Self-adjointe extensions for symmetric Laplacians on polygons.

UNIVERSIT ` A DEGLI STUDI DELL’INSUBRIA

Self-Adjoint Extensions for

Symmetric Laplacians on Polygons

Candidate: Luca Raimondi

Supervisor: Prof. Andrea Posilicano

Introduction

By the results contained in the paper by Birman and Skvortsov “On the square summability of the highest derivatives of the solution to the Dirichlet problem in a region with piecewise smooth boundary” (see reference [1]), the Laplace operator ∆

on a plane curvilinear polygon Ω with domain the Sobolev space H

(Ω) and homogeneous Dirichlet boundary conditions is a closed symmetric operator with deficiency indices (n, n), where n is the num- ber of non-convex corners. Therefore on a non-convex polygon, ∆

has infi- nite self-adjoint extensions. Such extensions have been recently determined by means of Kre˘ın’s resolvent formula in [8]. The purpose of this thesis is to extend such results to the case of different, more general, boundary conditions.

In the first part of the thesis (see Chapter 2) we consider the case of mixed Dirichlet-Neumann conditions, thus allowing each side Γ

of the polygon boundary to support either a Dirichlet or a Neumann homogeneous boundary condition. In this case, building on results by Grisvard (see [3], [4] and references therein), we have that, differently from the pure Dirichlet case, non-convexity is no more a necessary condition in order to have not zero deficiency indices. Indeed in this case the precise result is the following:

Let ω

the interior angle at the j-th vertex S

. If d

denotes the contribution to the dimension of the defect space due to the vertex S

, then

{

d

= 0 , 0 < ω

≤ π d

= 1 , π < ω

< 2π ,

both in the pure Dirichlet-Dirichlet and Neumann-Neumann cases, and

 



 

d

= 0 , 0 < ω

≤

π d

= 1 ,

π < ω

≤

π d

= 2 ,

π < ω

< 2π

(1)

in the mixed Dirichlet-Neumann case.

Thus while in the pure Neumann case the dimensions of the defect spaces is the same as in the case of the pure Dirichlet case already studied by Birman and Skvortsov, the mixed case has a different behavior, allowing both convex cases (with vertex contribution equal to one) and non-convex cases with double vertex contribution.

After explicitly characterizing the defect subspace we determined the self- adjoint extensions by a Kre˘ın’s resolvent formula proceeding analogously to the pure Dirichlet case given in [8], however taking into account the double contribution due to the vertices with mixed boundary conditions.

In the second part of the thesis we further extend our analysis by allowing some sides Γ

to support Robin boundary conditions of the kind (here n

denotes the exterior normal at the j-th side)

u(x) + α

∂u

∂n

(x) = 0 , x ∈ Γ

, α

> 0 .

While this is a deformation of the case considered in the first part, some not completely trivial calculations are necessary in order to get results similar to the ones concerning the mixed Dirichlet-Neumann case. By such calculations (which fill the entire Chapter 3) it turns out that in the Robin case the contribution d

of the j-th vertex to the dimension of the defect space is given by the number of eigenvalues λ

belonging to the interval (0, 1) of the 1-dimensional Robin boundary value problem

 



 

−u

(θ) = λ

u(θ) , θ ∈ (0, ω

) u(0) + α

u

(0) = 0 ,

u(ω

) − α

u

(ω

) = 0 .

Thus, by tuning the parameters α

, one can recover results anologous to the ones in (1). However also different behaviors are possible:

1. for any ϵ > 0, for any 0 < ω

< ϵ, there are parameter values which give d

= 1;

2. for any xπ < ω

≤ (3/2)π, x ≃ 1.43, there are parameter values which give d

= 2.

Moreover, as expected, the d

’s converge to the ones corresponding to the mixed Dirichlet-Neumann case as the α

’s converge to either 0 or ∞ accordingly to the different possible cases.

Again as in the mixed Dirichlet-Neumann a Kre˘ın’s formula giving the classification of all the self-adjoint extension is provided in Chapter 4.

