IN
DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS
STOCKHOLM SWEDEN 2019,
Regularity of semilinear elliptic partial differential equations with critical Sobolev exponents
JOHAN ERICSSON
KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES
INOM
EXAMENSARBETE TEKNIK, GRUNDNIVÅ, 15 HP
STOCKHOLM SVERIGE 2019,
Regularitet för semilinjära elliptiska partiella
differentialekvationer med kritiska Sobolevexponenter
JOHAN ERICSSON
KTH
SKOLAN FÖR TEKNIKVETENSKAP
Abstract
We study the regularity and existence of solutions in Sobolev spaces to semilinear elliptic partial differential equations in bounded domains.
We identify a growth condition on the nonlinear term that suffices to prove existence and uniqueness of solutions provided the domain and boundary conditions are adequately regular. There are two major steps. First we reduce the semilinear equation to a linear equation that we solve. Secondly we apply a topological fixed point theorem to prove the existence of a solution to the semilinear equation.
Keywords: semilinear elliptic partial differential equations, regu- larity, Sobolev spaces, measurable coefficients.
Sammanfattning
Vi studerar regularitet och existens av lösningar i Sobolevrum till semilinjära elliptiska partiella differentialekvationer i begränsade öpp- na områden. Vi identifierar ett tillväxtvillkor på den ickelinjära termen som är tillräckligt för att påvisa existens av lösningar givet att området och randvillkoren är regulära. Det är två huvudsakliga steg i metoden.
Först reducerar vi den semilinjära ekvationen till en linjär ekvation som vi löser. Sedan använder vi en topologisk fixpunktsats för att bevisa att det existerar en lösning till den semilinjära ekvationen.
Nyckelord: semilijnjära elliptiska partiella differentialekvationer, regularitet, Sobolevrum, mätbara koefficienter.
Contents
1 Introduction 3
2 Preliminiaries 3
2.1 Sobolev spaces . . . 4
2.2 The dual of W01,2(Ω) . . . 5
2.3 Elliptic operators and weak solutions . . . 6
2.4 Inequalities . . . 6
2.5 Functional Analysis . . . 7
2.6 Some fixed point results in Banach spaces . . . 8
3 Boundaries and Traces 11 3.1 Straightening of boundaries . . . 11
3.2 Traces of W1,2 functions . . . 11
3.3 Minimisers of the Dirichlet energy . . . 12
4 Existence and uniqueness of solutions 13 4.1 Weak coercivity and critical exponents . . . 14
4.2 A maximum principle . . . 17
4.3 Existence of solution to the linear equation . . . 19
4.4 Interior and boundary W2,2 regularity . . . 21
4.5 Existence of a solution to the nonlinear equation . . . 27
5 Remarks 29
6 Acknowledgements 29
1 Introduction
Our primary interest in this text is to study the existence and regularity of weak solutions to the semilinear elliptic partial differential equations of the form
∆u(x) + B(x, u) · ∇u(x) = 0, x ∈ Ω,
u = g, x ∈ ∂Ω, (1)
in open bounded domains Ω ⊂ Rn. Equations of this form may not always have twice differentiable solutions. However the requirements for a function to be a solution to a differential equation can be be weakened which leads to the concept of weak solutions. The appropriate spaces for studying these solutions are the Sobolev spaces. In section 2 we give a brief overview of Sobolev spaces and provide the required material we will use from functional analysis.
Our approach to equation (1) will also require us to reformulate the boundary condition and introduce the trace operator. The neccessary material for this is introduced in section 3 together with a discussion of how to extend a function defined on the boundary ∂Ω to Ω by methods from calculus of variations.
In section 4 we identify a growth condition on B that guarantees the existence of a solution to equation (1) in the Sobolev space W1,2(Ω) provided the boundary of Ω is of class C2 and g ∈ W2,2(Ω). The main result is the following theorem:
Theorem 21. Let Ω be of class C2, |B(x, v)| ≤ |v(x)|γ, for some γ < n−22 and g ∈ W2,2(Ω), then there exists a solution u ∈ W1,2(Ω) to the equation
∆u(x) + B(x, u) · ∇u(x) = 0, x ∈ Ω,
u = g, x ∈ ∂Ω. (2)
There are two main ideas in our approach to prove existence of solutions to equation (1). First we reduce the equation to a linear equation on the form
∆u(x) + b(x) · ∇u(x) = 0, x ∈ Ω,
u(x) = 0, x ∈ ∂Ω, (3)
which we solve by Hilbert space methods. Then we apply a topological fixed point theorem to show that there exists a solution to equation (1). In section 4 we also prove that the linear equation satisfy a maximum principle and that the solution of equation (3) is in W2,2(Ω) whenever ∂Ω is C2 and b ∈ Lˆn(Ω) for some ˆn > n.
2 Preliminiaries
We will assume familiarity with the basic concepts of real analysis, e.g. the properties of Lebesgue integrable functions and Lp spaces, all such material can be found in [Roy88].
2.1 Sobolev spaces
Let Ω ⊂ Rn be an open domain. Given a function u ∈ L1loc(Ω) and a multiindex α we say that v ∈ L1loc(Ω) is the αth weak derivative of u, written Dαu = v, provided
(−1)|α|
ˆ
Ω
u Dαϕ dx = ˆ
Ω
v ϕ dx, ∀ ϕ ∈ Cc∞(Ω).
We define the Sobolev space Wk,p(Ω) ⊂ Lp(Ω) as the space of functions for which all weak derivatives up to order k are in Lp(Ω). Sobolev spaces are Banach spaces under the norm
kukk,p=
X
|α|≤k
ˆ
Ω
|Dαu|pdx
1 p
.
