Jost J. Partial Differential Equations

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7.3 Brownian Motion 177

(reﬂection across the hyperplane through x

that is orthogonal to the jth

coordinate axis). We then have



|x−x

|≤r

− x

)P (t, x

,x)dx =



|x−x

|≤r

i(x

− x

)P (t, ix

,ix)dx

= −



|x−x

|≤r

− x

)P (t, x

,x)dx

because of (7.3.19) and the assumed invariance of P , and this indeed

implies (7.3.18).

Similarly, the invariance of P under rotations of R

yields



|x−x

|≤r

− x

)

P (t, x

,x)dx =



|x−x

|≤r

− x

)

P (t, x

,x)dx

for all x

∈ R

,r >0,t>0,j,k =1,...,d, (7.3.20)

and ﬁnally as in (7.3.18),



−x|≤r

− x

)(x

− x

)P (t, x

,x)dx =0 forj = k, (7.3.21)

if x

∈ R

, r>0, t>0, j, k ∈{1,...,d}.

(5) Let ϕ ∈ D(A) ∩ C

). We then obtain the existence of

Aϕ(x

) = lim

t0



P (t, x

,x)(ϕ(x) − ϕ(x

))dx

= lim

t0



|x−x

|≤ε

P (t, x

,x)(ϕ(x) − ϕ(x

))dx by (7.3.8)

= lim

t0



|x−x

|≤ε



j=1

− x

)

∂ϕ

∂x

)P (t, x

,x)dx

+ lim

t0



|x−x

|≤ε



j,k

− x

)(x

− x

)

∂

∂x

+ τ(x − x

))P (t, x

,x)dx

by Taylor expansion for some τ ∈ [0, 1), as ϕ ∈ C

The ﬁrst term on the right-hand side vanishes by (7.3.18). Thus, the

limit for t  0 of the second term exists, and it follows from (7.3.17) and

P (t, x

,x) ≥ 0that

lim sup

t0



|x−x

|≤ε



− x

)

P (t, x

,x)dx < ∞. (7.3.22)

By (7.3.8), this limit superior does not depend on ε>0, and neither does

the corresponding limit inferior.

178 7. The Heat Equation, Semigroups, and Brownian Motion

(6) Now let f ∈ D(A) ∩ C

). As in (5), we obtain, by Taylor expanding

f at x

f(x

) − f(x

))



(f(x) − f(x

))P (t, x

,x)dx



|x−x

|>ε

(f(x) − f(x

))P (t, x

,x)dx



|x−x

|≤ε



− x

)

∂f

∂x

)P (t, x

,x)dx



|x−x

|≤ε



j,k

− x

)(x

− x

)

∂

∂x

)P (t, x

,x)dx



|x−x

|≤ε



j,k

− x

)(x

− x

)σ

(ε)P (t, x

,x)dx

(where the notation suppresses the x-dependence of the remain-

der term σ

(ε), since this converges to 0 for ε → 0 uniformly

in x,sincef ∈ C

))



|x−x

|>ε

(f(x) − f(x

))P (t, x

,x)dx



|x−x

|≤ε



− x

)

∂

(∂x

)

)P (t, x

,x)dx



|x−x

|≤ε



j,k

− x

)(x

− x

)σ

(ε)P (t, x

,x)dx

by (7.3.18), (7.3.21). (7.3.23)

By (7.3.8), the ﬁrst term on the right-hand side tends to 0 as t → 0for

every ε>0. Because of (7.3.22) and lim

ε→0

(ε)=0(sincef ∈ C

), the

last term converges to 0 as ε → 0 for every t>0.Sincewehaveobservedat

the end of (5), however, that in the second term on the right-hand side, limits

can be performed independently of ε, for all ε>0, we obtain the existence

lim

t0



|x−x

|≤ε



− x

)

∂

(∂x

)

)P (t, x

,x)dx = Af(x

), (7.3.24)

by performing the limit t → 0 on the right-hand side of (7.3.23).