In the final chapter we give, to enhance the reader intuition, some simple

examples regarding the case in which Ω is a wedge.

Contents

1 Preliminary results 5

1.1 Sobolev spaces on Polygons . . . . 5 1.2 Self-adjoint extension and Kre˘ın’s formula . . . . 10 2 Self-Adjoint Extensions of Symmetric Laplacians with Mixed

Dirichlet-Neumann Boundary Conditions 15

3 Symmetric Laplacians with Mixed Robin Boundary Condi-

tions in a Polygon 27

4 Self-adjoint Extensions for Symmetric Laplacians with Mixed

Robin Boundary Conditions 45

5 Examples 49

5.1 Case 1: Dirichlet boundary conditions . . . . 49

5.2 Case 2: mixed Dirichlet-Neumann boundary conditions . . . . 51

5.3 Case 3: Robin boundary conditions . . . . 52

Chapter 1

Preliminary results

1.1 Sobolev spaces on Polygons

In this section we recall the basic definitions and general results

that will be used in the following part of this work. Let Ω be an arbitrary open subset of R

with boundary ∂Ω. Denoting by L

(Ω) the space of all square integrable (complex valued) functions for the Lebesgue measure on Ω, we denote by C

(Ω) (resp. C

( ¯ Ω)) the space of all infinitely differentiable unctions with compact support in Ω (resp. the restriction to Ω of functions in C

( R

)).

Given s any real number, we shall denote by m its integral part and by σ its fractional part. According one has s = m + σ with 0 ≤ σ ≤ 1. Also we denote by D

the differentiation with respect to x

for 1 ≤ i ≤ n and for an arbitrary multi-integer α = (α

, α

, ...α

) with nonnegative components we set D

= D

D

...D

.

Definition 1.1.1. We denote H

(Ω) the space of all distributions u defined in Ω such that

i) D

u ∈ L

(Ω) for |α| ≤ m when s = m is a nonnegative integer ii) u ∈ H

(Ω) and

∫

Ω×Ω

|D

u(x) − D

u(y) |

|x − y|

dxdy < ∞ ,

for |α| = m when s = m + σ is a nonnegative and non integral integer.

We define a Hilbert norm on H

(Ω) by

∥u∥

= ( ∑

|α|≤m

∫

Ω

|D

u(x) |

dx )

1Proofs can be found e.g. in [3], Chapter 1, and [4], Chapter 1.

in the case i) and by

∥u∥

= (

∥u∥

+ ∑

|α|=m

∫

Ω×Ω

|D

u(x) − D

u(y) |

|x − y|

dxdy )

,

in case ii).

Definition 1.1.2. We denote by H

(Ω) the closure of C

(Ω) in H

(Ω).

Definition 1.1.3. We denote by H

(Ω) the dual space of H

(Ω).

Definition 1.1.4. For every positive s we denote by ˜ H

(Ω) the space of all u defined in Ω such that ˜ u ∈ H

( R

) where ˜ u is the continuation of u by zero outside Ω.

On ˜ H

(Ω) we define the norm ∥˜u∥

. A simple calculation shows that this norm is equal to ∥u∥

when s is an integer and equivalent to

∥u∥

+ ∑

|α|≤m

(∫

Ω

|D

u(x) |

w(x)dx )

when s is not an integer, where w(x) is an appropriate weight. If Ω is a Lipschitz

bounded domain, the weight w(x) is equivalent to ρ(x)

where ρ(x) denotes the distance from x to the boundary.

Theorem 1.1.1. Let Ω be a Lipschitz open subset of R

. Then C

( ¯ Ω) is dense in H

(Ω) for all s ≥ 0.

Theorem 1.1.2. Let Ω be a Lipschitz open subset of R

. Then C

(Ω) is dense in ˜ H

(Ω) for all s ≥ 0. Moreover C

(Ω) is dense in H

(Ω) for s ∈ [0, 1/2).

Definition 1.1.5. We denote by ˜ H

(Ω) the dual space of ˜ H

(Ω).

