Part V, Chapter 24
Scalar second-order elliptic PDEs
This chapter deals with scalar-valued, second-order, elliptic PDEs. We focus
on weak formulations endowed with a coercivity property which makes wellposedness solely hinge on the Lax–Milgram Lemma. The key example is the
diffusion-advection-reaction problem with various types of boundary conditions. In particular, we study in some detail how the boundary conditions are
enforced in the weak formulation. Moreover, important regularity properties
of the exact solution, which will be useful later to establish error estimates
for the finite element approximation, are listed at the end of the chapter.
24.1 Model problem
Let D be a domain in Rd . Let d, β, and µ be functions defined over D that
take values in Rd×d , Rd , and R, respectively. Given a function f : D → R, the
unknown is a function u : D → R that solves the following linear PDE:
−∇·(d∇u) + β·∇u + µu = f
in D,
(24.1)
equipped with boundary conditions that are discussed later. Using Cartesian
coordinates, (24.1) amounts to solve
X
d
d
X
∂u
∂
∂u
dij
βi
+
+ µu = f in D.
(24.2)
∂xi
∂xj
∂xi
i,j=1
i=1
The above problem reduces to the Poisson equation −∆u = f when d is the
identity tensor in Rd and both β and µ vanish identically. More generally,
(24.1) is a diffusion-advection-reaction equation modeling for instance heat
or mass transfer or flows in porous media. The first term on the left-hand
side of (24.1) accounts for diffusion processes, the second one for advection
processes, and the third one for reaction processes (depletion occurs when µ
is positive).
320
Chapter 24. Scalar second-order elliptic PDEs
24.1.1 Model data
We assume that d ∈ L∞ (D), β ∈ W 1,∞ (D) (this assumption can be weakened
as long as both β and ∇·β are bounded in D), and µ ∈ L∞ (D). For later use,
we introduce the quantity
1
µ0 (x) := µ(x) − ∇·β(x),
2
a.e. x ∈ D.
(24.3)
Using the divergence formula for the field ( 12 v 2 )β, we observe that the following integration by parts formula holds:
Z
Z
Z
1
1
v(β·∇v) dx = −
(∇·β)v 2 dx +
(β·n)v 2 ds,
(24.4)
2 D
2 ∂D
D
for all smooth functions v. Let ℓD be a length scale associated with D, e.g.,
ℓD = diam(D); then we equip H 1 (D) with the norm kvkH 1 (D) = (kvk2L2 (D) +
1
ℓ2D k∇vk2L2 (D) ) 2 , so that both terms in the sum have the same dimension.
Formula (24.4) can be shown to hold in H 1 (D) by a density argument.
Concerning the second-order terms in the PDE (24.1), we assume that d
takes symmetric values, and we introduce the following important assumption:
Definition 24.1 (Ellipticity). For a.e. x ∈ D, let [λ♭ (x), λ♯ (x)] be the
(smallest) interval containing the eigenvalues of d(x). We say that the
PDE (24.1) is elliptic if
0 < λ♭,D := ess inf λ♭ (x),
x∈D
ess sup λ♯ (x) =: λ♯,D < +∞.
(24.5)
x∈D
Example 24.2 (Anisotropic diffusion). We say that the diffusion process
is anisotropic if the diffusion matrix is not proportional to the identity. A
2
2
2
u
− ∂∂xu2 = f for a real
two-dimensional example is the PDE − ∂∂xu2 + 2κ ∂x∂1 ∂x
2
1
2
number κ, which is elliptic under the assumption that |κ| < 1.
⊓
⊔
24.1.2 Towards a weak formulation
Proceeding formally as in §18.1, we assume that u is smooth enough, say
u ∈ H 2 (D), we multiply (24.1) by a test function w ∈ H 1 (D) and integrate
over D to obtain
Z
Z
(−∇·(d∇u)w + (β·∇u)w + µuw) dx =
f w dx.
(24.6)
D
D
Integrating by parts the first term on the left-hand side leads to
Z
Z
Z
∇u·d·∇w dx −
(n·d·∇u)w ds,
−∇·(d∇u)w dx =
D
D
(24.7)
∂D
where n denotes the outward unit normal to D, and where we have used the
symmetry of d to write ∇u·d·∇w = ∇u·(d∇w) = ∇w·(d∇u); using Cartesian
Part V. Elliptic PDEs
321
Pd
∂u ∂w
. Putting
coordinates, this quantity is equal to ∇u·d·∇w = i,j=1 dij ∂x
i ∂xj
everything together, we arrive at
Z
Z
a(u, w) −
(n·d·∇u)w ds =
f w dx,
∀w ∈ H 1 (D),
(24.8)
∂D
D
1
where a is defined for all (v, w) ∈ H (D)×H 1 (D) as follows:
Z
(∇v·d·∇w + (β·∇v)w + µvw) dx.
a(v, w) =
(24.9)
D
Using the Cauchy–Schwarz inequality three times leads to
|a(v, w)| ≤ M kvkH 1 (D) kwkH 1 (D) ,
∀v, w ∈ H 1 (D),
(24.10)
with M := ℓ2D kλ♯ kL∞ (D) + ℓD kβkL∞ (D) + kµkL∞ (D) , which proves that the
bilinear form a is bounded on H 1 (D)×H 1 (D). Equation (24.8) constitutes
the common starting point to derive weak formulations for the PDE (24.1)
with various types of boundary conditions.
