On the Regularity of Elliptic Differential Equations Using Symmetry Techniques and

On the Regularity of Elliptic Differential Equations Using Symmetry Techniques and

Suitable Discrete Spaces ^∗

Christoph Pflaum

Abstract

We present an elementary and short proof for regularity of second or- der elliptic differential equations with homogeneous Dirichlet boundary conditions. The proof uses a discrete function space with piecewise mul- tilinear functions and symmetry techniques on the unit cube.

1 Introduction

In this paper, we prove the regularity of second order elliptic differential equa- tions with homogeneous Dirichlet boundary conditions on domains whose bound- ary is locally a smooth deformation of the boundary of the unit cube. In the two dimensional case, this implies the regularity of Poisson’s equation on every domain with a piecewise smooth boundary and with no reentrant corner.

The usual approach to prove regularity is to first prove the regularity on domains with a smooth boundary and then to study each corner (see [2]). Es- pecially in more than two dimensions, this approach leads to very long proofs.

Here, we will present an elementary and short proof for the regularity of el- liptic equations for a certain class of domains with corners. However, we can not analyze every corner in more than two dimension, but we can show the W₂²-regularity of the solution near a lot of corners which appear in application.

The proof of regularity in this paper consists of two ideas. First, we assume that the domain Ω is the d-dimensional unit cube Ω^d :=]0,1[^d. Then, we can extend every function in a symmetric way to a function on a band or a torus.

Then, with the help of finite difference operators, it is no problem to prove the regularity of the solution. This is an old technique. But this approach can be used only for a certain class of elliptic equations (see section 2). The crucial restriction is that the values of some coefficients have to be zero at the boundary

∗1991 Mathematics Subject Classifications: 65D10.

Key words and phrases: regularity of elliptic differential equations c1996 Southwest Texas State University and University of North Texas.

Submitted: July 25, 1996. Published November 22, 1996.

1

last step is to generalize the regularity on the unit cube to the regularity on a certain class of domains with corners.

Now, let us describe an elliptic equation on the bounded domain Ω⊂R^d. Assume that

B=





b11 · · · b1d

... ... bd1 · · · bdd





is a matrix contained in (W_∞¹(Ω))^d×d, and thatB is a uniformly elliptic matrix.

Then, the bilinear form

a:W₂¹(Ω)×W₂¹(Ω) → R, (1) (u, v) 7→

Z

Ω

(∇u)^TB∇v dλ

isW₂¹(Ω)-elliptic. Furthermore, assume thatf ∈L²(Ω) and let us write f(v) :=

Z

Ω

f v dλ.

Then, there is a unique solutionu∈W₂¹(Ω) of the weak equation

a(u, v) =f(v) for every v∈W₂¹(Ω). (2) Our aim is to proveu∈W₂²(Ω) under suitable assumptions on Ω.

2 Regularity Using Symmetry

Using symmetry, very simple regularity proofs can be obtained for a restricted class of equations. Therefore we first assume Ω = Ω^d=]0,1[^d.

The idea of using symmetry is to extend functions defined on the unit cube to functions on a band. This band is

B^d=S¹×]0,1[^d−1

whereS¹ is the interval ]−1,1[ identified at the points{−1,1}. Formally, we also can defineS¹ by

S¹=R/(2Z).

-1 1

Figure 1: Extension Operator ˜.

-1 1

Figure 2: Extension Operator ˆ.

ThereforeS¹is a circle. Obviously, there is a natural embedding Ω^d,→B^d.

This shows that we can restrict every function defined onB^d to a function on Ω^d. But usually, we will omit the restriction operator.

Now, we define two extension operators:

˜:L²(Ω^d) → L²(B^d) and ˆ:L²(Ω^d) → L²(B^d).

The operator ˜ is the anti-symmetric operator extension in the direction of the first coordinate. This means that

˜

v(x1, x2,· · ·, xd) =−v(−x1, x2,· · ·, xd) for (x1, x2,· · ·, xd)∈Ω^d (see Figure 1).

