)We transform the problem via the bi-Lipschitz transformation (see Fig. 4)

def= 1/

2 1/

2 0

0 2 00 0 1

on{(x,y,z): y > x}, 2 0 01/2 1/2 00 0 1

on{(x,y,z): y x}.(24)

Fig. 3. Transformation of a model domain (see the left side of Fig. 1) to the generic domain, with Neumann boundary (hatched area) and material

interface (shaded area). (,,, andkeep their names after transformation.)


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Fig. 4. The piecewise linear transformation in the x y-plane. Triangle before (left) and after (right) transformation by .

(Please notice that the determinants of both matrices in (24) equal 1.)()is again a triangledenoted by and

has now the vertices (0, 0),(0,

2),(

2, 0), while the new domain is = ]1, 1[.equals the subinterval]0,

2[of the x -axis. The image ofconsists of two triangles having one common edge E {(x,x,z):

x > 0, z R}. (Of course, if was orthogonal to the z-axis, then both triangles are also orthogonal to thez-axis.) Clearly, the Neumann boundary part is now the rectangle with the vertices (0, 0, 1),(0, 0, 1),(

2, 0, 1),

(

2, 0, 1). The transformed matrix (see Proposition 16),

1/2 1/2 00 2 00 0 1

a11 a12 a13a12 a22 a23a13 a23 a33

1/2 1/2 00 2 00 0 1

T , (25)is calculated as

a11+2a12+a222 a12+ a22 a13+a232

a12+ a22 2a22

2a23a13+a23

2

2a23 a33

, (26)while the transformed matrix

2 0 01/2 1/2 00 0 1

a11 a12 a13a12 a22 a23a13 a23 a33

2 0 01/2 1/2 00 0 1

T , (27)is calculated as 2a11 a12+ a11

2a13

a12+ a11 a11+2a12+a222 a13+a2322a13

a13+a232

a33

(28)(and analogously for the matrix b). We reflect the problem at the x z-plane in the spirit of Proposition 17 and obtain

a new triangle with the vertices (0,

2 ), (

2, 0),(0,

2 ), a new domain def= ]1, 1[and the coefficientfunctionon is defined as in (22). Thus, we end up with a Dirichlet problem on . By Proposition 17 it sufficesto show that

: W1,p0 ( ) W1,p()is a topological isomorphism for a p > 3. For this, however, we may apply Proposition 11: we are done if we are able

to show that for all edges E the induced operators A have a trivial kernel for all with ]0, 1/3+[ ( >0arbitrarily small). The occurring edgesE are the following, see also Fig. 5:

geometric edges, bimaterial outer edges, the edgesE+z andEz lying betweenP and (0, 0, 1), or betweenPand (0, 0, 1), respectively,

the edgeExz , which is the intersection of the (transformed) with thex z-plane,

E and the reflectedE .


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Fig. 5. Generic model domain after transformation by with Neumann boundary (hatched area), material interface (shaded area), and the edges

E+z ,Exz ,E , as well as bimaterial outer edges. (keeps its name after transformation with .)

6.2. Reformulation of the transmission conditions

The aim of this subsection is to express the transmission conditions for the Ansatz functions (see (12)) in a

condensed manner in terms ofj,j+1,j.

Lemma 20.Let be defined by(13), and

m = m11 m12m12 m22

def= j11 j12

j

12 j

22

,u()

def= c+ei(e2i + 1) + cei(e2i + 1), (29)wherec+, c are arbitrary complex constants. Further, letb1, b2 be defined as in(7). Then

b2()u( ) + b1()u() = iD 1/2m

c+ei(e2i + 1) cei(e2i + 1)

, (30)

whereDm again denotes the determinant of the matrixm.

Proof. First, one easily verifies:

u( ) = c+ei(e2i + 1)(i) 1 e2i

1 + e2i+ cei(e2i + 1)i 1 e

2i

1 + e2i. (31)

Next we want to prove:

b2( )i 1 e2i

1 + e2i+ b1( ) = iD1/2 . (32)

For this we calculate:

i1 e2i1 + e2i= i

e2i e2i + , (33)

and abbreviate the denominator i(m22+ D1/2m ) + m12of by N . One has:e2i = e2ii(m22+ D1/2m ) + m12 i(m22 D1/2m ) + m12/N


13/24


and

e2i + = e2ii(m22+ D1/2m ) + m12+ i(m22 D1/2m ) m12/N,what leads to

i1

e2i

1 + e2i= im

12(e

2i

+1)

+m

22i(e

2i

1)

+D

1/2

m i(e

2i

+1)

m12(e2i 1) + m22i(e2i + 1) + D1/2m i(e2i 1) . (34)

We augment the last fraction by sin e2i1 ; exploiting the equation

e2i + 1e2i 1sin =

ei(ei + ei)ei(ei ei) sin =

cos

i =i cos ,

the right-hand side of (34) becomes:

iim12 cos + im22 sin D1/2m cos m12 sin m22 cos + iD1/2m sin

=m12 cos m22 sin iD1/2m cos

m12 sin m22 cos + iD1/2m sin

= (m12 cos m22 sin iD1/2

m cos )(m12 sin m22 cos iD1/2

m sin )(m12 sin m22 cos )2 + Dm sin2

= (m22 m11) cos sin + m12(cos2 sin2 ) + iD1/2m

m11 sin2 2m12 cos sin + m22 cos2

= b1( ) + iD1/2m

b2( ). (35)

Thus, (32) holds true. By complex conjugation one obtains from (32):

b2( )i1 e2i1 + e2i+ b1( ) =iD

1/2m ; (36)

(32) and (36) together with (31) give the assertion (30).