24.2 Dirichlet boundary condition
The goal of this section is to prove the well-posedness of the weak formulation
when a Dirichlet boundary condition is enforced.
24.2.1 Homogeneous Dirichlet condition
We consider the homogeneous Dirichlet condition
u = 0 on ∂D.
(24.11)
We enforce this condition strongly by using the functional space H01 (D) for
both solution and test spaces. Recall from the Trace Theorem B.59 that u ∈
1
H01 (D) implies that γ0 (u) = 0, where γ0 : H 1 (D) → H 2 (∂D) is the trace
map. The weak formulation is as follows:
Find u ∈ V := H01 (D) such that
R
(24.12)
a(u, w) = D f w dx, ∀w ∈ V.
′
In what follows, we identify L2 (D) with its dual space L2 (D) so that we are
′
′
in the situation where H01 (D) ֒→ L2 (D) ≡ L2 (D) ֒→ H −1 (D) = H01 (D)
with dense embeddings (recall that the notation V ֒→ W means that the
embedding of V into W is continuous).
Proposition 24.3 (Weak solution). Let u solve (24.12) with f ∈ L2 (D).
Then, the PDE (24.1) holds a.e. in D, and the boundary condition (24.11)
holds a.e. on ∂D.
322
Chapter 24. Scalar second-order elliptic PDEs
Proof. Let u be a weak solution. Testing the weak formulation (24.12) against
R
an arbitrary function ϕ ∈ C0∞ (D) ⊂ H01 (D) leads to h−∇·(d∇u), ϕi = D (f −
β·∇u − µu)ϕ dx since f ∈ L2 (D) and β·∇u + µu ∈ L2 (D) owing to the
assumptions on the data. Hence, −∇·(d∇u) defines a bounded linear form on
L2 (D) with Riesz–Fréchet representative equal to f − β·∇u − µu. This means
that the PDE (24.1) holds a.e. in D. Moreover, since u ∈ H01 (D), γ0 (u) = 0
1
in H 2 (∂D) ֒→ L2 (∂D) so that (24.11) holds a.e. on ∂D.
⊓
⊔
R
Remark 24.4 (f ∈ H −1 (D)). Whenever f ∈ H −1 (D), the term D f w dx
in (24.12) becomes hf, wiH −1 (D),H01 (D) . Then the weak solution u is such that
⊓
⊔
−∇·(d∇u) ∈ H −1 (D), and the PDE (24.1) holds in H −1 (D).
To prove the well-posedness of (24.12), we make sufficient assumptions
on the PDE coefficients to achieve V -coercivity. Recalling that the Poincaré
inequality (B.55) (with p = 2) states that there is CP,D > 0 such that
CP,D kvkL2 (D) ≤ ℓD k∇vkL2 (D) for all v ∈ V , we can equip the space V with
the norm kvkV := ℓD k∇vkL2 (D) .
Proposition 24.5 (Well-posedness). Let V = H01 (D). Assume the ellip2
ticity condition (24.5). Assume that µ0 (x) ≥ −(1 − θ)CP,D
ℓ−2
D λ♭ (x) for a.e.
x ∈ D for some θ > 0. Then, the bilinear form a is V -coervice with
a(v, v) ≥ λ♭,D min(1, θ)k∇vk2L2 (D) ,
∀v ∈ V.
(24.13)
Consequently, the problem (24.12) is well-posed.
Proof.
Let v ∈ V = H01 (D).Using (24.3) and (24.4), we infer that a(v, v) ≥
R λ♭,D k∇vk2ℓ2 (Rd ) + µ0 |v|2 dx. The assumptions on λ♭ and µ0 imply that
D
n
o
2
2
a(v, v) ≥ λ♭,D k∇vk2L2 (D) − (1 − θ)CP,D
ℓ−2
D kvkL2 (D) ,
and we conclude by distinguishing the cases θ ≤ 1 (where we use the Poincaré
inequality to absorb the last term on the right-hand side) and θ > 1 (where
the last term is nonnegative).
⊓
⊔
Example 24.6 (Pure diffusion). Coercivity for a purely diffusive problem
(i.e., µ0 ≡ 0) holds with θ = 1.
⊓
⊔
Remark 24.7 (Variational formulation). Assume that β is identically
zero in D. Then, owing to Proposition 19.8, u solves (24.12) if and only if u
is the minimizer in H01 (D) of the following energy functional:
Z
1
⊓
⊔
∇v·d·∇v + µv 2 − 2f v dx.