The operator ˆ is the symmetric extension operator in the direction of the first coordinate. This means that

ˆ

v(x1, x2,· · ·, xd) =v(−x1, x2,· · ·, xd) for (x1, x2,· · ·, xd)∈Ω^d (see Figure 2).

Furthermore, denote ˜W2¹(B^d) the Sobolev space of the symmetric functions W˜₂¹(B^d) :=n

˜

uu∈W₂¹(Ω^d)o .

Observe that ˜W₂¹(B^d) is a subspace ofW₂¹(B^d). The extended bilinear form ˜a is defined by

˜

a: ˜W₂¹(B^d)×W˜₂¹(B^d) → R ( ˜w,˜v) 7→ 1 2 Z

B^d

(∇u)^TB¯∇v dλ

Observe that the elements of the matrix ¯Bare the symmetric extended elements of the matrixB with the exception of the elements ˜b1i and ˜bi1 fori 6= 1. For example, this implies that

ˆbkk∈W_∞¹(B^d) for 1≤k≤d.

But the elements ˜b1i and ˜bi1 do not have such a property in general. A simple calculation shows

˜

a(˜u,˜v) =a(u, v) for every u, v∈W₂¹(Ω^d).

At last we need some difference operators. Let 0< h <1. Then define δ_h¹(w)(x1) := w(x1+^h₂)−w(x1−^h2)

h and δ_h²:=δ_h¹◦δ_h¹. So, these operators act in the direction of the first coordinate.

With these preliminaries, we obtain the following regularity result:

Theorem 1 (Regularity in Case of Diagonal Matrices B) Assume thatB is a diagonal matrix. Then, the solutionuof the equation (2) on the unit cube Ω^d satisfies the inequality

kukW₂²(Ω^d)≤CkfkL²(Ω^d) (3) whereC is a constant independent ofu.

Proof: By a symmetry argument we obtain (see Figure 1)

δh²(˜v)∈W˜2¹(B^d) (4) for every ˜v∈W˜2¹(B^d). We have to prove that

˜

a(δ¹_h(˜u), δ_h¹(˜v)) =−˜a(˜u, δ_h²(˜v)) +S(˜u, δ_h¹(˜v)) (5) whereS is a bilinear form with the property

S(˜u, δ_h¹(˜v)≤Cku˜kW₂¹(B^d)kδ_h¹(˜v)kW₂¹(B^d).

Equation (5) can be treated as a substitute for integration by parts. For the proof of equation (5) we need the following simple formula

Z

S¹

δ¹h(w)bvdx=− Z

S¹

wbδh¹(v)dx− Z

S¹M^h w, δ¹h

2(b)

vdx (6)

for everyw, v∈L²(S¹) andb∈L^∞(S¹), whereM^h is the Operator M^h(w, v)(x) := ¹₂ w x+^h₂

v x+^h₄

+w x−^h2

v x−^h4

.

Now, we obtain (5) by the fact that the non-diagonal elements of ¯B are zero and that the diagonal elements are contained inW_∞¹(B^d).

Equations (2), (4), and (5) imply

kδ¹_h(˜u)k²W₂¹(Ω^d) ≤ C1kδ¹_h(˜u)k²W₂¹(B^d)

≤ C2˜a(δ¹h(˜u), δh¹(˜u))

= −C2˜a(˜u, δ²_h(˜u)) +C2S(˜u, δ¹_h(˜u))

= −C2a(u, δ²_h(˜u)) +C2S(˜u, δ¹_h(˜u))

= −C2f(δ²_h(˜u)) +C2S(˜u, δ¹_h(˜u))

≤ C3

kfkL²(Ω^d)+ku˜kW₂¹(B^d)

kδ¹_h(˜u)kW₂¹(B^d)

≤ C4kfk^L2(Ωd)kδ_h¹(˜u)kW₂¹(Ω^d)

whereC1, C2,C3,C4 are suitable constants. Thus, we obtain kδ_h¹(˜u)kW₂¹(Ω^d)≤C4kfkL²(Ω^d). The limith→0 shows (see Satz 9.5 in [4])

∂

∂x1

(˜u) W₂¹(Ω^d)

≤C5kfkL²(Ω^d). A symmetry argument completes the proof.

q.e.d.