Theorem 1.1.3. Let Ω be a bounded Lipschitz open subset or R

then for every s > 0 there exists a continuous linear operator P

from H

(Ω) into H

( R

) such that

P

u

Ω

= u , for any u ∈ H

(Ω).

2By a Lipschitz domain we mean that the boundary of Ω is locally the graph of a Lipschitz function.

Theorem 1.1.4. Let s

> s

≥ 0 and assume that Ω is a bounded Lipschitz open subset or R

then the injection of H

(Ω) into H

(Ω) is compact.

Theorem 1.1.5. Let s

> s

> s

≥ 0 and assume that Ω s a bounded Lipschitz open subset or R

then there exists a constant C (which depends on Ω, s

, and s

) such that

∥u∥

≤ ϵ∥u∥

+ Cϵ

∥u∥

, for all u ∈ H

(Ω).

Theorem 1.1.6. Assume that Ω is any bounded open subset of R

then there exists a constant K(Ω) which depends only on the diameter of Ω such that

∥u∥

≤ K(Ω) ( ∑

1≤n

∫

Ω

|D

u |

dx )

for any u ∈ H

(Ω)

Theorem 1.1.7.

The following inclusions hold H

( R

) ⊂ L

( R

)

for s < n/2 and q ≥ 2 such that 1/q = 1/2 − s/n and H

( R

) ⊂ C

( R

) for any integers k < s − n/2.

Theorem 1.1.8. Given Ω be a bounded Lipschitz open subset of R

and denote by ρ(x) the distance from x to Γ. Then one has u/ρ

∈ L

(Ω) for all u ∈ H

(Ω) when 1 < s < 1/2 and for all u ∈ H

(Ω) when 1/2 < s < 1.

Let us now recall the well known trace theorem on an hyperplane. Given u a smooth function on R

we define the function γ

u by

γ

u(x

, . . . , x

) := u(x

, . . . , x

, 0) .

The density property allows one to extend γ

to a continuous linear operator from H

( R

) onto H

( R

) provided s > 1/2. As direct consequence one has

3Applying Theorem 1.1.3 one obtains the same inclusions for spaces over a bounded Lipschitz domain Ω.

Theorem 1.1.9. The mapping

u → (

γ

u, γ

∂u

∂x

, ..., γ

∂

u

∂x

)

defined on C

( R

) has for k < s − 1/2 an unique continuous extension as an operator from

H

( R

) onto ∏

0≤p≤k

H

( R

) .

From now on Ω will be a polygonal domain in R

where the boundary

∂Ω is given by the union the of sides Γ

, j = 1, . . . N . Given u ∈ H

(Ω) we define

γ

u := γ

P

u

by taking a system of orthogonal coordinates (x

, x

) ∈ R

with respect to which Γ

⊂ {(x

, 0) , x

∈ R}. Given such a definition one has the following Theorem 1.1.10. Let Ω be a bounded polygonal open subset of R

, then for each j the mapping

u → {γ

, 1 ≤ l ≤ k, 1 ≤ j ≤ N}

which is defined for u ∈ C

( ¯ Ω) has for k < s − 1/2 a unique continuous extension from

H

(Ω) into

∏

∏

0≤p≤k

H

(Γ

) .

As regards the range of the application given in the previous theorem one has the following

Theorem 1.1.11. Let Ω be a bounded polygonal open subset of R

. Then the mapping

u 7→ {γ

u , 0 ≤ l ≤ m − 1 , 1 ≤ j ≤ N } is linear continuous from H

(Ω) onto the subspace of

∏

1≤j≤N

∏

0≤l≤m−1

H

(Γ

)

defined by the following conditions. Let L be any differential operator with constant coefficients and order d ≤ m − 1. Denote by P

the differential operators tangential to Γ

such that

L = ∑

l

P

∂

∂n

,

where n

denotes the exterior normal at γ

. Then one has i) ∑

l

(P

f

)(S

) = ∑

l

(P

f

)(S

) for d ≤ m − 2 ii) ∑

l

(P

f

) ≡ ∑

l

(P

f

) at S

for d = m − 1.