ED (v) :=
2 D
Part V. Elliptic PDEs
323
24.2.2 Non-homogeneous Dirichlet condition
1
Let g ∈ H 2 (∂D). We consider the non-homogeneous Dirichlet condition
u=g
on ∂D.
(24.14)
1
2
Since the trace map γ0 : H 1 (D) → H (∂D) is surjective, there is ug ∈ H 1 (D)
for some uniform constant
such γ0 (ug ) = g and kug kH 1 (D) ≤ Cγ0 kgk 12
H (∂D)
Cγ0 ; see Theorem B.59 and Remark B.61. Setting u0 := u − ug , we obtain
γ0 (u − ug ) = g − g = 0, i.e., u0 ∈ H01 (D). This leads us to the following weak
formulation:
Find u ∈ H 1 (D) such that u0 := u − ug ∈ V := H01 (D) satisfies
R
(24.15)
a(u0 , w) = D f w dx − a(ug , w), ∀w ∈ V.
The right-hand side in (24.15) defines a bounded linear form on V owing to the
boundedness of a on H 1 (D)×H 1 (D) and the above bound on ug . Proceeding
as in the homogeneous case, one can prove
Proposition 24.8 (Weak solution, well-posedness). Let u solve (24.15)
1
with f ∈ L2 (D) and g ∈ H 2 (∂D). Then, the PDE (24.1) holds a.e. in D,
1
and the boundary condition (24.14) holds a.e. on ∂D (actually in H 2 (∂D)).
Moreover, under the assumptions of Proposition 24.5, (24.15) is well-posed.
24.3 Robin/Neumann conditions
The Dirichlet conditions are called essential boundary conditions since they
are imposed explicitly in the solution space. Instead, Robin and Neumann
conditions belong to the class of natural boundary conditions. These conditions
are not explicitly enforced in the solution space, but they are enforced in the
weak formulations. That test functions are nonzero at the boundary plays a
crucial role in this context.
24.3.1 Robin condition
Let ρ ∈ L∞ (∂D) and g ∈ L2 (∂D). We consider the Robin boundary condition
ρu + n·d·∇u = g
on ∂D.
(24.16)
Starting from (24.8), we consider test functions in H 1 (D) (i.e., not only in
H01 (D) as for Dirichlet conditions), and we use the Robin condition in the
boundary integral on the left-hand side of (24.8) by replacing n·d·∇u by
g − ρu. Re-arranging terms leads to the following weak formulation:
Find u ∈ V := H 1 (D) such that
R
R
(24.17)
ǎ(u, w) = D f w dx + ∂D gw ds, ∀w ∈ V,
324
Chapter 24. Scalar second-order elliptic PDEs
R
with a(v, w) := a(v, w) + ∂D ρvw ds for all (v, w) ∈ H 1 (D)×H 1 (D) (we
slightly abuse the notation by omitting the trace map γ0 in the boundary integrals). Owing to the Cauchy–Schwarz inequality and the Trace Theorem B.59,
we infer that
Z
ρvw ds ≤ kρkL∞ (∂D) Cγ20 kvkH 1 (D) kwkH 1 (D) .
(24.18)
∂D
Since a is bounded on H 1 (D)×H 1 (D), so is ǎ. Similarly, the right-hand side
in (24.17) defines a bounded linear form in H 1 (D).
To interpret the boundary condition satisfied by a weak solution, we identify L2 (∂D) with its dual space L2 (∂D)′ so that we are in the situation where
1
1
H 2 (∂D) ֒→ L2 (∂D) ≡ L2 (∂D)′ ֒→ H − 2 (∂D) with dense embeddings, where
1
1
H − 2 (∂D) is the dual space of H 2 (∂D).
Proposition 24.9 (Weak solution). Let u solve (24.17) with f ∈ L2 (D),
ρ ∈ L∞ (∂D), and g ∈ L2 (∂D). Then, the PDE (24.1) holds a.e. in D, and
the boundary condition (24.16) holds a.e. on ∂D.
Proof. As in the proof of Proposition 24.3 one can show that the PDE (24.1)
holds a.e. in D. In particular, introducing the diffusive flux φ := −d∇u, we obtain that φ ∈ L2 (D) and ∇·φ = f − β·∇u − µu ∈ L2 (D), i.e., φ ∈ H(div; D).
Theorem B.66 allows us to define the trace of the normal component of a field
1
in H(div; D) by means of the trace map γ·n : H(div; D) → H − 2 (∂D). This
1
yields γ·n (φ) ∈ H − 2 (∂D) with
Z
(φ·∇w + (∇·φ)w) dx,
hγ·n (φ), γ0 (w)i − 12 21 =
,H
H
D
for all w ∈ H 1 (D). Since u solves the weak formulation and since ∇·φ =
f − β·∇u − µu, we infer that
Z
(ργ0 (u) − g)γ0 (w) ds = 0.