We will use the convention that Dαu = u when α = (0, . . . , 0). We are going to denote the weak derivative of first order by Dju = Dαju for j = 1, . . . , n, where αj is the multiindex with zeros in every entry except for the jth posi- tion where it is 1. We define the weak gradient ∇u := (D1u, . . . , Dnu). When considering a differentiable function f we will sometimes use the notation
∂j or ∂x∂
j to denote the ordinary partial derivative given by the difference quotient
∂f
∂xj = lim
h→0
f (x + hej) − f (x)
h .
Note that if f is differentiable in Ω then an integration by parts shows that
∂jf = Djf , i.e. they belong to the same function class in Lp(Ω). Weak derivatives satisfy the following properties.
Lemma 1 (Properties of weak derivatives). Let u, v ∈ Wk,p(Ω), η ∈ Ck(Ω) and λ, ν ∈ R, then the weak derivative has the following properties
Linearity: Dj(λu + νv) = λDju + νDjv, Product rule: Dj(ηu) = Dj(η) u + Dj(u) η.
The space W0k,p(Ω) is defined as the closure of Cc∞(Ω) in Wk,p(Ω). A very important property of functions in the space Wk,p(Ω) is that they can ap- proximated by smooth functions. We state this fact in the next theorem.
Theorem 1. Let Ω be bounded and C1. If u ∈ Wk,p(Ω), then there exists functions uj ∈ C∞(Ω) such that
j→∞lim uj = u, in Wk,p(Ω).
The next two inequalities are very important for coming estimates and shows that the L2and L2∗norm of functions in W01,2(Ω) is bounded by the L2 norm of their weak derivatives.
Lemma 2 (Poincaré’s inequality). Let Ω be a bounded domain. Then there exists a constant Cp such that for every u ∈ W01,2(Ω)
kuk2 ≤ Cpk∇uk2.
Note that from Poincaré’s inequality we can conclude that ||∇ · ||2 is an equivalent norm to k · k1,2 on W01,2(Ω) for bounded domains. This is seen by
k∇uk2 ≤ kuk1,2= q
kuk22+ k∇uk22 ≤q
1 + Cp2k∇uk2.
Theorem 2 (Sobolev’s inequality). Let p < n. Then there exists a constant Cs such that for every u ∈ W01,p(Ω)
kukp∗≤ Csk∇ukp, (4)
where p∗:= n−pnp .
Sobolev spaces can be compactly embedded in Lp spaces. This will play an important role in the proof of existence of solutions to equation (1).
Theorem 3 (Rellich-Kondrachov). Let Ω be bounded, then W01,p(Ω) is com- pactly embedded in Lq(Ω) for every 1 ≤ p ≤ n and 1 ≤ q < p∗.
2.2 The dual of W01,2(Ω)
The Sobolev space W01,2(Ω) has even more structure than a general Banach space, it is a Hilbert space under the inner product
hu, vi1,2 = ˆ
Ω
uv dx + ˆ
Ω
∇u · ∇v dx.
We have the following characterisation of the dual space, H−1, of W01,2(Ω).
Theorem 4 (Dual space of W01,2(Ω)). F is an element of H−1 if and only there exists functions f0, . . . , fn in L2(Ω) satisfying
F u = ˆ
Ω
f0u dx +
n
X
j=1
ˆ
Ω
fjDju dx, ∀ u ∈ W01,2(Ω) (5)
Proof. Let F ∈ H−1. By the Riesz Representation theorem there exists a unique element f ∈ W01,2(Ω) such that
hf, ui1,2 = ˆ
Ω
f u dx +
n
X
j=1
ˆ
Ω
Djf Dju dx, ∀ u ∈ W01,2(Ω).
Since f and Djf are in L2(Ω) there exists functions fj = Djf in L2(Ω) satisfying equation (5).
Conversely assume f0, . . . , fnare functions in L2(Ω) satisfying equation (5).
The linearity of F follows from linearity of integration. It remains to show that F is bounded. We have
F u = ˆ
Ω
f0u dx+
n
X
j=1
ˆ
Ω
fjDju dx ≤
n
X
j=0
kfjk2kDjuk2 ≤
n
X
j=0
kfjk2
kuk1,2, proving that F is bounded operator on W01,2(Ω).
2.3 Elliptic operators and weak solutions
Consider a second order partial differential operator L of the form Lu =
n
X
i,j=1
Di(aijDju) +
n
X
i=1
biDiu + ciu] + d u.
We assume that the coefficients aij(x), bi(x), ci(x) and d(x) are measurable and lie in the proper spaces such that Lu is locally integrable whenever u ∈ W1,2(Ω). The operator L is said to be elliptic provided there exists a positive constant θ such that
n
X
i,j=1
aij(x)ξiξj ≥ θ|ξ|2, ∀ x ∈ Ω, ∀ ξ ∈ Rn.
Given an elliptic operator L and a function f ∈ L2(Ω) we say that u ∈ W1,2(Ω) is a weak solution to the equation Lu = f in Ω provided
ˆ
Ω n
X
i,j=1
aijDju Diϕ −
n
X
i=1
biDiu ϕ − ciu Diϕ] + d u ϕ dx = ˆ
Ω
f ϕ dx,
for every ϕ ∈ W01,2(Ω).
2.4 Inequalities
In this section we state, without proof, some inequalities frequently used throughout the text, which can be found in [Ada75].
Lemma 3 (Cauchy’s inequality). Let a, b, ε be positive real numbers, then ab ≤ ε
2a2+ 1 2εb2.
For a real number, 1 ≤ p ≤ ∞, we define the the Hölder conjugate, p0 of p, by
p0 =
p
p−1 if 1 < p < ∞, 1 if p = ∞,
∞ if p = 1.
By this definition p, p0 satisfy 1 p + 1
p0 = 1.
The next inequality is a more general version of Cauchy’s inequality.
Lemma 4 (Young’s inequality). Let a, b, ε be positive real numbers and 1 <
p < ∞, then
ab ≤ εap p + 1
εp0/p bp0
p0 .