As regards “half” and ’full” Green Formula the result is the following:

Theorem 1.1.12.

∫

Ω

u △v dx +

∫

Ω

∇v∇u dx = ∑

j

∫

Γj

γ

uγ

v dσ , u ∈ H

(Ω) , v ∈ H

(Ω) ,

∫

Ω

u △v dx−

∫

Ω

v △u dx = ∑

j

( ∫

Γj

γ

uγ

v dσ −

∫

Γj

γ

vγ

v dσ )

, u, v ∈ H

(Ω) . (1.1) The linear maps γ

and γ

can be extended to a larger domain and this implies an extension of Green’s folmulae also. Indeed, defining the maximal domain

D(△

) := {u ∈ L

(Ω) : △u ∈ L

(Ω) } , (1.2) one has

Theorem 1.1.13. Let Ω be a bounded polygonal open domain in R

. Then the maps γ

and γ

v have unique continuous extensions

ˆ

γ

: D(△

) → ˜ H

(Γ

) , ˆ

γ

: D(△

) → ˜ H

(Γ

) . Moreover

Theorem 1.1.14.

∫

Ω

v △udx +

∫

Ω

∇v∇udx = ∑

j

⟨γ

u, ˆ γ

v ⟩ ,

for every v ∈ D(△

) and every u ∈ H

(Ω) such that

γ

u ∈ ˜ H

(Γ

) .

Theorem 1.1.15.

∫

Ω

u △vdx −

∫

Ω

v △udx = ∑

j

( ⟨γ

u, ˆ γ

v ⟩ − ⟨ˆγ

v, γ

u ⟩ ) ,

for every v ∈ D(△

) and every u ∈ H

(Ω) such that γ

u ∈ ˜ H

(Γ

) , γ

u ∈ ˜ H

(Γ

) .

1.2 Self-adjoint extension and Kre˘ın’s formula

Let us consider the Hilbert space H equipped with the inner product ⟨·, ·⟩

and the self-adjoint operator

A : D(A) ⊆ H → H .

Then we define H

as the Hilbert space given by the domain D(A) equipped by the inner product

⟨ϕ, ψ⟩

= ⟨ϕ, ψ⟩ + ⟨Aϕ, Aψ⟩.

Now, given a closed subspace N ⊂ H

dense in H , let us define S as the closed, densely defined, symmetric operator obtained by restricting A to N . Our purpose is to describe all self-adjoint extensions of S together with their resolvent.

Since N is closed, H

= N ⊕N

, and then N coincides with the kernel of the orthogonal projection onto N

. Now, since N

≃ H

/ N is an Hilbert space, without lost of generality we can always suppose that N coincides with the kernel of a surjective bounded linear operator:

τ : H

→ h,

with h an auxiliary Hilbert space. Since this suffices for our purposes, we will suppose that h is finite dimensional, thus we can pose

h = C

. This meas that S has finite deficiency indices.

In the following

K (L), R(L), ρ(L),

will be respectively the kernel, the range and the resolvent set of a given linear operator L. Given τ as above one has

S = A |K (τ), R(τ) = C

, K (τ) = H . (1.3)

For any z ∈ ρ(A) we define the resolvent of A, i.e. the bounded linear operator from H

to H as:

R

:= ( −A + z)

(1.4)

For any element of ρ(A) we also consider the continuous linear map:

G

:= (τ R

)

: C

→ H , (1.5) that is injective being τ surjective. One has that (1.3) are equivalent to:

R(G

) ∩ D(A) = {0}. (1.6)

Furthermore, by the first resolvent identity one has ([6], lemma 2.1):

(z − w)R

G

= G

− G

, (1.7)

R(G

− G

) ∈ D(A). (1.8)

Let us now consider a family of linear operators Γ : C

→ C

such that

(Γ

)

= Γ

, (1.9)

Γ

− Γ

= (z − w)G

G

. (1.10) Let us observe that this class is nonempty, in fact by (1.7) and the definition of Γ(z) one can prove (see [6], lemma 2.2) that each of these families differs by a z-independent, symmetric operator from the family ˆ Γ

(z) defined as

Γ ˆ

(z) := τ

( G

+ G

2 − G

)

, w ∈ ρ(A).