−hγ·n (φ), γ0 (w)i − 21 21 +
H
,H
∂D
1
Since the trace operator γ0 : H 1 (D) → H 2 (∂D) is surjective and the above
equality holds for all w ∈ H 1 (D), we infer that γ·n (φ) defines a bounded
linear form on L2 (∂D) with Riesz–Fréchet representative equal to ργ0 (u) − g.
Hence, the boundary condition holds in L2 (∂D), i.e., a.e. on ∂D.
⊓
⊔
Remark 24.10 (Data regularity). The fact that f ∈ L2 (D) is needed
in the proof of Proposition 24.9 to establish that ∇·φ ∈ L2 (D). It is also
1
possible to assume that g is only in H − 2 (∂D). In this case, the boundary
term in (24.17) takes the form hg, wi − 12 21 , and a weak solution satisfies
H
1
the boundary condition in H − 2 (∂D).
,H
⊓
⊔
Part V. Elliptic PDEs
325
We now address the well-posedness of (24.17). Let µ0,D := ess infx∈D µ0 (x)
with µ0 defined in (24.3). Let also ν0 (x) := ρ(x) + 21 (β·n)(x) for a.e. x ∈ ∂D
and define ν0,D := ess infx∈∂D ν0 (x). One can show (see Exercise 24.1) that
there is čD > 0 such that
−1
čD kvkL2 (D) ≤ k∇vkL2 (D) + ℓD 2 kvkL2 (∂D) ,
∀v ∈ H 1 (D).
(24.19)
Proposition 24.11 (Well-posedness). Let V = H 1 (D). Assume the ellipticity condition (24.5). Assume that µ0,D > 0, ν0,D ≥ 0 or that µ0,D ≥ 0,
ν0,D > 0. Then, the bilinear form ǎ is V -coercive, and the problem (24.17) is
well-posed.
Proof. Let v ∈ V = H 1 (D). Using (24.4), we infer that
ǎ(v, v) ≥ λ♭,D k∇vk2L2 (D) + µ0,D kvk2L2 (D) + ν0,D kvk2L2 (∂D) .
(24.20)
If µ0,D > 0, we can drop the term multiplied by ν0,D in (24.20), and coercivity
2
follows from ǎ(v, v) ≥ min(ℓ−2
D λ♭,D , µ0,D )kvkH 1 (D) . If ν0,D > 0, we can drop
the term multiplied by µ0,D in (24.20), and coercivity follows from ǎ(v, v) ≥
2
min(λ♭,D , ℓD ν0,D )(k∇vk2L2 (D) +ℓ−1
D kvkL2 (∂D) ) and the fact that the right-hand
1
side defines a norm on H (D) equivalent to k·kH 1 (D) owing to (24.19).
⊓
⊔
Example 24.12 (Pure diffusion). Coercivity for a purely diffusive problem
holds provided ρ is uniformly bounded away from zero from below.
⊓
⊔
Remark 24.13 (Variational formulation). Assume that β is identically
zero in D. Then, owing to Proposition 19.8, u solves (24.17) if and only if u
minimizes the following energy functional in H 1 (D):
Z
Z
1
1
ER (v) :=
∇v·d·∇v + µv 2 − 2f v dx +
ρv 2 − 2gv ds.
⊓
⊔
2 D
2 ∂D
24.3.2 Neumann condition
The Neumann condition is a particular case of the Robin condition in which
ρ vanishes identically on ∂D, i.e., we want to enforce
n·d·∇u = g
on ∂D.
(24.21)
The following weak formulation is obtained by setting ρ to zero in (24.17):
Find u ∈ V := H 1 (D) such that
R
R
(24.22)
a(u, w) = D f w dx + ∂D gw ds, ∀w ∈ V.
Proposition 24.14 (Weak solution, well-posedness). Let u solve (24.22)
with f ∈ L2 (D) and g ∈ L2 (∂D). Then, the PDE (24.1) holds a.e. in D,
and the boundary condition (24.21) holds a.e. on ∂D. If the ellipticity condition (24.5) holds and if µ0,D > 0 and ess infx∈∂D (β·n)(x) ≥ 0, the bilinear
form a is V -coercive, and problem (24.22) is well-posed.
326
Chapter 24. Scalar second-order elliptic PDEs
Proof. Set ρ ≡ 0 in Propositions 24.9 and 24.11.
⊓
⊔
The assumption of Proposition 24.14 fails when, e.g., µ and β vanish identically in D, i.e., for the purely diffusive problem
−∇·(d∇u) = f
in D,
n·d·∇u = g
on ∂D.
(24.23)
In this situation, we observe that if u is a solution, then u + c is also a solution for any constant c ∈ R. A simple way to deal with this arbitrariness is to
restrict the solution space to
over D, i.e., Rwe con
functions of zero mean-value
sider the space H∗1 (D) := v ∈ H 1 (D) | v D = 0 where v D = |D|−1 D v dx.