Corollary 21.Letu be the function on[

0, n]

which coincides on]

j, j+

1

[withujdefined in(14).

(i) Assume firstj {1, . . . , n 1}and letDj andDj+1 denote the determinants of the matrices:

j

11 j

12

j

12 j

22

and

j+111

j+112

j+112

j+122

,

respectively. If we abbreviatedef= j and def= j+1, then the transmission conditions in the point= j,

[u]= [b2u+ b1u]=0, (37)express as

cj,+

ei(e2i+

1)

+cj,

ei(

e2i

+1)

=cj

+1,

+ei(e2i

+1)

+cj

+1,

ei(

e2i

+1) (38)

and

D1/2j

cj,+ei(e2i + 1) cj,ei(e2i + 1)

= D1/2j+1

cj+1,+ei(e2i + 1) cj+1,ei(e2i + 1)

, (39)

respectively. Thus, in case ofDj= Dj+1 for(37)it is necessary and sufficient thatcj,+(e2i + 1) = cj+1,+(e2i + 1) (40)

and

cj,

(

e2i

+1)

=cj

+1,

(e2i

+1) (41)

hold.


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(ii) Assume nown= 0+ 2 . Then the corresponding transmission conditions in 0 express asc1,+ei0 (1e2i0 + 1) + c1,ei0 (1e2i0 + 1)

= cn,+ein (ne2i0 + 1) + cn,ein (ne2i0 + 1) (42)

and

D1/21

c1,+ei0 (1e2i0 + 1) c1,ei0 (1e2i0 + 1)

= D1/2n

cn,+ein (ne2i0 + 1) cn,ein (ne2i0 + 1)

, (43)

respectively.

6.3. Discussion of the edge singularities

For geometric edges and bimaterial outer edges we show in Appendix A that the operators Ahave a trivial kernel

if ]0, 1/2].Next we consider the edgesE+z andEz : starting withE+z , one has to deal with the coefficient matrices:

mdef= a11+2a12+a22

2 a12 a22

a12 a22 2a22

if ]/2, /4[,

odef=

2a11 a12 a11a12 a11 a11+2a12+a222

if ]/4, 0[,

odef=

2a11 a12+ a11a12+ a11 a11+2a12+a222

if ]0, /4[,

mdef=

a11+2a12+a222 a12+ a22

a12+ a22 2a22

if ]/4, /2[.

Thus, one has to consider the ansatz functions (see Remark 13):

udef=

w

def= c+ei(e2i+ 1) + cei(e2i+ 1) on]/2, /4[,v

def=d+ei(e2i+ 1) +dei(e2i+ 1) on]/4, 0[,v

def= d+ei(e2i+ 1) + dei(e2i+ 1) on]0, /4[,w

def= c+ei(e2i+ 1) + cei(e2i+ 1) on]/4, /2[,with defined by (13) (and analogously from the entries of the matrix o). Please notice that the determinants of the

matricesm,m,o,oall equal the determinant of the matrixa11 a12a12 a22

,

the value of which we denote by D in this proof. Taking this into account, the transmission conditions in = /4read in view of (40)/(41),

c+(1 i) =d+(1 i), (44)and

c(1 + i) =d(1 + i). (45)Analogously, the transmission conditions in 0 equivalently express as

d+(1 +) = d+(1 + ) (46)and

d(1 + ) = d(1 +), (47)


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while those in/4 can be written as

d+(1 + i) = c+(1 + i) (48)and

d(1 i)

= c(1 i)

. (49)The boundary conditionu(/2) = w(/2) =0 leads to

c+ei/2(1 ) + cei/2(1 ) =0, (50)or, in other words,

c+= cei(1 )(1 ) , (51)

while the boundary conditionu(/2) = w(/2) =0 gives:c+ei/2(1 ) + cei/2(1 ) =0,

or, alternatively,

c= c+ei(1 )(1 ) . (52)

Combining (51), (49), (47), (45), (52), (44), (46), (48), one ends up with the characteristic equation for :

(1 )(1 )

(1 i)(1 i)

(1 + )(1 +)

(1 + i)(1 + i)

(1 )(1 )

(1 i)(1 i)

(1 + )(1 +)

(1 + i)(1 + i)

=

(1 + i)(1 )

(1 )(1 i)