Notice that ˆ Γ

(z) is well defined by (1.8).

In the case 0 ∈ ρ(A), the easiest choice (the one we will take in the following chapters) is

Γ

= τ (G

− G

) ≡ zG

G

. (1.11)

Given the orthogonal projector

Π : C

→ C

we pose

C

:= R(Π),

and for any symmetric operator:

Θ : C

→ C

, we can define the linear operator

Γ

:= (Θ + Π Γ

Π) : C

→ C

, (1.12) and the open set

Z

:= {z ∈ ρ(A) : det Γ

(z) ̸= 0}.

Now we have the following result

Theorem 1.2.1.

Let A, τ , S Π, Θ and Γ

as above. Then C \ R ⊆ Z

and the bounded linear operator:

R

:= R

+ G

Π Γ

ΠG

, z ∈ Z

, (1.13) is the resolvent of the self-adjoint extension A

of S defined as

A

: D(A

) ⊆ H → H , ( −A

+ z)ϕ := ( −A + z)ϕ

, D(A

) := {ϕ ∈ H : ϕ = ϕ

+ G

Π Γ

Πτ ϕ

, ϕ

∈ D(A)}.

The definition is z-independent and the decomposition appearing in D(A

) is univocal.

Proof. By (1.9) e (1.10) we have

|ζ · Γ

ζ |

≥ Im (z)

∥G

ζ ∥

for any ζ ∈ C

. So det Γ

̸= 0 for any z ∈ C\R. Now by (1.10), from [6]

we have that R

satisfies the resolvent identity

(z − w)R

R

= R

− R

(1.14) and by (1.9),

R

= R

. (1.15)

Furthermore by (1.6), R

is injective. Then the extension A

:= z − R

is well defined on

D(A

) := R(R

),

and z-independent, respectively symmetric, by (1.14), respectively by (1.15).

Finally it is self-adjoint since R(−A

± i) = H by construction.

4See [6], theorem 2.1

Corollary 1.2.2. Suppose that 0 ∈ ρ(A). Then

A

: D(A

) ⊆ H → H , A

ϕ = Aϕ

,

D(A

) := {ϕ ∈ H : ϕ = ϕ

+ G

ξ

, ϕ

∈ D(A) , ξ

∈ C

, Πτ ϕ

= Θξ

}.

Notwithstanding the easy proof the self-adjoint extensions provided above exhaust the class of all self-adjoint extension of the symmetric operator S:

Theorem 1.2.3.

The family of A

, given by Theorem 1.2.1, coincides with the family F of all self-adjoint extensions of the symmetric operator S.

Thus F can be parameterized by bundle

p : E( C

) → P (C

),

where P ( C

) denotes the set of orthogonal projectors in C

and p

(Π) de- notes the set of symmetric operators in C

. In particular the set of sym- metric operators on C

, i.e. p

(1), parameterize the extensions such that D(A

) ∩ D(A) = N , also called relatively prime extensions.

The next result

give us informations about the spectrum and eigenfunc- tions of A

Theorem 1.2.4.

λ ∈ σ

(A

) ∩ ρ(A) ⇔ 0 ∈ σ

(Γ

), where σ

( ·) denotes the point spectrum. Moreover

G

: K (Γ

) → K (−A

+ λ) is a bijection for any λ ∈ σ

(A

) ∩ ρ(A).

The next theorem provides the quadratic forms corresponding to the self- adjoint extension given above. Since this suffices for our purposes we suppose here that 0 ∈ ρ(A).

Theorem 1.2.5. Let

F : D(F ) × D(F ) ⊆ H × H → R be the quadratic form associated to −A and suppose that

R(G

) ∩ D(F ) = {0} .