Note also that a necessary condition for a solution to exist is the following
compatibility condition on f and g:
Z
Z
f dx +
g ds = 0.
(24.24)
D
∂D
Indeed, owing
to the
R if (24.23) has a solution
R divergence Rformula, we infer that
R
u, then D f dx + ∂D g ds = − D ∇·(d∇u) dx + ∂D n·d·∇u ds = 0.
We now consider the following weak formulation:
Find u ∈ V := H∗1 (D) such that
R
R
(24.25)
ad (u, w) = D f w dx + ∂D gw ds, ∀w ∈ V.
R
with the bilinear form ad (v, w) := D ∇v·d·∇w dx. Note that the test functions in (24.25) have also zero mean-value over D.
Proposition 24.15 (Well-posedness). Let u solve (24.25) with f ∈ L2 (D)
and g ∈ L2 (∂D) satisfying (24.24). Then, the PDE (24.23) holds a.e. in D
and the boundary condition holds a.e. on ∂D. Moreover, under the ellipticity
condition (24.5), the bilinear form ad is V -coercive, and problem (24.25) is
well-posed.
⊓
⊔
Proof. See Exercise 24.4.
24.3.3 Mixed Dirichlet–Neumann conditions
It is possible to combine the Dirichlet and the Neumann conditions. Let ∂Dd
be a closed subset of ∂D and set ∂Dn = ∂D \ ∂Dd . We assume that both
subsets ∂Dd and ∂Dn have positive (surface) measures, and we enforce a
Dirichlet and a Neumann condition on ∂Dd and ∂Dn , respectively:
u = gd on ∂Dd ,
n·d·∇u = gn on ∂Dn ,
(24.26)
with functions gd and gn defined on ∂Dd and ∂Dn , respectively. We assume
that gd can be extended over ∂D and the extension in question, say ged , is
1
in H 2 (∂D). As a result, there is ud ∈ H 1 (D) such that γ0 (ud ) = ged , which
implies that ud = gd a.e. on ∂Dd . Assume also that gn ∈ L2 (∂Dn ). The
Part V. Elliptic PDEs
327
space V := {v ∈ H 1 (D) | v = 0 a.e. on ∂Dd } is a closed subspace of H 1 (D).
Indeed, if (vn )n∈N is a Cauchy sequence in V , vn → v in H 1 (D) as n → ∞;
since γ0 (vn ) → γ0 (v), we infer that v = 0 a.e. on ∂Dd . Consider the weak
formulation:
(
Find u0 ∈ V such that the following holds for all w ∈ V ,
R
R
(24.27)
a(u0 , w) = D f w dx + ∂Dn gn w ds − a(ud , w).
If f ∈ L2 (D) and the above assumptions on gd and gn hold, then u = u0 + ud
solves the PDE (24.1) a.e. in D. The Dirichlet condition holds a.e. on ∂Dd
1
(actually in H 2 (∂Dd ) which is the space spanned by the restrictions to ∂Dd
1
of functions in H 2 (∂D)). Furthermore, letting
1
2
H00
(∂Dn ) := {v ∈ L2 (∂Dn ) | ∃w ∈ V s.t. γ0 (w) = ve on ∂D},
(24.28)
where ve is the zero-extension of v to ∂D, one can show that the Neumann
1
2
(∂Dn )′ . Since gn ∈
condition γ·n (d∇u) = gn holds in the dual space H00
2
L (∂Dn ), we infer that the Neumann condition holds a.e. on ∂Dn . Finally,
under the assumptions of Proposition 24.5, problem (24.27) is well-posed. A
key argument in the proof is the following Poincaré inequality on V : There
is CP,∂Dd > 0 such that CP,∂Dd kvkL2 (D) ≤ ℓD k∇vkL2 (D) for all v ∈ V . This
inequality is (for instance) a consequence
of Lemma B.73 applied with p = 2
R
and the linear form f (v) = ∂Dd v ds for all v ∈ H 1 (D) (f restricted to
constant functions is nonzero since ∂Dd has positive measure).
24.4 Elliptic regularity
We have seen that the natural solution space V for (24.14), with Dirichlet conditions, or (24.16), with Robin conditions, is such that H01 (D) ⊂ V ⊂ H 1 (D).
Since functions in H 1 (D) are not smooth, in particular they are not twice
differentiable, a natural question is whether it is possible to prove existence
of solutions in spaces of higher regularity for smooth data. Beside purely academic considerations, regularity estimates are important for the finite element
error analysis, since convergence rates directly depend on the regularity of solutions.
Elliptic problems have this particular property that stronger regularity results can indeed be derived if the domain D and the data are smooth enough.
We collect in this section some theoretical results regarding the so-called elliptic regularity. We assume that the basic hypotheses on the PDE coefficients
stated in §24.1.1 hold, and only additional assumptions are stated explicitly.