Lemma 5 (Hölder’s inequality). Let u ∈ Lp(Ω) and v ∈ Lp0(Ω), then uv is
integrable and ˆ
Ω
|uv|dx ≤ kukpkvkp0.
Furthermore if Ω is bounded, and u ∈ Lq(Ω), then for every 1 ≤ p ≤ q ≤ ∞ we have
kukp ≤ µ(Ω)1p−1qkukq.
Hölder’s inequality can be generalised to cases when the integrand is a prod- uct of more than two functions.
Lemma 6 (Generalised Hölder’s inequality). Let 1
p1 + · · · + 1 pk = 1.
If uj ∈ Lpj(Ω), then ˆ
Ω
|u1| · · · |uk|dx ≤
k
Y
j=1
kujkpj.
2.5 Functional Analysis
In this section we state some results from functional analysis, which we need to prove existence of a solution to equation (1). These results can be found in [Yos95] and [GT01]. Throughout this text whenever we consider a Banach space it will be over the real number field.
Definition 1 (Compact operator). A linear operator T : V1 → V2 between two linear normed spaces is compact provided T maps bounded sets in V1 into relatively compact sets.
Thus an operator is compact if it maps bounded sequences into sequences with a convergent subsequence. Note that if K is a continuous operator and T is compact, then KT is also compact.
Compact operators will be important for the developments in later sections.
The two most important results, to be used later, from this section is Theo- rem 5 and Theorem 7.
Theorem 5 (Fredholm Alternative). If T is a compact linear mapping from a normed linear vector space, V , into itself, then either
u − T u = 0,
has a nontrivial solution u ∈ V , or for each y ∈ V there exists a unique solution u ∈ V to the equation
u − T u = y.
Furthermore, in the second case the operator (Id−T )−1exists and is bounded, where Id is the identity operator.
Let H be a Hilbert space andL be a bilinear mapping from H × H to R.
We say thatL is bounded if there exists a constant β > 0 such that
|L (u, w)| ≤ βkukHkwkH, ∀u, v ∈ H. (6) We say thatL is coercive if there exists a constant α > 0 such that
L (w, w) ≥ αkwk2H, ∀w ∈ H. (7)
Theorem 6 (Riesz Representation Theorem). Let H be a hilbert space. For every linear functional, F ∈ H∗ there exists a unique element such that
F (u) = (f, u), for every u in H.
Theorem 7 (Lax-Milgram). Let H be a Hilbert space andL : H × H → R be a bounded, bilinear, coercive mapping. Then for every F ∈ H∗ there exists a unique u ∈ H such that
L (u, v) = F (v), ∀ ∈ H.
Note that if we fix an element, u ∈ H, and L is a bounded bilinear form, then L (u, ·) is an element of H∗. The linearity and boundedness follows directly from the bilinearity of L and equation (6). Thus every bilinear form defines an operator Λ : H → H∗ by u 7→L (u, ·). Furthermore if L is coercive, then equation (7) implies that Λ has a continuous inverse.
2.6 Some fixed point results in Banach spaces
We will use a topological fixed point theorem to handle the nonlinearity in equation (1) which is presented in Theorem 8 below. It is a fixed point theo- rem for infinite dimensional Banach spaces that can be applied to operators provided they are compact and all possible fixed points satisfy a uniform bound. The proofs in this section can all be found in [GT01], but we still include them here, and with more detail, as they will be central to our later developments.
Theorem 8 (Schaefer’s fixed point theorem). Let B be a a Banach space and T : B → B be compact. If there exists a constant, M , such that
kukB < M,
for every u ∈ B and λ ∈ [0, 1] for which λT u = u, then T has a fixed point.
Before we prove Theorem 8 we state Brouwer’s fixed point theorem and Schauder’s fixed point theorem which we will need for the proof.
Theorem 9 (Brouwer’s fixed point theorem). Let D be the closed unit sphere in Rn for some n ∈ N. Then every continuous map F : D → D has at least one fixed point.
Theorem 10 (Schauder’s fixed point theorem). Let K be a compact convex subset of a Banach space B and T : K → K be continuous, then T has a fixed point.
Proof. Let n ∈ N consider a cover of K by the balls {B(x, 1/n)}x∈K. By compactness there exists a finite subcover {Bj}Nj=1 = {B(xnj, 1/n)}N (n)j=1 for some N (n) depending on n. Let Hnbe the convex hull of {xn}N (n)j=1 , i.e.
Hn=
N (n)
X
j=1
cjxnj : n ∈ N, cj ≥ 0,
N (n)
X
j=1
cj = 1
.
We define the mapping Sn: K → Hn by Snx :=
PN
j=1d(x, K \ Bj) xj PN
j=1d(xj, K \ Bj) . We note that Snx is an element of Hn as
1 PN
j=1d(xj, K \ Bj)
N
X
j=1
d(xj, K \ Bj) = 1.
Furthermore Sn is continuous. This easily seen once we establish that dj(·) := d(·, K \ Bj) : K → R is continuous. This follows from the defi- nition of the distance function and the triangle inequality
|dj(x) − dj(y)| = | inf{|x − z| : z ∈ K \ Bj}
− inf{|y − z| : z ∈ K \ Bj}|
≤ inf{| inf{|x − y| + |y − z| : z ∈ K \ Bj} − inf{|y − z| : z ∈ K \ Bj}|
= |x − y|.
From this we deduce that dj is Lipschitz continuous with Lipschitz constant 1. Since the ball {Bj} cover K the denominator of Snwill always be nonzero.
By considering Sn as the product of the two continuous functions f1:=
N
X
j=1
dj(·) : K → R, and f2 :=
N
X
j=1
dj(·) : K → B
the continuity can be proved by simple calculus, in the same way the continu- ity of product of continuous real valued functions is proven. The convex hull Hn is homeomorphic to a closed unit sphere, D in RM for some M ≤ N (n).