3 Discrete Regularity on Discrete Spaces

The proof of Theorem 1 can not be extended to general matricesB, because the extended matrix elements ˜b1i and ˜bi1are not very smooth at the boundary of Ω^d. A different situation appears, if we study the discrete regularity in a suitable discrete space. Then, the idea of the proof of Theorem 1 can be used for general matricesB. This we will show now.

Forh= _N¹ andN ∈NletVh be the finite element space with the following properties:

• every function in Vh has homogeneous Dirichlet boundary conditions,

• every function in Vh is a piecewise multilinear function on the uniform tensor product grid of mesh size h.

kδ_h(˜uh)kW₂¹(Ω^d)≤CkfkL²(Ω^d)

whereC is a constant independent ofuh.

Proof: The proof of Theorem 1 shows, that we only have to prove

˜

a(δ¹_h(˜u), δ_h¹(˜v)) =−˜a(˜u, δ_h²(˜v)) +S(˜u, δ_h¹(˜v)) whereS is a bilinear form with the property

S(˜u, δ_h¹(˜v)≤Cku˜kW₂¹(B^d)kδ_h¹(˜v)kW₂¹(B^d).

Furthermore observe that we only have to study the following terms of ˜a(δ_h¹(˜u), δ¹_h(˜v))

: Z

B^d

˜b1i

∂δ¹_h(˜u)

∂x1

∂δ¹_h(˜v)

∂xi

dλ and Z

B^d

˜bi1

∂δ_h¹(˜u)

∂xi

∂δ_h¹(˜v)

∂x1

dλ.

fori6= 1. The other terms can be treated like in the proof of Theorem 1. By integration by parts we obtain

Z

B^d

˜b1i

∂δ¹_h(˜u)

∂x1

∂δ_h¹(˜v)

∂xi

dλ=

= Z

B^d

˜b1i

∂δ_h¹(˜u)

∂xi

∂δ¹_h(˜v)

∂x1

dλ−Z

B^d

∂˜b1i

∂x1

δ_h¹(˜u)∂δ_h¹(˜v)

∂xi

dλ+ Z

B^d

∂˜b1i

∂xi

δ¹_h(˜u)∂δ¹_h(˜v)

∂x1

dλ.

The last two terms of this sum are of low order. This shows that we only have to prove for a functionb∈W_∞¹(Ω^d)

Z

B^d

˜b∂δ_h¹(˜u)

∂xi

∂δ¹_h(˜v)

∂x1

dλ=−Z

B^d

˜b∂u˜

∂xi

∂δ_h²(˜v)

∂x1

dλ+S⁰(˜u, δ¹_h(˜v)) whereS⁰ is a bilinear form with the property

S⁰(˜u, δ_h¹(˜v)≤Cku˜kW₂¹(B^d)kδ_h¹(˜v)kW₂¹(B^d). By the formula (6), we obtain

Z

B^d

˜b∂δ_h¹(˜u)

∂xi

∂δ_h¹(˜v)

∂x1

dλ=

= −

Z

B^d

˜b∂u˜

∂xi

∂δ²_h(˜v)

∂x1

dλ− Z

B^dM^h ∂u˜

∂xi

, δ¹h 2(˜b)

∂δ_h¹(˜v)

∂x1

dλ=

= −Z

B^d

˜b∂u˜

∂xi

∂δ²_h(˜v)

∂x1

dλ−2 Z

Ω^d

M^h ∂u˜

∂xi

, δ¹h 2(˜b)

∂δ_h¹(˜v)

∂x1

dλ.

B

B

B

B

B

B

B

@

@

@

A

A

A

A

A

J

J

J

J

h 1 h 1 h 1

Figure 3: Functions ˜v,δ¹_h(˜v), and _∂x^∂

1δ_h¹(˜v) in x-direction and restricted on the interval [0,1].