In what follows, the treatment of the lower-order terms is straightforward: we
make the necessary additional assumptions of β and µ so that, for u ∈ H 1 (D),
328
Chapter 24. Scalar second-order elliptic PDEs
the advection-reaction term β·∇u + µu has the same regularity as that requested for the source f . For instance, if β ∈ L∞ (D) and µ ∈ L∞ (D) as
per §24.1.1, then β·∇u + µu ∈ L2 (D).
24.4.1 Interior regularity
We first present a general result concerning interior regularity, i.e., regularity
in any subset ω ⊂⊂ D (meaning that ω ⊂ D). Notice that we do not make
any assumption on the boundary condition satisfied by the weak solution or
on the smoothness of D.
Theorem 24.16 (Interior regularity). Let D be a bounded open set. Assume that d ∈ C1 (D) and f ∈ L2 (D). Let u ∈ H 1 (D) be any of the above
weak solutions. Then, for any open subset ω ⊂⊂ D,
kukH 2 (ω) ≤ C kf kL2 (D) + kukL2 (D) ,
(24.29)
the constant C depending on ω, D, and the PDE coefficients.
Proof. See Evans [229, §6.3.1]. The main tool is the technique of difference
quotients by Nirenberg [374], Agmon et al. [6].
⊓
⊔
Remark 24.17 (Sharper bound). If the weak formulation is well-posed,
the bound (24.29) is of the form kukH 2 (ω) ≤ Ckf kL2 (D) owing to the a priori
estimate kukH 1 (D) ≤ C ′ kf kL2 (D) .
⊓
⊔
Remark 24.18 (Higher-order interior regularity). Let m be a nonnegative integer. Assume that d ∈ Cm+1 (D), that the coefficients βi and
µ are in C m (D), and that f ∈ H m (D). Let u ∈ H 1 (D) be any of the
above weak solutions. Then, for any open subset ω ⊂⊂ D, kukH m+2 (ω) ≤
C(kf kH m (D) + kukL2 (D) ), the constant C depending on ω, D, m, and the
PDE coefficients; see [229, §6.3.1].
⊓
⊔
24.4.2 Regularity up to the boundary
We are now concerned with the regularity of the weak solution up to the
boundary. In this context, the smoothness of D and the boundary condition
enforced at ∂D play a role. The following theorems gather results established
over the years by many authors; we refer in particular to the textbooks by
Grisvard [257, 258], Dauge [183]. We consider three situations: domains having
a smooth boundary, convex domains and Lipschitz domains. In what follows,
we assume that the weak formulations are well-posed.
Theorem 24.19 (Smooth domain). Let D be a domain with a C 1,1 boundary. Assume that d is Lipschitz in D, i.e., there is L so that kd(x) −
d(y)kℓ2 (Rd×d ) ≤ Lkx − ykℓ2 (Rd ) for all x, y ∈ D. Let p ∈ (1, ∞) and assume
that f ∈ Lp (D). Then,
Part V. Elliptic PDEs
329
(i) The weak solution of the Dirichlet problem with boundary data g ∈
1
W 2− p ,p (∂D) is in W 2,p (D);
(ii) The weak solution of the Neumann problem with boundary data g ∈
1
W 1− p ,p (∂D) is in W 2,p (D); the same conclusion holds for the Robin
problem if ρ is Lipschitz in ∂D.
Proof. See Theorems 2.4.2.5-2.4.2.7 in [257] (see also [229, §6.3.2] for the
Dirichlet problem and p = 2).
⊓
⊔
Remark 24.20 (Neumann problem). In the literature, elliptic regularity
for the Neumann problem is often considered with the assumptions of Proposition 24.14, that is to say, the coefficient µ is uniformly bounded from below
away from zero. The Neumann problem (24.23) with the compatibility condition (24.24) can be treated by obsvering that if u is the weak solution of
this problem, then u is also the weak solution of the Neumann problem set in
H 1 (D) with coefficient µ ≡ 1 and source term f replaced by f + u.
⊓
⊔
Remark 24.21 (Higher-order regularity). Let m be a positive integer.
Assume that ∂D is of class C m+1,1 , d ∈ Cm,1 (D), and f ∈ W m,p (D). Assume
that the coefficients βi and µ are in C m (D). Then, for the Dirichlet problem
1
with g ∈ W m+2− p ,p (∂D), the weak solution is in W m+2,p (D). The same
1
conclusion holds for the Robin and Neumann problems if g ∈ W m+1− p ,p (∂D)
m,1
and ρ ∈ C (∂D). See [257, p. 129].
⊓
⊔
The smoothness assumption on the boundary of the domain can be relaxed
if the domain D is convex. Notice that a convex, bounded, open set enjoys a
uniform cone property, see [257, Cor. 1.2.2.3]).
Theorem 24.22 (Convex domain). Let D be a convex domain. Assume
that d is Lipschitz in D. Let f ∈ L2 (D). Then,
(i) The weak solution of the Dirichlet problem with g = 0 is in H 2 (D);
(ii) The weak solution of the Robin or Neumann problem with g = 0 is in
H 2 (D).