5See [7], section 3

6See [7], section 2

Then

F

: D(F

) × D(F

) ⊆ H × H → R ,

D(F

) = {ϕ ∈ H : ϕ = ϕ

+ G

ξ

, ϕ

∈ D(F ) , ξ

∈ C

} , F

(ϕ, ψ) = F (ϕ

, ψ

) + Θξ

·ξ

,

is the quadratic form associated to −A

.

Proof. Let L : D(L) ⊆ H → H be the linear operator associated to F

, i.e.

D(L) :=

{ϕ ∈ D(F

) : ∃˜ϕ ∈ H s.t. ∀ψ ∈ D(F

) , F

(ϕ, ψ) = ⟨˜ϕ, ψ⟩

} , Lϕ := ˜ ϕ .

Since D(F ) ⊆ D(F

), D(F

) is dense and so L is well-defined.

By the definition of D(F

) and by taking, in the definition of D(L), at first ξ

= 0 and then ψ = G

ξ

, one gets that ϕ = ϕ

+ G

ξ

∈ D(L) if and only if there exists ˜ ϕ such that

∀ψ

∈ D(F ) , F (ϕ

, ψ

) = ⟨˜ϕ, ψ

⟩

and

∀ξ ∈ C

, Θξ

·ξ = ⟨˜ϕ, G

ξ ⟩

. Thus ϕ

∈ D(A), Lϕ = −Aϕ

, and

⟨˜ϕ, G

ξ ⟩

= −⟨Aϕ

, G

ξ ⟩

= −(τR

Aϕ

) ·ξ = (τϕ

) ·ξ = (Πτϕ

) ·ξ .

This gives Πτ ϕ

= Θξ

, and so L = −A

.

Chapter 2

Self-Adjoint Extensions of Symmetric Laplacians with Mixed Dirichlet-Neumann Boundary Conditions

We start this sections recalling some important results about the Laplace operator on a polygon with mixed boundary conditions at the boundary

. Let Ω ⊂ R

be a plane bounded open curvilinear polygon. This means that the boundary ∂Ω is a piecewise smooth closed curve with no cups points.

The point where such a curve fails to be differentiable are called vertices. To simplify the exposition we further suppose that a such curve coincides with a broken line in a neighborhood of each vertex. If the whole boundary is made of broken lines we says that Ω is a classical polygon.

We also assume that Ω is connected and simply connected domain and we will denote each open smooth segment of ∂Ω (i.e. its sides) by Γ

where the index j ranges from 1 to some integer N . These segments are numbered in such a way that Γ

follows Γ

according to the positive orientation. We also denote by S

the vertex which is the endpoint of Γ

.

Furthermore we define n

(resp. t

) as the unit outward normal (resp. tan- gent) vector on Γ

and by ω

the measure of the interior angle at S

. Polar coordinates (r

, θ

) with origin at S

will be used. Such coordinates are choosen in such a way that Γ

is on the half-axis θ = ω

and Γ

is on the half-axis θ = 0. Then we introduce the cartesian coordinates attached to each corner with vertex S

as

x

= r

cos θ , y

= r

sin θ ,

1All the missing proofs can be found in [3], Sections 2.1 - 2.3.

accordingly Γ

is a subsegment of the line y

= 0.

In considering mixed boundary conditions, it is useful to fix a partition of {1, ..., N} (the set numbering the vertices) by the subsets D and N defined according to the following rule:

• j ∈ D if Γ

supports a Dirichlet boundary condition;

• j ∈ N if Γ

supports a Neumann boundary condition.

We also will consider the sets D

, N

, ( N , D), (D, N ), defined by

• j ∈ D

if both Γ

and Γ

support Dirichlet boundary conditions;

• j ∈ N

if both Γ

and Γ

support Neumann boundary conditions;

• j ∈ (N , D) if Γ

supports a Neumann boundary condition and Γ

supports a Dirichlet boundary condition;

• j ∈ (D, N ) if Γ

supports a Dirichlet boundary condition and Γ

supports a Neumann boundary condition.