The argument of (3) shows that for f ∈ D(A),

∂

(∂x

)

7.3 Brownian Motion 179

may approximate arbitrary values, and so in particular, we infer the existence

lim

t0



|x−x

|≤ε



− x

)

P (t, x

,x)dx

independently of ε. By (7.3.20), for each j =1,...,d,

lim

t0



|x−x

|≤ε

− x

)

P (t, x

,x)dx

exists and is independent of j and by translation invariance independent of

as well. We thus call this limit c. By (7.3.24), we then have

Af(x

)=cΔf(x

The rest follows from Theorem 7.2.3. 

Remark: If we assume only spatial homogeneity, i.e., translation invariance,

but not invariance under reﬂections and rotations, the inﬁnitesimal generator

still is a second-order diﬀerential operator; namely, it is of the form

Af(x)=



j,k=1

(x)

∂

∂x

(x)+



j=1

(x)

∂f

∂x

(x)

with

(x) = lim

t0



|y−x|≤ε

− x

)(y

− x

)P (t, x, y)dy,

and thus in particular,

= a

≥ 0 for all j, k,

and

(x) = lim

t0



|y−x|≤ε

− x

)P (t, x, y)dy,

where the limits again are independent of ε>0. The proof can be carried

out with the same methods as employed for demonstrating Theorem 7.3.2.

A reference for the present chapter is Yosida [23].

Summary

The heat equation satisﬁes a Markov property in the sense that the solution

u(x, t)attimet

+ t

with initial values u(x, 0) = f (x) equals the solution at

time t

with initial values u(x, t

). Putting

180 7. The Heat Equation, Semigroups, and Brownian Motion

f)(x):=u(x, t),

we thus have

f)(x)=P

f)(x);

i.e., P

satisﬁes the semigroup property

= P

◦ P

for t

≥ 0.

Moreover, {P

}

t≥0

is continuous on the space C

in the sense that

lim

tt

= P

for all t

≥ 0 (in particular, this also holds for t

=0,withP

=Id).

Moreover, P

is contracting because of the maximum principle, i.e.,

P

f

≤f

for t ≥ 0,f ∈ C

The inﬁnitesimal generator of the semigroup P

is the Laplace generator, i.e.,

Δ = lim

t0

− Id).

Upon these properties one may found an abstract theory of semigroups in

Banach spaces. The Hille–Yosida theorem says that a linear operator A :

D(A) → B whose domain of deﬁnition D(A) is dense in the Banach space B

and for which Id −

A is invertible for all n ∈ N and

)

(Id −

−1

)

≤ 1

generates a unique contracting semigroup of operators

: B → B (t ≥ 0).

For a stochastic interpretation, one considers the probability density

P (t, x, y) that some particle that during the random walk happened to be

at the point x at a certain time can be found at y at a time that is larger by

the amount t. This constitutes a Markov process inasmuch as this probability

density depends only on the time diﬀerence, but not on the individual values

of the times involved. In particular, P (t, x, y) does not depend on where the

particle had been before reaching x (random walk without memory). Such a

random walk on the set S satisﬁes the Chapman–Kolmogorov equation

P (t

+ t

,x,y)=



P (t

,x,z)P (t

,z,y)dz

Exercises 181

and thus constitutes a semigroup.

If such a process on R

is spatially homogeneous and satisﬁes

lim

t0



|x−y|>ρ

P (t, x, y) dy =0

for all ρ>0andx ∈ R

, it is called a Brownian motion. One shows that

up to a scaling factor, such a Brownian motion has to be given by the heat

semigroup, i.e.,

P (t, x, y)=

(4πct)

d/2

−

|x−y|

4ct

Exercises

7.1 Let f ∈ C

) be bounded, u(x, t) a solution of the heat equation

(x, t)=Δu(x, t)forx ∈ R

,t>0,

u(x, 0) = f(x).

Show that the derivatives of u satisfy

∂

∂x

u(x, t)|≤const sup |f|·t

−1/2

(Hint: Use the representation formula (4.2.3) from Section 4.2.)

7.2 As in Section 7.2, we consider a continuous semigroup

exp(tA):B → B (t ≥ 0),B a Banach space.

Let B

be another Banach space, and for t>0 suppose

exp(tA):B

→ B

is deﬁned, and we have for 0 <t≤ 1 and for all ϕ ∈ B

exp(tA)ϕ

≤ const t

−α

ϕ

for some α<1.

Finally, let

Φ : B → B

be Lipschitz continuous.