Proof. See Theorems 3.2.1.2, 3.2.1.3, and 3.2.3.1 in [257].
⊓
⊔
Finally, let us consider Lipschitz domains. Elliptic regularity in Lipschitz
domains has been deeply studied in the literature; see, e.g., Kondrat′ ev [321],
Maz′ ja and Plamenevskiı̆ [360], Jerison and Kenig [303, 304]. We first consider
polygons in R2 . This case has been throughly investigated in Grisvard [257,
Chap. 4].
Theorem 24.23 (Polygon). Let D ⊂ R2 be a polygon with boundary vertices
{Sj }1≤j≤J where the segment joining Sj to Sj+1 corresponds to the boundary
face denoted by Fj (setting conventionally J + 1 = 1). Let ̟j ∈ (0, 2π) be the
interior angle formed by the faces Fj and Fj+1 . Assume that d is the identity
matrix and that ̟j 6= π for all j ∈ {1: J}. Let f ∈ L2 (D). Then,
330
Chapter 24. Scalar second-order elliptic PDEs
(i) There is s0 ∈ ( 12 , 1] such that the weak solution of the Dirichlet problem
3
enforcing u = gj on Fj , with gj ∈ H 2 (Fj ) and gj (Sj ) = gj+1 (Sj ) for all
j ∈ {1: J}, is in H 1+s (D) for all s ∈ [0, s0 ] and s0 = 1 if D is convex;
(ii) The same conclusion holds for the weak solution of the Neumann problem
1
∂u
= gj on Fj with gj ∈ H 2 (Fj ) for all j ∈ {1: J}.
enforcing ∂n
Proof. See [257, Cor. 4.4.4.14] (which treats mixed Dirichlet–Neumann conditions and Lp -Sobolev spaces). Note that the weak solution is in H 2 (D) up to
π
singular perturbations behaving in radial coordinates as r ̟j in the vicinity
of the polygon vertices Sj .
⊓
⊔
Remark 24.24 (Variable coefficients). The case of variable coefficients
can be treated by freezing the diffusion tensor at each polygon vertex and
applying locally a coordinate transformation so as to recover the Laplace
operator; see [257, §5.2].
⊓
⊔
The study of elliptic regularity in a polyhedron is more intricate since vertex, edge, and edge-vertex singularities can occur; see Grisvard [257, §8.2],
Dauge [183, §5], Lubuma and Nicaise [348, 349], Nicaise [373], Guo and
Babuška [274, 275], Costabel et al. [176, 177]. For s ∈ (0, 1), the space
H −1+s (D) is defined by interpolation between L2 (D) and H −1 (D); this space
can also be defined as the dual space of H01−s (D) (the subspace of H 1−s (D)
spanned by functions with zero trace on ∂D) for s ∈ (0, 12 ).
Theorem 24.25 (Polyhedron). Let D ⊂ R3 be a polyhedron. There exists
s0 > 21 , depending on D, such that the Laplace operator is an isomorphism
from H 1+s (D) ∩ H01 (D) to H −1+s (D) for all s ∈ [0, s0 ].
Proof. This is a consequence of Theorem 18.13 in Dauge [183, p. 158].
Let us now consider the more general case of a Lipschitz domain with
Lipschitz diffusion tensor.
Theorem 24.26 (Lipschitz domain). Let D be a Lipschitz domain. Assume that d is Lipschitz in D. There is s0 ∈ (0, 21 ) such that, for all s ∈ [0, s0 ],
(i) The weak solution of the Dirichlet problem with f ∈ L2 (D) and g ∈
1
H 2 +s (∂D) is in H 1+s (D).
(ii) The weak solution of the Neumann problem with f ∈ L2 (D) and g ∈
1
H − 2 +s (∂D) is in H 1+s (D).
Proof. See Theorems 3 and 4 in Savaré [419]. Notice also that the lowest-order
terms in the PDE are in L2 (D) and that L2 (D) ⊂ H −1+s (D) for all s ≤ 1, so
that f can be replaced by f − β·∇u − µu.
⊓
⊔
Remark 24.27 (Very weak solution). It is possible to extend the notion
of elliptic regularity to the so-called very weak solutions. Such solutions do not
necessarily belong to the space H 1 (D). For instance, using the transposition
technique from Lions and Magenes [346, Chap. 2], it is shown in Savaré [419]
that the statement of Theorem 24.26 also holds for all s ∈ (− 21 , 0).
⊓
⊔
Part V. Elliptic PDEs
331
The Lipschitz regularity of d is rather restrictive since it excludes domains
composed of different materials. Following Jochmann [307], it is possible to
replace this hypothesis by a (so-called) multiplier assumption, which consists
of assuming that there is s0 ∈ (0, 21 ) such that
the map H s0 (D) ∋ ξ 7−→ d·ξ ∈ H s0 (D) is bounded.