2(1 + )(1 + i)

(1 i)(1 +)

2=e2i . (53)

Let us remark that c+ cannot vanish unless also the other coefficients vanish. Moreover, we notice that all the terms1 + i, 11 , 1 , 11i , 1 + , 11+i , 1 i, 11+ have positive real part because||, || < 1. Hence, we have:

(1 + i)(1 )=

1 + i1

and

(1 )(1 i)=

1 1 i

, (54)

as well as

(1 + )(1 + i)=

1 + 1 + i

and

(1 i)(1 +)=

1 i1 +

, (55)

if 1. Further, observing the relations,

1 + i1

= 111i

and 1 + 1 + i

= 11i1+

, (56)

and putting=arg1+i1 and=arg 1+1+i , this altogether enables us to rewrite (53) as

e2i(+2(+)) =1. (57)It is obvious that all satisfying (57) must be real. Our claim is now: + equals /2 or3/2. For this, wemention that, by definition,, ], ]; thus the claim is true, if we can show

1 + i1

1 + 1

+i

=i. (58)

This we will do now: exploiting the definitions of, we get:


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1 + i1 =

1

2

D1/2 + m12 m22+ i(D1/2 + m22 m12)m12+ iD1/2

= 12m11

D1/2 + m11 m12+ i(D1/2 m11+ m12)

=

1

2m11D1/2 + a11 a222 + iD1/2 + a22 a112 .Analogously, we calculate:

1 + 1 + i =

2io22

D1/2 + o12 o22+ i(D1/2 + o22 o12)= 2io22

D1/2 + a11a222 + i(D1/2 + a22a112 ).

Taking into account o22=m11, this gives (58). Hence, the transcendental equation (57) for reads in any case ase4i = 1. Trivially, the smallest positive possible is 0=1/2. Thus, the edge E+z meets the preconditions ofProposition 11. The considerations for the edge Ez are the same, word by word.

Next we consider the edge Exz , lying in the xz-plane. The coefficient matrices belonging to its neighbouring

sectors are:

Q = q11 q12 q13q21 q22 q23q31 q32 q33

def= a11+2a12+a22

2

a12+

a22a13+a23

2a12+ a22 2a22 2a23

a13+a232

2a23 a33

, (59)

R= r11 r12 r13r21 r22 r23

r31 r32 r33

def=

b11+2b12+b222 b12+ b22 b13+b232

b12+ b22 2b22

2b23b13+b23

2

2b23 b33

, (60)ify > 0 and their reflected counterparts,

Q

=q11 q12 q13

q12 q22

q23

q13 q23 q33 and R=r11 r12 r13

r12 r22

r23

r13 r23 r33 ,if y


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obeys the transmission conditions in, , 0, . Then / ]0, 1/2].

Proof. The transmission condition[b2u+ b1u]0=0 together with Corollary 21 (see in particular (39)) implies:c+(+ 1) c(+ 1) = c+(+ 1) c(+ 1). (62)

On the other hand, the transmission condition forb2u+ b1uin/ (see (43)) gives:d+ei (+ 1) dei (+ 1) = d+ei (+ 1) dei (+ 1). (63)

Let us first consider the case, where

c+= c, c= c+, d+= d, d= d+. (64)Inserting these relations in (62) and (63) one obtains that both sides of (62) and (63) in fact have to vanish. But this

means in view of Lemma 20 nothing else but

b2()u( ) + b1()u() =0 for=0, .

Thus, the restriction ofu to the interval

]0,

[leads to a bimaterial problem including a Neumann condition on both

interval ends. Then / ]0, 1/2], see Theorem 25 below.Assume now that (64) is not satisfied. Then we introduce the function:

udef=

w def= cei(e2i+ 1) + c+ei(e2i+ 1) on]0, [,v

def=dei(e2i+ 1) +d+ei(e2i+ 1) on], [,(65)

on[0, ]and consider the function:

udef= u|[0, ] u =

(c+ c)ei(e2i+ 1) + (c c+)ei(e2i+ 1) on]0, [,(d+d)ei(e2i+ 1) + (dd+)ei(e2i+ 1) on], [.

It is straightforward to verify that the condition[u]0=0 implies u(0)=0 and the periodicity condition in/yields u()

=0. Next we intend to show the transmission conditions

[u

]

= [b2()u

+b1()u

]

=0. Because we

already know by supposition[u]= [b2()u+ b1()u]=0 it remains to show[u]= [b2( )u+ b1( )u]=0.One easily verifies[u]= [u], and the latter is zero by supposition. Finally, by Lemma 20 we have:

b2( )u+ b1( )u

= (b2w+ b1w)| (b2v+ b1v)|= iD

cei(e2i + 1) c+ei(e2i + 1)

+ iD

dei(e2i + 1) d+ei(e2i + 1)

= (b2w+ b1w)|+ (b2v+ b1v)|= [b2u+ b1u].