Show that for every f ∈ B there exists T>0 with the property that the

evolution equation

∂v

∂t

= Av + Φ(v(t)) for t>0,

v(0) = f,

182 7. The Heat Equation, Semigroups, and Brownian Motion

has a unique, continuous solution v :[0,T] → B.

(Hint: Convert the problem into the integral equation

v(t)=exp(tA)f +



exp((t − s)A)Φ(v(s))ds

and use the Banach ﬁxed-point theorem (as in the standard proof of the

Picard–Lindel¨of theorem for ODEs) to obtain a solution of that integral

equation.)

7.3 Apply the results of Exercises 6.1, 6.2 to the initial value problem for the

following semilinear parabolic PDE:

∂u(x, t)

∂t

= Δu(x, t)+F (t, x, u(x),Du(x)) for x ∈ R

,t>0,

u(x, 0) = f(x),

for compactly supported f ∈ C

). We assume that F is smooth with

respect to all its arguments.

7.4 Demonstrate the assertion in the remark at the end of Section 7.3.

8. The Dirichlet Principle.

Variational Methods for the Solution of PDEs

(Existence Techniques III)

8.1 Dirichlet’s Principle

We consider the Dirichlet problem for harmonic functions once more:

We want to ﬁnd a solution u : Ω → R, Ω ∈ R

a domain, of

Δu =0 inΩ,

u = f on ∂Ω,

(8.1.1)

with given f .

Dirichlet’s principle is based on the following observation: Let u ∈ C

(Ω)

be a function with u = f on ∂Ω and



|∇u(x)|

dx =min





|∇v(x)|

dx : v : Ω → R with v = f on ∂Ω



(8.1.2)

We now claim that u then solves (8.1.1). To show this, let

η ∈ C

∞

(Ω).

According to (8.1.2), the function

α(t):=



|∇(u + tη)(x)|

possesses a minimum at t = 0, because u + tη = f on ∂Ω,sinceη vanishes

on ∂Ω. Expanding this expression, we obtain

α(t)=



|∇u(x)|

dx +2t



∇u(x) ·∇η(x)dx + t



|∇η(x)|

dx. (8.1.3)

In particular, α is diﬀerentiable with respect to t, and the minimality at t =0

implies

˙α(0) = 0. (8.1.4)

∞

(A):={ϕ ∈ C

∞

(A) : the closure of {x : ϕ(x) =0} is compact and contained

in A}.

184 8. Existence Techniques III

By (8.1.3) this implies



∇u(x) ·∇η(x)dx =0, (8.1.5)

and this holds for all η ∈ C

∞

(Ω).

Integrating (8.1.5) by parts, we obtain



Δu(x)η(x)dx =0 forallη ∈ C

∞

(Ω). (8.1.6)

We now recall the following well-known and elementary fact:

Lemma 8.1.1: Suppose g ∈ C

(Ω) satisﬁes



g(x)η(x)dx =0 for all η ∈ C

∞

(Ω).

Then g ≡ 0 in Ω. 

Applying Lemma 8.1.1 to (8.1.6) (which is possible, since Δu ∈ C

(Ω)by

our assumption u ∈ C

(Ω)), we indeed obtain

Δu(x)=0 inΩ,

as claimed.

This observation suggests that we try to minimize the so-called Dirichlet

integral

D(u):=



|∇u(x)|

dx (8.1.7)

in the class of all functions u : Ω → R with u = f on ∂Ω. This is Dirichlet’s

principle.

It is by no means evident, however, that the Dirichlet integral assumes

its inﬁmum within the considered class of functions. This constitutes the

essential diﬃculty of Dirichlet’s principle. In any case, so far we have not

speciﬁed which class of functions u : Ω → R (with the given boundary values)

we allow for competition; the possibilities include functions of class C

∞

,which

would be natural, since we have shown already in Chapter 1 that any solution

of (8.1.1) automatically is of regularity class C

∞

; functions of class C

,which

would be natural, since then the diﬀerential equation Δu(x)=0wouldhave

a meaning; and functions of class C

because then at least (assuming Ω

bounded and f suﬃciently regular, e.g., f ∈ C

)theDirichletintegralD(u)

would be ﬁnite. Posing the question somewhat diﬀerently, should we try to

minimize D(U) in a space of functions that is as large as possible, in order to

increase the chance that a minimizing sequence possesses a limit in that space

that then would be a natural candidate for a minimizer, or should we rather

8.1 Dirichlet’s Principle 185

select a smaller space in order to facilitate the veriﬁcation that a tentative

solution is a minimizer?