Finally we introduce the set corresponding to mixed boundary conditions:

M := (D, N ) ∪ (N , D) and for later convenience we also define

M

:= {j ∈ D

∪ N

: ω

> π } ∪ {j ∈ M : ω

> π/2 } , M

:= {j ∈ M : ω

> 3

2 π } . We will always suppose that

D ̸= ∅ (2.1)

Let △

be the distributional Laplace operator on the curvilinear polygon Ω and let us define

△

: D(△

) ⊂ L

(Ω) → L

(Ω) , △

u := △

u where D(△

) is defined in (1.2).

We introduce the spaces V (Ω) and V

(Ω), respectively containing the variational and the strong solutions of the boundary value problem

 



 

∆

u = f , f ∈ L

(Ω)

γ

u = 0 , j ∈ D

γ

u = 0 , j ∈ N ,

defined by

V (Ω) = {u ∈ H

(Ω) : γ

u = 0 , j ∈ D} (2.2) and

V

(Ω) = {u ∈ H

(Ω) : γ

u = γ

u = 0 , j ∈ D , i ∈ N } . (2.3) The results given by the next theorems will be useful in the following:

Theorem 2.0.6. The space H

(Ω) ∩V (Ω) is dense in V (Ω) for every m > 1.

Theorem 2.0.7. The space H

(Ω) ∩ V

(Ω) is dense in V

(Ω) for every m > 1.

For any function in V

(Ω) a Caccioppoli’s type inequality holds true:

Theorem 2.0.8. Assume that Ω is a bounded polygonal open subset of R

and that (2.1) holds. Then there exists a constant C

such that

∀u ∈ V

(Ω) , ∥u∥

≤ C

∥∆

u ∥

. (2.4) As a direct consequence of Poincar´ e inequality (see Theorem 1.1.6) V (Ω) is a Hilbert space for the scalar product induced by the bilinear form

F (u, v) :=

∫

Ω

∇u · ∇v dx , u , v ∈ V (Ω) .

This allow us to apply the Lax-Milgram Theorem and to conclude

that there exists a unique self-adjoint operator

△

: D(△

) ⊆ V (Ω) ⊂ L

(Ω) → L

(Ω) such that

F (u, v) = ⟨−∆

u, v ⟩

, u ∈ D(∆

) , v ∈ V (Ω) .

On the other hand by Theorem 2.0.8 and by Green’s Formula the linear operator

△

: V

(Ω) ⊆ L

(Ω) → L

(Ω) , △

u := △u ,

is closed and symmetric. Thus a natural question arises: is △

self-adjoint?

Equivalently: does △

coincide with △

?

The answer to the previous question depends on the shape of Ω and in order

2see e.g. [2], Chapter IV, Section 1.

to answer to this question we need some more definition. For any vertex S

we consider the measure ω

of the corresponding interior angle and define

n

:= # M

, n

:= # M

In the following we will see that the deficiency indices of ∆

are both equal to n

+ n

.

Consequently if n

+ n

̸= 0 we have D(△

) ̸= V

(Ω). This is an immediate consequence of the fact that for every j ∈ D

(respectively j ∈ N

) the function

r

sin π ω

θ

, (respectively r

cos

j

θ

) belongs to K (△

) ∩ H

(W ), where W is the wedge

W = {(x, y) ≡ (r

cos θ

, r

sin θ

) : 0 ≤ r

< R , 0 < θ

< ω

} , but fails to be in H

(W ) when π/ω

< 1.

In the case j ∈ M the conclusion is quite similar considering for example the pair of functions

u

= r

(m−1/2)π ωj

j

sin (m − 1/2)π

ω

θ , m = 1, 2 . From now on we will suppose that n

+ n

̸= 0 so that

V

(Ω) ≡ D(△

) ( D(△

) .

Thus any self-adjoint extension of △

will be a restriction of its adjoint

△

. Since △

is the adjoint of the restriction of △

to C

(Ω), one has

△

⊂ △

.