(24.30)
It is shown in Jochmann [307], (see also Bonito et al. [68]) that this assumption
holds if D is partitioned into M connected Lipschitz domains D1 , . . . , DM
and if there is a real number s0 ∈ (0, 21 ) and there are diffusion tensors dm ∈
PM
C0,α (Dm ), for all m ∈ {1: M }, so that d = m=1 1Dm dm , where 1Dm is the
characteristic function of Dm .
Theorem 24.28 (Piecewise smooth coefficients). Assume that there is
s0 ∈ (0, 21 ) such that (24.30) holds. Then, there is s ∈ (0, s0 ), depending on D
and d, so that the weak solution of the homogeneous Dirichlet or Neumann
problem with f ∈ L2 (D) (and g = 0) is in H 1+s (D).
Proof. See Theorem 3 in [307] or Lemma 3.2 in [68]. The statement also holds
for f in the dual space of H01−s (D) for the Dirichlet problem and in the dual
space of H 1−s (D) for the Neumann problem.
⊓
⊔
The above result also holds for the mixed Dirichlet-Neumann problem. We
refer to Jochmann [307] for more details on this question.
Exercises
Exercise 24.1 (Poincaré–Friedrichs). Prove (24.19). (Hint: use (B.56).)
Exercise 24.2 (Complex-valued diffusion). Assume that the domain D
is partitioned into two disjoint subdomains D1 and D2 . Let κ1 , κ2 be two
complex numbers, both with positive modulus and such that κκ12 6∈ R− . Set
κ(x) = κ1 1D1 (x) + κ2 1D2 (x) for all x ∈ D. Let f ∈ L2 (D).
R Show that the
1
κ∇u·∇w dx =
problem
of
finding
u
∈
V
:=
H
(D;
C)
such
that
a(u,
w)
:=
0
D
R
f
w
dx
for
all
w
∈
V
is
well-posed.
(Hint:
use
(19.7).)
D
Exercise 24.3 (Potential flow). Consider the PDE ∇·(−κ∇u + βu) = f
in D with homogeneous Dirichlet conditions and assume that κ is a positive
real number. Assume that β = ∇ψ for some smooth function ψ; we say that
β is a potential flow. Find a functional E : H01 (D) → R of which the weak
solution u is a minimizer on H01 (D). (Hint: consider the function e−ψ/κ u.)
Exercise 24.4 (Purely diffusive Neumann). Prove Proposition 24.15.
(Hint: observe that for w ∈ H 1 (D), the function w̃ = w − wD is in H∗1 (D);
for well-posedness, use the Poincaré–Wirtinger Lemma 13.16.)
332
Chapter 24. Scalar second-order elliptic PDEs
Exercise 24.5 (Mixed Dirichlet–Neumann). The goal is to show by a
counter-example that there is in general no regularity result implying that
the weak solution is in H 2 (D) for the mixed Dirichlet–Neumann problem even
if the domain and the boundary data are smooth. Using polar coordinates,
set D = {(r, θ) ∈ (0, 1) × (0, π)}, ∂Dn = {0 < r < 1, θ = π}, and ∂Dd =
1
∂D\∂Dn . Verify that the function u(r, θ) = r 2 sin( 12 θ) satisfies −∆u = 0
1
∂u
|n = 0, and u|d = r 2 sin( 21 θ). (Hint: in polar coordinates, ∆u =
in D, ∂n
∂u
1 ∂2u
1 ∂
2
r ∂r r ∂r + r 2 ∂θ 2 .) Verify that u 6∈ H (D).
Exercise 24.6 (H 2 (Rd )-seminorm). Prove that |φ|H 2 (Rd ) = k∆φkL2 (Rd ) for
all φ ∈ C0∞ (Rd ). (Hint: use Lemma B.12.)
Exercise 24.7 (Counter-example to elliptic regularity in W 2,∞ (D)).
Let D be the unit disk in R2 . Consider the function u(x1 , x2 ) = x1 x2 log(r)
with r2 = x21 + x22 (note that u|∂D = 0). Verify that ∆u ∈ L∞ (D), but that
u 6∈ W 2,∞ (D). (Hint: consider the cross-derivative.)
Exercise 24.8 (Embedding). Let D be a bounded open set in Rd and let
α ∈ (0, 1]. Prove that C 0,α (D) ֒→ W s,p (D) for all s ∈ [0, α) and all p ∈ [1, ∞).
(Hint: use Definition 1.33.)
Exercise 24.9 (Domain with slit). Let D = {0 ≤ r < 1, 0 < θ < 2π},
where (r, θ) are the polar coordinates, so that D is the closed ball of radius
1 centered at 0. Let B = int(D). Let u(r, θ) = r cos( 12 θ). Is u in W 1,p (D)? in
W 1,p (B)? Is C 1 (D) dense in W 1,p (D)? (Hint: C 1 (B) is dense in W 1,p (B).)