Let F denote a homeomorphism F : Hn→ D and consider the composition F ◦ Sn◦ T|Hn : Hn→ D. This map is continuous, hence Brouwer’s fixed point theorem implies that F ◦ Sn◦ T|Hn ◦ F−1 has at least one fixed point in D.
From this it follows that the map Sn◦ T|
Hn : Hn→ Hn has at least one fixed point, ξn, and since K is compact the sequence {ξn}∞n=1 of fixed points has a convergent subsequence {ξnm}∞m=1 that converges to some point x0 ∈ K.
We will show that x0 is a fixed point of T . To see this note that x =
PN
j=1d(x, K \ Bj) x PN
j=1d(xj, K \ Bj), for every x ∈ K.
Which combined with triangle inequality yields the estimate kξn− Snξnk =
PN
j=1d(ξn, K \ Bj) (ξn− xj) PN
j=1d(ξn, K \ Bj)
≤ PN
j=1d(ξn, K \ Bj) kξn− xjk PN
j=1d(ξn, K \ Bj) < 1 n.
(8)
Next we use that ξn are fixed points of the maps Sn◦ T and the estimate from equation (8) to get
kT ξn− ξnk = kT ξn− Sn◦ ξnk < 1 n.
And since T is continuous and ξn → x0 it follows that T ξn converges to x0 as n → ∞.
Note that if K is a closed convex set in a Banach space B and T : K → K is precomact, i.e. has compact closure then Theorem 10 implies that T has a fixed point. We are now ready to prove Theorem 8.
Proof Theorem 8. We assume without loss of generality that M = 1. Let B denotes the closed unit ball in B and define J : B → B by
J x =
(T x, if kT xk ≤ 1,
T x
kT xk, if kT xk > 1.
It follows from the continuity of T that J is continuous. Furthermore J B = B which has compact closure. Thus Theorem 10 implies that J has a fixed point x0. Hence x0 is fixed point for T if kT x0k ≤ 1. Assume for contradiction that kT x0k > 1. Then we get
x0 = J x0= T x0
kT x0k = λT x0, with λ := 1
kT x0k < 1.
and kx0k = kJ x0k = 1, which contradicts the assumptions of the theorem when M = 1. It follows that kT x0k < 1 and that x0 is a fixed point of T .
3 Boundaries and Traces
In this section we define how to straighten boundaries and how to relate a function in W1,2(Ω) to a L2 function defined on the boundary ∂Ω. We will show how to extend a function defined on the boundary ∂Ω to the entire domain Ω.
3.1 Straightening of boundaries
The boundary, ∂Ω, of an open bounded domain Ω is said to be of class Ck provided for each point x ∈ ∂Ω there exists a ball B(x, r) centered at x and a function π ∈ Ck(Rn−1); R satisfying
Ω ∩ B(x0, r) = {x ∈ B(x0, r) : xn> π(x1, . . . , xn−1)},
after relabelling and reorienting of the coordinate axes. Let ∂Ω be of class Ck, and x0 ∈ ∂Ω. Then we may define a Ck diffeomorphism, Ψ, defined by
x7−→ (xΨ 1, . . . , xn−1, xn− π(x1, . . . , xn−1)).
Let y = Ψ(x), then we can easily show Ψ has a well-defined Ck inverse on Ψ(B(x, r)) given by
y Ψ
−1
7−→ (y1, . . . , yn−1, yn+ π(y1, . . . , yn−1)).
Note that DΨ = DΨ−1 = I, where I denotes the identity matrix. Thus
|DΨ| = |DΨ−1| = 1 and (DΨ)−1= DΨ−1.
3.2 Traces of W1,2 functions
In this section we introduce the trace operator which is an important for deal- ing with boundary data of partial differential equations in Sobolev spaces.
We start by citing two theorems from [Eva10].
Theorem 11. Let Ω be a bounded domain with C1 boundary. Then there exists a bounded linear operator T r : W1,2(Ω) → L2(∂Ω) with the properties T r(u) = u|∂Ω, if u ∈ W1,2(Ω) ∩ C(Ω)
kT r(u)kLp(∂Ω) ≤ Ckuk1,2 f or every u ∈ W1,2(Ω) where the constant C only depends on Ω.
The function T r(u) is called the trace of u. Let g ∈ L2(Ω), then we say that a function u ∈ W1,2(Ω) equals g in the trace sense provided T u = g in L2(∂Ω). The next theorem shows the relation between functions in W01,2(Ω) and and functions with trace zero.
Theorem 12. Let Ω be a C1 domain and u ∈ W1,2(Ω). Then T u = 0 if and only if u ∈ W01,2(Ω)
3.3 Minimisers of the Dirichlet energy
The trace operator is not an onto mapping, thus if equation (1) is to have a unique solution we will need to impose some conditions on our boundary term g. For this we will need some results from the calculus of variations.
We will see that under the right conditions there exists unique minimisers to the Dirichlet energy integral
J [u] = 1 2
ˆ
Ω
|∇u|2dx. (9)
Let
I[u] :=
ˆ
Ω
L(∇u, x) dx .
From now on let g ∈ L2(Ω) and A := {w ∈ W1,2(Ω) : T r(w) = g}. The next two theorems can be found in [Eva10].
Theorem 13. Assume A is nonempty. If L(p, x) : Rn× Ω → R is convex in the first variable and there exists two constants α > 0 and β ≥ 0 such that
L(p, x) ≥ α|p|2− β, ∀ p ∈ Rn, x ∈ Ω, then there exists a minimiser u ∈ A to I.
The next theorem imposes one more condition on L which asserts that the minimiser u will be the unique minimiser of I[u] in A.
Theorem 14. If L : Rn× Ω → R and there exists a positive constant θ such
that n
X
i,j=1
Lpjpi(p, x)ξjξi ≥ θ|ξ|2, ∀ p, ξ ∈ Rn, x ∈ Ω, then a minimiser u ∈ A to I is unique.