But the right-hand side of this equation is zero in view of the transmission condition [b2u+ b1u]=0. Thus, thissecond case leads to a bimaterial Dirichlet problem, for which also Theorem 25 gives / ]0, 1/2].

Remark 23.The key point is here that we can invest estimates offor bimaterial edges in both, the Neumann andDirichlet case (see Theorem 25 below), despite the fact that in the original problem the edge is situated on a Neumann

boundary plane.

It remains to consider the edgeE (and its reflected counterpart). Let first tR be a number such that(0, 0, t) + Ehas its endpoint in0R3 and O be a rotation of the plane{(x,x,z): x, zR}which transforms (0, 0, t) + Etothez-axis. Suppose that for onewith ]0, 1/2]there is a (nontrivial) functionvfrom the kernel of the resultingoperator A. If one takes the coefficient function defined in (15) as

(x,y,z)def

=

j

11 j

12 j

13

j

12

j

22

j

23

j

13 j

23 j

33 if(x,y) K j+1j , (66)


18/24


then, by Lemma 14, there is a compactly supported element f W1,6(R3) such that thealso compactlysupportedvariational solution W1,2(R3) of =f does not belong to W1,4(R3). Because the sup-port of is compact, it can then (the more) not belong to W1,6(R3). Now we revoke the transformations O, the shift

(0, 0, t)and . Applying Proposition 16, one obtains a fW1,6(R3)and a W1,2(R3) \ W1,4(R3)satisfying

=f

, or, equivalently,

+

=f

+

W1,6(R3). It is not hard to see that the matrix val-

ued functionequals abovethe matrix+and belowthe matrix(see (23)). But a result of [14, Thm. 3.11],see also [3, Ch. 4.5], says that

+ 1 : W1,p(R3) W1,p(R3)is a topological isomorphism for any p ]1, [. This contradicts the above supposition. The proof for the reflected Eruns along the same lines; thus the proof of Theorem 1 is complete.

7. Proof of Theorem 2 and of Corollary 3

First we consider the case where is one side of the triangle . Modulo an affine transformation in R2 we

may focus on the case where is identical with the interval

]0, 1

[on the x-axis, see Proposition 16. We reflect

symmetrically at thexz-plane and obtain a domain and a reflected coefficient function. The resulting boundaryconditions are then homogeneous Dirichlet on all . By Proposition 17 it is sufficient to show that

: W1,p0 ( ) W1,p( )is a topological isomorphism for a p >3. Of course, we will again apply Proposition 11 and have, hence, to discuss

the edge singularities. The occurring edges are:

(i) geometric edges,

(ii) bimaterial outer edges,

(iii) the intersection of thex z-plane with, in particular, the parts of the z-axis below and above the intersection

point withis a bimaterial outer edge.

For all these edges we already know that the corresponding operators Ahave a trivial kernel provided ]0, 1/2];namely: the claim for geometric edges and bimaterial outer edges is shown in the next section (see Theorem 24 and

Theorem 25) while the situation of (iii) is exactly the same as treated in Lemma 22.

Let us now regard the second case: modulo an affine transformation in R2 we may restrict ourself to the case

whereis the union of the interval]0, 1[ on the x-axis and the interval[0, 1[ on the y-axis. Again we reflect theproblem at the x z-plane, but afterwards a second time at the y z-plane. Thus, we end up with a Dirichlet problem

on def= V ]1, 1[, whereV R2 is the square with the vertices (0, 1), (1, 0), (0,1), (1, 0). Denoting the newcoefficient function by, it suffices by Proposition 17 to show that

: W

1,p0 (

)

W1,p(

)

is a topological isomorphism for a p >3. According to Proposition 11, it remains to show that for every edge E

the kernels of the corresponding operators A are trivial if ]0, 1/3+[ ( arbitrarily small). If(0, 0, t ) is theintersection point ofwith the z-axis, then the occurring edges are:

(i) geometric edges,

(ii) bimaterial outer edges,

(iii){(0, 0, s): s ]1, t[},(iv){(0, 0, s): s ]t, 1[},(v) the intersection of thex z-plane with,

(vi) the intersection of thex z-plane with the reflected,

(vii) the intersection of they z-plane with,

(viii) the intersection of they z-plane with the reflected.


19/24


The geometric and bimaterial outer edges are treated in Appendix A. (iii), (iv), (v), (vi) lead again to a constella-

tion (61), which was treated in Lemma 22. This is also true for (vii) and (viii), but requires here an additional moments

thought: let us denote the value of the coefficient function above by+and below by. Concerning (vii),the reflected matrices then equal