In order to analyze this question, we consider a minimzing sequence

)

n∈N

for D, i.e.,

lim

n→∞

D(u

)=inf{D(v):v : Ω → R,v = f on ∂Ω} =: κ, (8.1.8)

where, of course, we assume u

= f on ∂Ω for all u

. To ﬁnd properties of

such a minimizing sequence, we shall employ the following simple lemma:

Lemma 8.1.2: Dirichlet’s integral is convex, i.e.,

D(tu +(1− t)v) ≤ tD(u)+(1− t)D(v) (8.1.9)

for all u, v and all t ∈ [0, 1].

Proof:

D(tu +(1− t)v)=



|t∇u +(1− t)∇v|

≤





t |∇u|

+(1− t) |∇v|



because of the convexity of w →|w|

= tD(u)+(1− t)Dv.



Now let (u

)

n∈N

be a minimizing sequence. Then

D(u

− u



|∇(u

− u



|∇u



|∇u

− 4





∇



+ u





=2D(u

)+2D(u

) − 4D



+ u



. (8.1.10)

We now have

κ ≤ D



+ u



by deﬁnition of κ ((8.1.8))

≤

D(u

) by Lemma 8.1.2

→ κ for n, m →∞, (8.1.11)

since (u

) is a minimizing sequence. This implies that the right-hand side of

(8.1.10) converges to 0 for n, m →∞, and so then does the left-hand side.

186 8. Existence Techniques III

This means that (∇u

)

n∈N

is a Cauchy sequence with respect to the topology

of the space L

(Ω). (Since ∇u

has d components, i.e., is vector-valued, this

says that

∂u

∂x

is a Cauchy seqeunce in L

(Ω)fori =1,...,d.) Since L

(Ω)

is a Hilbert space, hence complete, ∇u

thus converges to some w ∈ L

(Ω).

The question now is whether w can be represented as the gradient ∇u of

some function u : Ω → R. At the moment, however, we know only that

w ∈ L

(Ω), and so it is not clear what regularity properties u should possess.

In any case, this consideration suggests that we seek a minimum of D in the

space of those functions whose gradient is in L

(Ω). In a subsequent step we

would then have to analyze the regularity proprties of such a minimizer u.

For that step, the starting point would be relation (8.1.5), i.e.,



∇u(x) ·∇η(x)dx =0 forallη ∈ C

∞

(Ω), (8.1.12)

which continues to hold in the context presently considered. By Corol-

lary 1.2.1 this already implies u ∈ C

∞

(Ω). In the next chapter, however,

we shall investigate this problem in greater generality.

Dividing the problem into two steps as just sketched, namely, ﬁrst proving

the existence of a minimizer and afterwards establishing its regularity, proves

to be a fruitful approach indeed, as we shall ﬁnd in the sequel. For that

purpose, we ﬁrst need to investigate the space of functions just considered in

more detail. This is the task of the next section.

8.2 The Sobolev Space W

1,2

Deﬁnition 8.2.1: Let Ω ⊂ R

be open and u ∈ L

loc

(Ω). A function v ∈

loc

(Ω) is called weak derivative of u in the direction x

(x =(x

,...,x

) ∈

)if



φv = −



∂φ

∂x

dx (8.2.1)

for all φ ∈ C

(Ω).

We write v = D

A function u is called weakly diﬀerentiable if it possesses a weak derivative

in the direction x

for all i ∈{1,...,d}.

It is obvious that each u ∈ C

(Ω) is weakly diﬀerentiable, and the weak

derivatives are simply given by the ordinary derivatives. Equation (8.2.1) is

then the formula for integrating by parts. Thus, the idea behind the deﬁnition

of weak derivatives is to use the integration by parts formula as an abstract

axiom.

(Ω):={f ∈ C

(Ω) : the closure of {x : f(x) =0} is a compact subset of Ω}

(k =1, 2,...).