We note that for the dirichlet energy defined in equation (9) we have L(p, x) =
1
2|p|2, hence by Theorem 13 there exists minimizer u ∈ A to J . Further- more as L satisfy the conditions of Theorem 14 the minimiser u is unique.
Minimisers of variational problems are closely related to solutions of partial differential equations. We define the the Euler Lagrange equations for L by
− ∇x· ∇p(L(∇u, x)) = 0, x ∈ Ω. (10) It is a known fact from the calculus of variations that any minimiser of J is also a solution to equation (10). For the Dirichlet energy the Euler Lagrange equations takes the form
− ∆u = 0, x ∈ Ω. (11)
From this it is possible to deduce regularity properties of minimizers. It can be shown with the same methods we use in section 4.4 that a minimizer, u ∈ A, of the Dirichlet energy integral J satisfy u ∈ W2,2(Ω0) for every
Ω0 ⊂⊂ Ω. Furthermore it is possible to show that u satisfy much stronger regularity properties. Actually a weak solution of equation (11) is smooth.
Regularity results of this form can be showed for solutions of equation (10) as well. De Giorgi, Nash and Moser, independently showed that solutions to the equation
n
X
i,j=1
Di aij(x) Dju = 0, (12) are Hölder continuous in Ω for some Hölder exponent γ > 0, when the coefficients aij ∈ L∞(Ω). It can be shown that when L is sufficiently smooth the weak formulation of equation (10) is equivalent to an equation of the same form as (12). A more thourough account of their methods and more general results of this nature can be found in the monograph [MP97].
4 Existence and uniqueness of solutions
We are interested in solving the nonlinear equation
∆u(x) + B(x, u) · ∇u(x) = 0, x ∈ Ω,
u = g, x ∈ ∂Ω. (13)
Regarding the boundary condition we apply a slight abuse of notation. What we mean more specifically is that u = g on ∂Ω in the trace sence. When we solve equation (13) we will assume that the boundary term g ∈ W2,2(Ω). But if g ∈ L2(∂Ω) only was to be defined on ∂Ω we could use the results of the previous section to define a functiong to be the unique function minimising the Dirichlet energy functional and satisfying T r(g) = g. The existence results of this section could then be applied to show that the equation
∆u(x) + B(x, u) · ∇u(x) = 0, x ∈ Ω,
T r(u) = T r(g) = g, x ∈ ∂Ω. (14) has a solution, u, that also solves equation (13).
We will investigate under what assumptions on B there exists a solution in W1,2(Ω). A function u ∈ W1,2(Ω) is a weak solution to equation (13)
provided ˆ
Ω
∇u · ∇ϕ − B(x, u) · ∇u ϕ dx = 0, (15) for every ϕ ∈ W01,2(Ω).
We will apply the Schaefer’s theorem to the operator T v = u, where u is the unique solution of the linear equation
∆u(x) + B(x, v) · ∇u(x) = 0, x ∈ Ω,
u = g, x ∈ ∂Ω. (16)
We will always assume that g ∈ W2,2(Ω), and consider the equivalent equa- tion with zero boundary values
∆w(x) + B(x, v) · ∇w(x) = f (x), x ∈ Ω,
w = 0, x ∈ ∂Ω, (17)
where f (x) = −∆g(x) − B(x, v) · ∇g(x). Fix v ∈ W01,2(Ω) and let b(x) :=
B(x, v). Then w is a solution to equation (17) provided L (w, ϕ) :=
ˆ
Ω
∇u · ∇ϕ − b(x) · ∇u ϕ dx = ˆ
Ω
f ϕ dx, ∀ϕ ∈ W01,2(Ω). (18) Any function w that solves equation (17) yields a solution of equation (16), given by u = w + g. Let L denote the partial differential operator corre- sponding to equation (17), i.e.
L := ∆ + b(x) · ∇.
The proof of existence to the linear equation is similar to the proof of Theo- rem 8.3 in [GT01]. The genereal idea is to consider the weak formulation as a Hilbert space equation. The existence will then follow from an application of the Fredholm alternative and Lax-Milgrams’ theorem. However we have a weaker condition on b(x) which leads to more technicalities. We prove a more general result that holds whenever the functions b are in the space Lnˆ for ˆn > n, where n is the dimension of the space.
4.1 Weak coercivity and critical exponents
In order to apply Lax-Milgrams’ theorem we need to show that the bilinear formL is bounded and coercive. In this section we present an appropriate growth condition on b such thatL is bounded and weakly coercive. This will suffice to prove existence of solutions to equation (17) when Lax-Milgrams’
theorem is combined with the Fredholm alternative.
Lemma 7. If b ∈ Ln, then the bilinear form,L , is bounded on W01,2(Ω).
Proof. We have
|L (w, ϕ)| ≤ ˆ
Ω
|∇u · ∇ϕ| + |b(x)| |∇u| |ϕ| dx.
Hölder’s inequality yields
|L (w, ϕ)| ≤ k∇wk2k∇ϕk2+ ˆ
Ω
|b||∇w||ϕ| dx.
By the definition of 2∗ we have 1 n+ 1
2∗ + 1 2 = 1.
Thus we may apply the generalised Hölder’s inequality to estimate the re- maining integral which yields
|L (w, ϕ)| ≤ k∇wk2k∇ϕk2+ kbknk∇wk2kϕk2∗. (19) Then it follows from equation (19) and Theorem 2 that
|L (w, ϕ)| ≤ k∇wk2k∇ϕk2+ Cskbknk∇wk2k∇ϕk2.
Hence
|L (w, ϕ)| ≤ (1 + 2Cskbkn+ Cs2)k∇wk22k∇ϕk22.