+11 +12 +13+12 +22 +23+13 +23 +33

and 11 12 1312 22 2313 23 33

.We perform now a rotation within thex y-plane which transforms the (positive) y -axis into the (positive)x -axis and

the (positive)x -axis into the negative y -axis; clearly the transformed edge lies then in the x z-plane. One obtains the

transformed coefficient matrices, 0 1 01 0 00 0 1

+11 +12 +13+12 +22 +23+13

+23

+33

0 1 01 0 00 0 1

= +22 +12 +23+12 +11 +13

+23 +13 +33

,

0 1 01 0 00 0 1

11

12

13

12 22 2313

23

33

0 1 01 0 00 0 1

= 22 12 23

12 11 1323 13 33

,while the reflected matrices transform as follows: 0 1 01 0 0

0 0 1

+11 +12 +13+12 +22 +23+13 +23 +33

0 1 01 0 00 0 1

= +22 +12 +23+12 +11 +13

+23 +13

+33

, 0 1 01 0 0

0 0 1

11 12 1312 22 2313 23 33

0 1 01 0 00 0 1

= 22 12 2312 11 13

23 13

33

.Thus, from this point on we are in the same situation as in the discussion for the edge Exz (see page 40) and everythingruns completely the same way. (viii) is analogous to (vii).

We come to the proof of Corollary 3: because we demanded that the plane should not touch the upper plate nor

the ground plate it is possible to divide the problem by a suitable partition of unity into one which affects the upper

(lower) part and is separated from and one which contains but has only a Dirichlet condition on its upper(ground) plate. The latter is already treated in Theorem 2. The first can be reflected at the upper (ground) plate and

one ends up again with the setting which is treated in Theorem 2.

8. Concluding remarks

The results of this paper easily carry over to problems with Robin boundary conditions. Indeed, one can prove that

ifis the surface measure on and

L

(, d ), then the linear mapT: W1,p

()

W1,p

()given by:

T , W

1,p

=

d

(and representing the Robin boundary condition) is infinitesimally small with respect to the operator . Thus,the domains of both operators are the same by classical perturbation theory, see [31, Ch. IV.1].

The reader has possibly asked himself why the results are deduced from [39] and why the concept of that paper does

not work for boundary conditions which are not Dirichlet. One problem consists in finding an adequate energy space

in case of edges on Neumann boundary parts which, additionally, has to be in correspondence with the properties of

the Mellin transform. Our attempts to find such an energy space have failed up to now.

Alternatively, the question arises whether it is possible to discard vertices from the analysisby reflection

argumentsalso for the Neumann case. It turns out that this can be done in relevant cases, but seems to be deli-

cate in general.


20/24


In principle it is possible to generalize our results to the case where not only one plane intersects the domain, but

severals do. In order to classify the singularities stemming from the additional inner edges (where the planes meet)

one can apply the result [13, Thm. 2.5]. We have not carried out this here only for technical simplicity, see also [32].

Acknowledgement

We gratefully acknowledge that some of the ideas from Appendix A are due to our colleagues J. Elschner and

G. Schmidt.

Appendix A. The transcendental equation for geometric edges and bimaterial outer edges

It is the aim of this section to discuss the edge singularities for geometric edges and bimaterial outer edges;

precisely, we intend to show the following two theorems:

Theorem 24.For any geometric edgeE the kernels of the associated operators A are trivial in each of the following

two cases:

(a) the opening angle1 0 is not larger than and ]0, 1[,(b) 1 0 ], 2 [and ]0, 1/2].

Theorem 25.LetK10

, K21

be two neighbouring sectors in R2 with1 0,2 1 and2 0< 2 . Let 1, 2be two real, positive definite22matrices corresponding to the sectors K 10,K

21

. Let t be the form defined in (8)

either on H10 (]0, 2[) or on H1(]0, 2[). Then there is an > 0such that the kernel of the corresponding operatorA(see Definition6)is trivial if ]0, 1/2 + ].

We will prove the theorems in several steps, starting with the following:

Lemma 26.LetC with|| < 1, and define for ], ]the number:

def= arge

2i + 1+ 1 ], ].

Then either , + ], 0[or= =0or,+ ]0, [, or= + = .

Proof. The cases=0 and= are straightforward. In the remaining cases one has:

ei(+ ) =ei e2i + 1+ 1

|+ 1||e2i + 1|=

(ei + ei)(+ 1)|+ 1||e2i + 1| =

||2ei + ei + 2(ei)|+ 1||e2i + 1| .

Thus, the imaginary part of ei(+ ) equals (1||2) sin

|1+

||e

2i

+1|, and its sign depends in an obvious way only on .

It follows the proof of Theorem 24; without loss of generality we may assume 1= . Again exploiting the Ansatzfunctions (14), the Dirichlet conditions in 0, 1 ], ]reads

c+ei0 (e2i0 + 1) + cei0 (e2i0 + 1) =0, (A.1)c+ei (+ 1) + cei (+ 1) =0. (A.2)

These equations are nontrivially solvable in c+, ciff

1=e2i e2i0(e2i0 + 1)

(+ 1)(+ 1)

(e2i0 + 1)

=e2i e2i0e2i0 + 1+ 1

+ 1e2i0 + 1

, (A.3)


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compare the considerations in Section 6.3, in particular (54). Putting def= arge2i0 +1

+1 , we may write (A.3)as e2i(0+) = 1. Obviously, must be real and|| = || < 1. Hence, in case (a), where 0 [0, [, weobtain 0+ [0, ] by Lemma 26, which excludes ]0, 1[. If 0 ], 0[, then, by Lemma 26, we have0+ ], 0[, which shows the assertion in case (b).