Inequality (2.4) shows that △

is injective and has a closed range and, posing N := R(∆

)

= K ((∆

)

)

one has the following results, which completely characterize the linear set N : Lemma 2.0.9. Let v ∈ N. Then v belongs to D(△

) and solves the (ad- joint) boundary value problem

 



 

△v = 0 in Ω ˆ

γ

v = 0 j ∈ D ˆ

γ

v = 0 i ∈ N .

3Here #S denotes the cardinality of the set S.

Lemma 2.0.10. Every v ∈ N is such that

∫

Ω

v △ η

dx = 0 , (2.5)

for any j ∈ N

, ∫

Ω

v △(y

η

)dx = 0 , (2.6)

for any j ∈ (N , D) with either ω

= π/2 or ω

= 3π/2,

∫

Ω

v △(x

η

)dx = 0 , (2.7)

for any j ∈ (D, N ) with either ω

= π/2 or ω

= 3π/2.

Here η

∈ C

(Ω) is a truncation function which depends only on the dis- tance to S

and such that η

≡ 1 near S

and vanishes near all sides Γ

, k ̸= j.

Theorem 2.0.11. Let v ∈ D(△

) be such that

 



 

△v = 0 in Ω ˆ

γ

v = 0 j ∈ D ˆ

γ

v = 0 i ∈ N .

and that it fulfill the conditions in Lemma 2.0.10. Then v ∈ N.

Lemma 2.0.12. Let v ∈ N then v ∈ C

( ¯ Ω \V ) where V is any neighborhood of the corners S

.

Denoting by I ⊂ L

(Ω) the set of function satisfying the conditions ap- pearing in Lemma 2.0.10, we can resume the results above by stating the following

Theorem 2.0.13.

K ((△

)

) = K (△

) ∩ K ∩ I ,

where K := {u ∈ D(△

) : ˆ γ

u = ˆ γ

u = 0 , j ∈ D , i ∈ N } Moreover, by Theorem 2.3.7 in [3], one has that

Theorem 2.0.14.

dim K ((△

)

) = n

+ n

. (2.8)

More precisely each corner contributes to the dimension of K (△

) as fol-

lows:

• the contribution of a Dirichlet corner (i.e. j ∈ D

) is 0 if ω

≤ π and 1 if ω

> π;

• the contribution of a Neumann corner (i.e. j ∈ N

) is 0 if ω

≤ π and 1 if ω

> π;

• the contribution of a mixed-corner (i.e. j ∈ M ) is 0 if ω

≤ π/2, 1 if π/2 < ω ≤ 3/2π and 2 if ω

> 3/2π.

In order to better characterize the above kernel we introduce some more definitions. We consider function C

(Ω) depending only on the radial vari- able as follows: given R

< R

< R,

χ

= 1 , if r < R

, χ

= 0 , if r > R

∀k ∈ D ∪ N , For any vertex S

∈ ∂Ω such that j ∈ D ∪ N we consider the disc

D

= {x ∈ R

: ∥x − S

∥ < R } and define the wedge

W

:= Ω ∩ D

≡ {(x, y) ≡ (r cos θ

, r cos θ

) : 0 ≤ r ≤ R, 0 < θ

< ω

} , where we choose R in such way that W

∩ W

= ∅ for j ̸= h.

On any disk D

centered at S

we define the functions u

u

= 1

√ π r

sin ( π ω

θ

)

, ∀j ∈ D

,

u

= 1

√ π r

cos ( π ω

θ

)

, ∀j ∈ N

, and (here m = 1, 2)

u

= C

r

(m−1/2) ωj π

sin

( (m − 1/2) ω

π θ

)

, ∀j ∈ (D, N ) , u

= C

r

(m−1/2) ωj π

sin

( (m − 1/2)

ω

π(ω

− θ

) )

, ∀j ∈ (N , D) ,

with

C

=

√ 2(4 − m)

3π , m = 1, 2 . (2.9)

With such a choice we have that the functions χ

u

are in L

(Ω) and L

(Ω)-

orthogonal.