And since the norm k∇ · k2 is equivalent to the norm k · k1,2 on W01,2(Ω) the bilinear formL is bounded.
The bilinear form, L , corresponding to equation (16) will not be coercive under our assumptions on b. Though we will show thatL is weakly coercive and for this we need the following estimate.
Lemma 8. Let w ∈ W01,2(Ω) and b ∈ L2s0(Ω), where s0 = (2∗2−2)r∗ . Then for every r ∈ (0, 1) the following estimate holds
ˆ
Ω
|b|2|w|2dx ≤ kbk22s0kwk2r2∗kwk2(1−r)2 . (20) Proof. Hölder’s inequality yields
ˆ
Ω
|w|2|b|2dx ≤
ˆ
Ω
|w|2sdx
1s ˆ
Ω
|b|2s0dx
1
s0
.
For every r ∈ (0, 1) we have ˆ
Ω
|w|2sdx = ˆ
Ω
|w|2rs|w|2(1−r)sdx.
It is clear that if s, t > 1 satisfy the equations
2rst = 2∗, (21)
2(1 − r)st0= 2, (22)
then we may apply Hölder’s inequality again which yields ˆ
Ω
|w|2sdx ≤
ˆ
Ω
|w|2∗dx
1t ˆ
Ω
|w|2dx
t01
. (23)
Given a fixed r ∈ (0, 1) we can solve for s and t. equation (21) yields t = 2∗
2r. Substituting this into equation (22) we get
(1 − r)2∗ 2rt
t
t − 1 = 1 ⇐⇒ t = 1 + 2∗− 2∗r 2r , and equation (21) then yields
s = 2∗
2rt = 2∗
2r 1 + 2∗−22r∗r =
2∗ 2r + 2∗− 2∗r.
Thus we get the expressions for s and t
s = 2∗
2r + 2∗− 2∗r, t = 1 + 2∗− 2∗r 2r .
It is clear that for every r ∈ (0, 1) both s and t are greater than 1. Further- more 2s < 2∗ since
2s = 2∗
r + 22∗ −2∗2r < 2∗ ⇐⇒ r + 2∗ 2 − 2∗r
2 > 1 ⇐⇒
r
1 −2∗
2
| {z }
=ν<0
> 1 −2∗ 2
| {z }
=ν<0
⇐⇒ rν > ν.
Which clearly holds for every r ∈ (0, 1). Hence, Theorem 2, Sobolev’s in- equality, implies that w ∈ L2s(Ω). We now find s0
s0= 1
1 −1s = 1 1 −2r+22∗∗−2∗r
= 2∗ (2∗− 2)r.
By our assumption b ∈ L2s0(Ω) which yields ˆ
Ω
|b|2|w|2dx ≤ kbk22s0
ˆ
Ω
|w|2∗dx
st1 ˆ
Ω
|w|2dx
1
st0
= kbk22s0kwk
2∗
st
2∗kwk
2 st0
2 = kbk22s0kwk2r2∗kwk2(1−r)2 . This proves the lemma.
Note that s0 seen as a function of r is decreasing on (0, 1). Furthermore the assumption that b ∈ L2s0(Ω) implies that b is in Ln for every r in (0, 1).
This is seen by considering the limit of 2s0 as r → 1,
r→1lim2s0 = 2 2∗
(2∗− 2) = 2
2n n−2 2n
n−2 − 2= n,
which is the same constant as appears in the bound for the bound of b in Lemma 7. Furthermore in Theorem 11 we will show that a weak maximum principle holds whenever b ∈ Ln(Ω)ˆ and ˆn > n. If we assume that there exists a constant, C, such that |B(x, v)| = C|v(x)|γ, then
kB(x, v)k2s0 ≤ Ckvkγ2s0.
By Theorem 2 any v ∈ W01,2(Ω) is en element of L2∗(Ω), which gives the bound 2s0γ ≤ 2∗. And since 2s0→ n as r → 1, we get
γ < 2∗ n.
This yields a growth condition on B(x, v) of the form B(x, v) ≤ C|v(x)|γ, γ < 2∗
n = 2
n − 2.
We shall refer to γ = n−22 as the critical Sobolev exponent. We will now prove thatL is weakly coercive whenever b ∈ Lˆn(Ω).
Lemma 9 (Weak coercivity). Let b ∈ Lnˆ(Ω) for some ˆn > n, then there exists a constant σ > 0 such that the bilinear form,
Lσ(w, ϕ) = ˆ
Ω
∇u · ∇ϕ − b(x) · ∇u ϕ + σwϕ dx, is coercive.
Proof. Let ˆn > n, and let 2s0 = ˆn. We use the same notation as in the proof of Lemma 8. By Lemma 8 we have the following estimate
Lσ(w, w) ≥ 1
2k∇wk22− kbk22s0kwk2r2∗kwk2(1−r)2 + σkwk22.
We infer from Theorem 2 and the Young’s inequality with p = 1r that for every ε > 0
kwk2r2∗kwk2(1−r)2 ≤ Cs2rk∇wk2r2 kwk2(1−r)2 ≤ Cs2r
εk∇wk22+ 1 ε1−rr
kwk22
.
By choosing
ε = 1
4Cs2kbk22s0
. we get that
kLσ(w, w)k ≥ 1
4k∇wk22+ (σ − C)kwk22.
Where the constant C depends on b, ε, r and Ω. It follows that we may choose σ large enough to makeLσ coercive.
4.2 A maximum principle
In this section we show that equation (16) satisfies a maximum principle. As a consequence the only solution to the homogenous equation
∆u(x) + b(x) · ∇u(x) = 0, x ∈ Ω,
u(x) = 0, x ∈ ∂Ω, (24)
is the trivial solution, u = 0, in W1,2(Ω). We define u+ = max{u, 0} and u−= max{−u, 0}. We are going to use the following lemma, a proof can be found in [MP97].