Concerning Theorem 25, we may apply a rotation (corresponding to a shift in the angle space) and thus reduce the

general case to that one where 0= ,1=0 and2= . Again using the Ansatz functions (14) we are getting thefollowing equations expressing the transmission conditions in 0, see Corollary 21,

c+(+ 1) + c(+ 1) = d+(+ 1) + d(+ 1), (A.4)and

D1/2m

c+(+ 1) c(+ 1)

= D1/2o d+(+ 1) d(+ 1). (A.5)We define:

def=1 if Dirichlet in ,

1 if Neumann in ,

and analogously for . In this convention (see Lemma 20), the boundary condition in

yields

c+ei(e2i + 1) cei(e2i + 1) =0or, what is the same,

c= c+e2i(e2i + 1)

(e2i + 1) . (A.6)

On the other hand, the corresponding boundary condition in implies:

d+ei (e2i + 1) dei(e2i + 1) =0or, equivalently,

d+=

d

e2i(e2i + 1)

(e2i + 1) . (A.7)

We insert (A.6) and (A.7) in (A.4) and (A.5) and obtain:

c

(+ 1) + e2i

(e2i + 1)(e2i + 1) (+ 1)

d

e

2i (e2i + 1)(e2i + 1) (+ 1)

+ (+ 1)

=0, (A.8)

and

D1/2m c

(+ 1) e2i

( e2i + 1)(e2i + 1) (+ 1)

+ D1/2o d

(+ 1) e2i (

e2i + 1)(e2i + 1) (+ 1)

=0, (A.9)

forc= c+and d= d. (A.8), (A.9) are nontrivially solvable iffD

1/2o

1 + e2i

( e2i + 1)(+ 1)

(+ 1)(e2i + 1)

1 e2i

(e2i + 1)(+ 1)

(+ 1)(e2i + 1)

+ D1/2m

1 + e2i

(e2i + 1)(+ 1)

(+ 1)(e2i + 1)

1 e2i

( e2i + 1)(1 + )

(+ 1)(e2i + 1)

=0. (A.10)

Putting

=arge2i + 1+ 1 , =arg

e2i + 1+ 1 ,

and arguing as in (54)(56), this altogether enables us to rewrite (A.10) as

D1/2o [1 + e2i(+ )][1 e2i(+) ] + D1/2m [1 + e2i(+) ][1 e2i(+ )] =0,


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or, what is the same,

D1/2o [ei(+ ) + ei(+ )][ei(+) ei(+) ]+ D1/2m [ei(+) + ei(+) ][ei(+ ) ei(+ )] =0. (A.11)

This means that in the pure Dirichlet case (with= = 1) (A.10) can be written equivalently asD

1/2o sin (+ ) cos (+ ) + D1/2m cos (+ ) sin (+ ) =0 (A.12)

and in the pure Neumann case (with = =1) asD

1/2o cos (+ ) sin (+ ) + D1/2m sin (+ ) cos (+ ) =0. (A.13)

BecauseDm and Doare arbitrary positive constants it suffices to focus the following discussion on (A.12).

Lemma 27.If , with+ < 2, then any solution of (A.12)satisfies / ]0, 1/2 + ]for an > 0.

Proof. Since, by Lemma 26, sin (+ ) =0 and sin (+ ) =0, if 0 < < 1, we can rewrite (A.12) asD1/2o cot (+ ) + D1/2m cot (+ ) =0. (A.14)

Note that

cot(+ i) = (cosh2 sinh2 ) sin cos

(sin cosh )2 + (cos sinh )2= sin2

2(sin2 + sinh2 ) ,

hence, with = + i, the real part of (A.14) satisfies:D

1/2o sin2 (+ )

sin2 (+ ) + sinh2 (+ ) + D

1/2m sin2(+ )

sin2 (+ ) + sinh2 (+ ) =0. (A.15)

If 0 < 1/2, then Lemma 26 shows that 0 < 2 ( + ), 2(+ ) , and therefore both terms on the left-handside of (A.15) are nonnegative. Due to

+ < 2 , at most one of them may be zero. This proves the assertion.

Remark 28. If one is confronted with a bimaterial outer edge supplemented by a Dirichlet condition in and aNeumann condition in (what means= =1), then (A.11) reads as

D1/2o sin

(+ ) sin(+ )+ D1/2m cos(+ ) cos(+ )=0. (A.16)If we again suppose , ]0, [, then we may divide (A.16) by sin((+ )) sin((+ )) (provided ]0, 1[)and obtain the equivalent condition:

cot

(+ )cot(+ )= D1/2oD

1/2m

. (A.17)

It is not hard to see that there are parameter configurations , , , , Do, Dm such that (A.17) is fulfilled for witharbitrarily small (positive) real part; see also [40], where the case of scalar multiples of the Laplacian already was

treated.