Thus, it is enough to prove Z

Ω^dM^h ∂u˜

∂xi

, δ¹h 2(˜b)

∂δ_h¹(˜v)

∂x1

dλ

≤Cku˜kW₂¹(Ω^d)kδ¹h(˜v)kW₂¹(Ω^d). Now, look to Figure 3. The function ^∂δ_∂x^h1(˜v)

1 is zero on the domain Φh:=]0,^h₂[×]0,1[^d−1 [

]1−^h2,1[×]0,1[^d−1. Therefore, we get

Z

Ω^d

M^h ∂u˜

∂xi

, δ¹h 2(˜b)

∂δ_h¹(˜v)

∂x1

dλ =

= Z

Ω^d\Φ_h

M^h ∂u˜

∂xi

, δ¹h 2(˜b)

∂δ_h¹(˜v)

∂x1

dλ ≤

≤ δ¹h 2(˜b)

L^∞

Ω^d\Φ_h

2

ku˜kW₂¹(Ω^d)kδ¹_h(˜v)kW₂¹(Ω^d)≤

≤ kbkW_∞¹(Ω^d)ku˜kW₂¹(Ω^d)kδ¹_h(˜v)kW₂¹(Ω^d). This completes the proof.

q.e.d.

4 General Regularity

Theorem 3 (Regularity in Case of General Matrices B) The solutionu of the equation (2) on the unit cubeΩ^d satisfies the inequality

kukW₂²(Ω^d)≤CkfkL²(Ω^d)

whereC is a constant independent ofu.

hlim→0ku−uhkW₂¹(Ω^d)= 0.

Therefore, we obtain Z

Ω^d

u ∂²ϕ

∂xk∂xl

dλ =

lim

h→0

Z

Ω^d

uh

∂²ϕ

∂xk∂xl

dλ =

=

lim

h→0

Z

Ω^d

∂²uh

∂xk∂xl

ϕ dλ

≤CkfkL²(Ω^d)kϕkL²(Ω^d)

for everyϕ∈ C^∞(Ω^d). This shows that ∂²u

∂xk∂xl

L²(Ω^d)

≤CkfkL²(Ω^d). (8) Now, we writeB =B^diag+B^rest, whereB^diagis the diagonal matrix ofB. Let us define

F(v) :=f(v)− Z

Ω^d

(∇u)^TB^rest∇v dλ and a^diag(u, v) :=

Z

Ω^d

(∇u)^TB^diag∇v dλ.

By (2) and (8), we obtain

a^diag(u, v) = F(v) for every v∈W₂¹(Ω^d) and

|F(v)| ≤ CkvkL²(Ω^d). Theorem 1 implies that

kukW₂²(Ω^d)≤CkfkL²(Ω^d). q.e.d.

Theorem 3 shows the regularity of elliptic equations on the unit cube Ω^d. Now, we generalize this result to more general domains. Let us assume that Ω is a domain with the following properties:

• Ω⊂R^d is open and ¯Ω is compact,

• for everyx∈Ω exists an open neighborhoodUxofxin Ω, a pointx⁰∈Ω^d, an open neighborhoodU_x⁰0 ofx⁰ in Ω^d, and aC²-diffeomorphism

Φx: ¯Ux7→U¯_x⁰0

such that

Φx(∂Ux∩∂Ω) =∂U¯_x⁰0∩∂Ω^d.

Now, we obtain the following Corollary:

Corollary 1 (Regularity in Case of General Domains Ω) Let us assume that the domain Ω satisfies the above assumptions. Let u be the solution of equation (2). Then, there is a constantC independent of usuch that

kukW₂²(Ω)≤CkfkL²(Ω).

The proof of this Corollary is analogous to the proof of Satz 9.1.4 in [3] or Theorem 8.12. in [1].

References

[1] D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Heidelberg, New York, 1977.

[2] P. Grisvard. Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics 24. Pitman, Boston, London, Melbourne, 1985.

[3] W. Hackbusch. Theorie und Numerik elliptischer Differentialgleichungen.

Teubner, Stuttgart, 1986.

[4] J. Wloka. Partielle Differentialgleichungen. Teubner, Stuttgart, 1982.

Christoph Pflaum

Institute of Applied Mathematics and Statistics University of W¨urzburg

Am Hubland 97074 W¨urzburg Germany

E-mail address: e-mail: [email protected]