Lemma 10. Let p ≥ 1. If u, v ∈ W1,p(Ω), then
∇ max{u, v} =
(∇u a.e. on {u ≥ v},
∇v a.e. on {v ≥ u},
∇ min{u, v} =
(∇u a.e. on {u ≤ v},
∇v a.e. on {v ≤ u}.
(25)
For a function u ∈ W1,2(Ω) we define sup
Ω
u := ess sup
Ω
u.
We say that a function u ∈ W1,2(Ω) satisfies u ≤ 0 on ∂Ω if u+is in W01,2(Ω).
For a function u ∈ W1,2(Ω) we define sup
∂Ω
u := inf{k : u − k ≤ 0 on ∂Ω}. (26) Similarly we say that u ≥ 0 on ∂Ω if −u ≤ 0 on ∂Ω. We define
inf∂Ωu := − sup
∂Ω
(−u). (27)
Lemma 11 (Weak maximum principle). Let u ∈ W1,2(Ω) and b ∈ Lˆn(Ω) where ˆn > n. If L (u, ϕ) ≤ 0 for every nonnegative ϕ ∈ Cc∞(Ω), then
sup
Ω
u ≤ sup
∂Ω
u+,
and if L (u, ϕ) ≥ 0 for every nonnegative ϕ ∈ Cc∞(Ω), then infΩ u ≥ inf
∂Ωu−. Proof. For every nonnegative ϕ ∈ Cc∞(Ω) we have
L (w, ϕ) :=
ˆ
Ω
∇u · ∇ϕ − b(x) · ∇u ϕ dx ≤ 0,
thus ˆ
Ω
∇u · ∇ϕ dx ≤ ˆ
Ω
|b||∇u||ϕ| dx.
Assume for contradiction that there exists a constant k ∈ R such that sup
∂Ω
u+≤ k < sup
Ω
u =: S. (28)
Let ϕk = (u − k)+. Then ϕk is nonnegative and in W01,2(Ω). Furthermore Lemma 10 implies that
∇ϕk=
(∇u a.e. on {u > k},
0 a.e. on {u ≤ k}, where ϕk= 0.
By the definition of ϕk we get ˆ
Ω
|b||∇u||ϕk| dx = ˆ
{u>k}
|b||∇ϕk||ϕk| dx,
and ˆ
Ω
∇u · ∇ϕkdx = ˆ
Ω
|∇ϕk|2dx ≤ ˆ
{u>k}
|b||∇ϕk||ϕk| dx.
Let Ωk:= supp(∇ϕk) ⊂ supp(ϕk) ⊂ Ω. Let ˆn > n and define ˆ2 = ˆn−22ˆn , then 1
2 +1 ˆ2 + 1
ˆ
n = 1, 2 < ˆ2 < 2∗.
By applying the generalised Hölder inequality to the above equation we get k∇ϕkk22≤ kbkn;Ωˆ kk∇ϕkk2;Ωkkϕkkˆ2;Ω
k ≤ kbknˆk∇ϕkk2kϕkkˆ2 By Theorem 2 and Hölder’s inequality we get
kϕkk2∗ ≤ Cskbknˆkϕkk2,Ω≤ Cskbknˆµ(Ωk)12−1ˆ2kϕkk2∗. Hence Ωk must satisfy
µ(Ωk) = µ[supp(∇ϕk)] ≥ 1 C
1 2−1ˆ s 2
. (29)
This must hold for every k < S. Thus we may consider a countable sequence {ϕk} defined as before, where the indices k are increasing and converges to S. Then the sequence {Ωk} is decreasing, hence converges to a set Ω0. and by the continuity of Lebesgue measure we get
k→Slimµ(Ωk) = µ
k→SlimΩk
. But equation (29) yields
k→Slimµ(Ωk) ≥ 1 C
1 2−1ˆ s 2
> 0.
At the same time ∇ϕk= 0 on the set Ω0, which implies that µ(Ω0) = 0. We arrive at a contradiction, implying that no k satisfying equation (28) exists, proving the statement for the supremum. The proof of the statement for the infimum follows along the same lines.
4.3 Existence of solution to the linear equation
We are now ready to prove the existence of solutions to the linear equation.
Theorem 15. If b ∈ Lˆn(Ω) for some ˆn > n and g ∈ W2,2(Ω), then there exists a unique solution, u in W1,2(Ω) to the equation
∆u(x) + b(x) · ∇u(x) = 0, x ∈ Ω,
u = g, x ∈ ∂Ω. (30)
Proof. Let g ∈ W2,2(Ω) and we define w = u − g. This yields the equation w + b · ∇w = −(∆g − b · ∇g), x ∈ Ω,
w = 0, x ∈ ∂Ω. (31)
Let f = −(∆g − b · ∇g). Note that f ∈ L2(Ω) and is well defined a.e. in Ω.
Consider the the partial differential operator Lσ = L − σ. The bilinear form corresponding to the operator
Lσw = Lw − σw,
is given by
Lσ(w, ϕ) = ˆ
Ω
∇u · ∇ϕ − b(x) · ∇u ϕ + σwϕ dx.
With this in mind we can solve the equivalent equation Lσw − σw = f.
It follows from Lemma 9, that we can choose σ such that the bilinear form, Lσ will be coercive. Let H01 = W01,2(Ω) and H−1 denote the dual space of H01. We define the operator I : H01 → H−1 by
H013 ϕ7−→Iw ˆ
Ω
wϕ dx.
We claim that the operator I is compact. To see this we can write I as a composition of a compact and a continuous operator. By the Rellich- Kondrachov compactness Theorem, there is a compact inclusion operator I2 : H01→ L2(Ω). Next we consider the operator I1: L2(Ω) → H−1 by
I2w(ϕ) = ˆ
Ω
wϕ dx.