Remark 29.In fact, the results of Theorem 24 and Theorem 25 are already proved in [13] (see Lemmas 2.9 and 2.5)

by completely different methods and based on the results of Ilyin [28,29]. Our intention was here to give a proof

which is straightforward and self-contained.

References

[1] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in: H.-J. Schmeisser, et al. (Eds.), Func-

tion Spaces, Differential Operators and Nonlinear Analysis, in: Teubner-Texte Math., vol. 133, 1993, pp. 9126.

[2] L. Adams, Z. Li, The immersed interface/multigrid methods for interface problems, SIAM J. Sci. Comput. 24 (2002) 463479.[3] P. Auscher, P. Tchamitchian, Square root problem for divergence operators and related topics, Astrisque 249 (1998).


23/24


[4] U. Bandelow, H.-C. Kaiser, T. Koprucki, J. Rehberg, Modeling and simulation of strained quantum wells in semiconductor lasers, in:

W. Jger, H.-J. Krebs (Eds.), MathematicsKey Technology for the Future. Joint Projects Between Universities and Industry, Springer,

Berlin/Heidelberg/New York, 2003, pp. 377390.

[5] H. Beirao da Veiga, On the W2,p -regularity for solutions of mixed problems, J. Math. Pures Appl. 53 (1974) 279290.

[6] H. Berestycki, F. Hamel, L. Roques, Analysis of the periodically fragmented environment model: Ispecies persistence, J. Math. Biol. 51

(2005) 75113.

[7] R. Bey, J.-P. Loheac, M. Moussaoui, Singularities of the solution of a mixed boundary problem for a general second order elliptic equationand boundary stabilization of the wave equation, J. Math. Pures Appl. 78 (1999) 10431067.

[8] R. Brown, The mixed problem for Laplaces equation in a class of Lipschitz domains, Commun. Partial Differential Equations 19 (1994)

12171223.

[9] Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175202.

[10] M. Costabel, M. Dauge, Y. Lafranche, Fast semi-analytic computation of elastic edge singularities, Comput. Meth. Appl. Mech. Engrg. 190

(2001) 21112134.

[11] M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integral Equations Oper. Theory 15 (1992) 227261.

[12] C. Ebmeyer, J. Frehse, Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains, Math.

Nachr. 203 (1999) 4774.

[13] J. Elschner, H.-C. Kaiser, J. Rehberg, G. Schmidt, W1,q regularity results for elliptic transmission problems on heterogeneous polyhedra,

Math. Models Methods Appl. Sci. 17 (2007) 593615.

[14] J. Elschner, J. Rehberg, G. Schmidt, Optimal regularity for elliptic transmission problems including C 1 interfaces, Interfaces Free Bound. 9

(2007) 233252.

[15] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, NewYork, London, Tokyo, 1992.

[16] P.C. Franzone, L. Guerri, S. Rovida, Wavefront propagation in an activation model of the anisotropic cardiac tissue: asymptotic analysis and

numerical simulation, J. Math. Biol. 28 (1990) 121176.

[17] H. Gajewski, Analysis und Numerik von Ladungstransport in Halbleitern (Analysis and numerics of carrier transport in semiconductors), Mitt.

Ges. Angew. Math. Mech. 16 (1993) 3557.

[18] E. Giusti, Metodi diretti nel calcolo delle variazioni, Unione Matematica Italiana, Bologna, 1994.

[19] A. Glitzky, R. Hnlich, Global estimates and asymptotics for electroreactiondiffusion systems in heterostructures, Appl. Anal. 66 (1997)

205226.

[20] J.A. Griepentrog, L. Recke, Linear elliptic boundary value problems with non-smooth data: Normal solvability on SobolevCampanato spaces,

Math. Nachr. 225 (2001) 3974.

[21] J.A. Griepentrog, Linear elliptic boundary value problems with non-smooth data: Campanato spaces of functionals, Math. Nachr. 243 (2002)

1942.

[22] K. Grger, AW1,p -estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann. 283

(1989) 679687.

[23] J.A. Griepentrog, K. Grger, H.-C. Kaiser, J. Rehberg, Interpolation for function spaces related to mixed boundary value problems, Math.

Nachr. 241 (2002) 110120.

[24] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.

[25] G. Harutjunjan, B.-W. Schulze, Parametrices of mixed elliptic problems, Math. Nachr. 277 (2004) 5682.

[26] M. Hieber, J. Rehberg, Quasilinear parabolic systems with mixed boundary conditions, WIAS-Preprint 1194, 2006.

[27] T. Hwang, W. Lin, J. Liu, W. Wang, JacobiDavidson methods for cubic eigenvalue problems, Numer. Linear Algebra Appl. 12 (2005)

605624.

[28] E.M. Ilyin, Singularities for solutions of elliptic boundary value problems with discontinuous highest order coefficients, Zap. Nauchn. Semin.