By the Riesz representation theorem it follows that I1 is continuous and therefore I = I1I2 is a compact operator. A solution to our equation should satisfy
Lσ(w, ϕ) − σIw(ϕ) = F (ϕ), ∀ ϕ ∈ H01, (32) where
F (ϕ) = ˆ
Ω
f ϕ.
Let Λσ : H01 → H−1 be the operator induced by Lσ. Then we can rewrite equation (32) as
Λσw + σIw = F. (33)
The operator Λσ has a well defined continuous inverse, thus equation (33) is equivalent to the equation
w + σΛ−1σ Iw = Λ−1σ F.
Note that y := Λ−1σ F is an element of H01. Since I is compact and Λ−1σ is continuous the operator T := −σΛ−1σ I is compact, thus we may apply the Fredholm alternative, Theorem 5. By the weak maximum principle, Theorem 11, the trivial solution w = 0 in H01 is the unique solution to
(w − T w = 0, Ω,
w = 0, ∂Ω.
Hence the Fredholm alternative implies that a there exists a unique solution to equation (31). From this we can construct the unique solution to equation (30) which is given by u = w + g.
4.4 Interior and boundary W2,2 regularity
In this section we prove that solutions to equation (16) are in W2,2(Ω) under the assumptions of Theorem (15). The main result is Theorem (16) below.
The proof relies on difference quotient estimates.
Theorem 16 (Boundary regularity). Let ∂Ω be of class C2, f ∈ L2 and b ∈ Lnˆ(Ω) for some ˆn > n. Assume that w ∈ W01,2(Ω) solves
∆w + b · ∇w = f, x ∈ Ω,
w = 0, x ∈ ∂Ω. (34)
Then w ∈ W2,2(Ω) and there exists a positive constant C such that kwk2,2 ≤ Cp
kbknˆk∇wk2+ kf k2+ kwk1,2
. (35)
Before we are ready to prove Theorem 16 we need the following result con- cerning difference quotients of Sobolev functions and their derivatives. Let Ω0 ⊂⊂ Ω. For 0 < |h| < dist(Ω0, ∂Ω) and k ∈ {1, . . . , n} we define the the difference quotient Dkh for a function w ∈ L1loc(Ω) as
Dkhw(x) := w(x + hek) − w(x)
h ,
and
Dhw := (Dh1w, . . . , Dnhw).
The proof of the following two theorems can be found in [Eva10].
Theorem 17. If w ∈ W1,2(Ω), and Ω0 ⊂⊂ Ω, then there exists a constant M such that
kDhwkL2(Ω0)≤ M k∇wkL2(Ω), for every 0 < |h| < 12d(Ω0, ∂Ω).
Theorem 18. Let w ∈ L2(Ω), and Ω0 ⊂⊂ Ω, if there exists a constant C such that
kDhwkL2(Ω0)≤ C,
for every 0 < |h| < 12d(Ω0, ∂Ω), then w ∈ W1,2(Ω0) and k∇wkL2(Ω0) ≤ C.
Lemma 12. Let w and ϕ be in W01,2(Ω), then ˆ
Ω
ϕDk−hw = − ˆ
Ω
wDkhϕ, and
Dhk(ϕw) = ϕ(x + hek)Dhkw + wDkhϕ.
We start by showing that w satisfy an interior regularity property, that is w ∈ W2,2(Ω0) for every Ω0⊂⊂ Ω. But before we are ready to do so we state the following lemma from chapter 2 of [Lad68].
Lemma 13. Let u ∈ W01,2(Ω) and ˆn ≥ n, then for every ε > 0 kuk22ˆn
ˆ n−2
≤ εk∇uk22+ c(ε)kuk22, where c(ε) is a constant of the form
c(ε) = n − nˆ n
n ˆ nε
ˆn−nn
C(n, ˆn)ˆn−n2ˆn , and the constant C(ˆn, n) only depends on ˆn and n.
Theorem 19 (Interior regularity). Let Ω0 ⊂⊂ Ω, f ∈ L2 and b ∈ Lnˆ for some ˆn > n. Assume that w ∈ W01,2(Ω) solves
∆w + b · ∇w = f, x ∈ Ω,
w = 0, x ∈ ∂Ω. (36)
Then w ∈ W2,2(Ω0) and there exists a positive constant C such that kwk2,2;Ω0 ≤ Cp
kbknˆk∇wk2+ kf k2+ kwk1,2
. (37)
for some constant C, not depending on w.
Proof. The weak formulation of the equation yields the equality ˆ
Ω
∇w · ∇ϕ = ˆ
Ω
(f + b · ∇w)ϕ, ∀ϕ ∈ W01,2(Ω). (38) Given a test function z ∈ W01,2(Ω) let |h| ≤ 12d(supp(z), ∂Ω). Fix k ∈ {0, . . . , n} and let D−h = Dk−h. Then the difference quotient D−hz is well defined in in Ω. By Lemma 12 we get
ˆ
Ω
∇w · ∇(D−hz) = − ˆ
Ω
Dh(∇w)∇z.
Thus ˆ
Ω
Dh(∇w) · ∇z = − ˆ
Ω
(f + b · ∇w) · D−h(z). (39) We know that b ∈ Lnˆ(Ω) and D−hz ∈ Lˆ2(Ω) since z ∈ L2∗(Ω), where we used the same notation as in the proof of Theorem 11, i.e.
1 2 +1
ˆ2 + 1 ˆ
n = 1, 2 < ˆ2 < 2∗. By the the generalised Hölder inequality we get
ˆ
Ω
Dh(∇w) · ∇z ≤ kf k2kD−h(z)k2+ kbknˆk∇wk2kD−h(z)kˆ2. (40) Let η ∈ Cc∞(Ω) be a cutoff function η satisfying 0 ≤ η ≤ 1 and
(η = 1, x ∈ Ω0⊂⊂ Ω,
∇η < µ := d(Ω02,∂Ω).