LOMI 38 (1973) 3345 (in Russian).

[29] E.M. Ilyin, Singularities of the weak solutions of elliptic equations with discontinuous higher coefficients. II. Corner points of the lines of

discontinuity, Zap. Nauchn. Semin. LOMI 47 (1974) 166169 (in Russian).

[30] D. Jerison, C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995) 161219.

[31] T. Kato, Perturbation Theory for Linear Operators, Reprint of the corr. print of the second ed. Classics in Mathematics, Springer,

Berlin/Heidelberg/New York, 1980.

[32] D. Knees, On the regularity of weak solutions of quasi-linear elliptic transmission problems on polyhedral domains, Zeitschr. Anal. An-

wend. 23 (2004) 509546.

[33] T. Koprucki, H.-C. Kaiser, J. Fuhrmann, Electronic states in semiconductor nanostructures and upscaling to semi-classical models, in:

A. Mielke (Ed.), Analysis, Modeling and Simulation of Multiscale Problems, Springer, Berlin/Heidelberg/New York, 2006, pp. 367396.

[34] O.A. Ladyzhenskaya, N.N. Uraltseva, Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering, Academic Press,

New York/London, 1968.

[35] D. Leguillon, E. Sanchez-Palenzia, Computation of Singular Solutions in Elliptic Problems and Elasticity, Wiley, Chichester, 1987.

[36] Y. Li, J. Liu, O. Voskoboynikov, C. Lee, S. Sze, Electron energy level calculations for cylindrical narrow gap semiconductor quantum dot,

Comp. Phys. Comm. 140 (2001) 399404.

[37] V. Mazya, Sobolev Spaces, Springer, Berlin/Heidelberg/New York, 1985.

[38] V. Mazya, J. Rossmann, Weighted Lp estimates of solutions to boundary value problems for second order elliptic systems in polyhedral

domains, Z. Angew. Math. Mech. 83 (2003) 435467.

[39] V. Mazya, J. Elschner, J. Rehberg, G. Schmidt, Solutions for quasilinear evolution systems in Lp

, Arch. Rat. Mech. Anal. 171 (2004)219262.


24/24


[40] D. Mercier, Minimal regularity of the solutions of some transmission problems, Math. Meth. Appl. Sci. 26 (2003) 321348.

[41] I. Mitrea, M. Mitrea, The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains, Trans.

Amer. Math. Soc. 359 (2007) 41434182.

[42] J. Rehberg, Quasilinear parabolic equations in Lp , in: M. Chipot, et al. (Eds.), Nonlinear elliptic and parabolic problems. A special tribute

to the work of Herbert Amann, in: Progress in Nonlinear Differential Equations and their Applications, vol. 64, Birkhuser, Basel, 2005,

pp. 413419.

[43] S. Rempel, B.-W. Schulze, Asymptotics for Elliptic Mixed Boundary Problems (Pseudo-Differential and Mellin Operators in Spaces withConormal Singularity), Math. Res., vol. 50, Akademie-Verlag, Berlin, 1989.

[44] G. Savar, Regularity and perturbation results for mixed second order elliptic problems, Commun. Partial Differential Equations 22 (1997)

869899.

[45] G. Savar, Regularity results for elliptic equations in Lipschitz domains, J. Funct. Anal. 152 (1998) 176201.

[46] S. Selberherr, Analysis and Simulation of Semiconductors, Springer, Wien, 1984.

[47] E. Shamir, Mixed boundary value problems for elliptic equations in the plane. The Lp theory, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III.

Ser. 17 (1963) 117139.

[48] E. Shamir, Regularization of mixed second-order elliptic problems, Isr. J. Math. 6 (1968) 150168.

[49] A. Sommerfeld, Electrodynamics, Lectures on Theoretical Physics, vol. III, Academic Press, New York, 1952.

[50] A. Sommerfeld, Thermodynamics and Statistical Mechanics, Lectures on Theoretical Physics, vol. V, Academic Press, New York, 1956.

[51] B.A. Szabo, Z. Yosibash, Numerical analysis of singularities in two dimensions. Part II: Computation of generalized flux/stress intensity

factors, Int. J. Numer. Meth. Engrg. 39 (1996) 409434.

[52] W. Wang, T. Hwang, W. Lin, J. Liu, Numerical methods for semiconductor heterostructures with band nonparabolicity, J. Comp. Phys. 190

(2003) 141158.

[53] W. Wang, A jump condition capturing finite difference scheme for elliptic interface problems, SIAM J. Sci. Comp. 25 (2004) 14791496.

[54] C. Weisbuch, B. Vinter, Quantum Semiconductor Structures: Fundamentals and Applications, Academic Press, Boston, 1991.

[55] D. Zanger, The inhomogeneous Neumann problem in Lipschitz domains, Commun. Partial Differential Equations 25 (2000) 17711808.

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Elliptic model problems including mixed boundary conditions and material heterogeneities