Kazuaki Taira. Boundary Value Problems and Markov Processes

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9.4 Proof of Part (ii) of Theorem 1.3 159

By assertions (9.56) and (9.60), we obtain that the last term of formula (9.53)

also tends to zero:

lim

α→+∞





−1



α(1 − μ(x



))G

∂D







∞

=0. (9.61)

Therefore, the desired assertion (9.52) follows by combining assertions

(9.54) and (9.61).

The proof of assertion (iii) is complete.

Step 6: Summing up, we have proved that the operator A, deﬁned by

formula (9.39), satisﬁes conditions (a) through (d) in Theorem 2.16. Hence,

in view of assertion (8.2), it follows from an application of part (ii) of the

same theorem that the operator A is the inﬁnitesimal generator of some Feller

semigroup {T

}

t≥0

on D \ M.

The proof of Theorem 9.18 and hence that of Theorem 1.4 is now complete.



9.4 Proof of Part (ii) of Theorem 1.3

We apply Theorem 2.2 to the operator A.

In the proof of Theorem 9.18, we have proved that the domain D(A)is

dense in the space C

(D \ M ). Furthermore, part (i) of Theorem 1.3 veriﬁes

condition (2.1). Therefore, it follows from an application of Theorem 2.2 that:

The semigroup T

can be extended to a semigroup T

which is analytic

in the sector Δ

= {z = t + is : z =0, |arg z| <π/2 − ε} for any

0 <ε<π/2.

This (together with Theorem 1.4) proves part (ii) of Theorem 1.3. 

Application to Semilinear Initial-Boundary

Value Problems

This chapter is devoted to the semigroup approach to a class of initial-

boundary value problems for semilinear parabolic diﬀerential equations. We

prove Theorem 1.5 by using the theory of fractional powers of analytic semi-

groups (Theorems 10.1 and 10.2). To do this, we verify that all the conditions

of Theorem 2.8 are satisﬁed. Our semigroup approach here can be traced back

to the pioneering work of Fujita–Kato [FK]. For detailed studies of semilinear

parabolic equations, the reader is referred to Friedman [Fr1], Henry [He] and

also [Ta4].

10.1 Local Existence and Uniqueness Theorems

Let D be a bounded domain of Euclidean space R

, with smooth boundary

∂D;itsclosure

D = D ∪ ∂D is an N -dimensional, compact smooth manifold

with boundary. We let

A =



i,j=1

(x)

∂

∂x



i=1

(x)

∂

∂x

+ c(x)

be a second-order, elliptic diﬀerential operator with real smooth coeﬃcients

D such that:

(1) a

(x)=a

(x) for all x ∈ D and 1 ≤ i, j ≤ N .

(2) There exists a positive constant a

such that



i,j=1

(x)ξ

≥ a

|ξ|

for all (x, ξ) ∈ D × R

(3) c(x) ≤ 0on

K. Taira, Boundary Value Problems and Markov Processes, 161

Lecture Notes in Mathematics 1499, DOI 10.1007/978-3-642-01677-6

10,

 Springer-Verlag Berlin Heidelberg 2009

162 10 Application to Semilinear Initial-Boundary Value Problems

As an application of Theorem 1.2, we consider the following semilinear

initial-boundary value problem: Given functions f (x, t, u, ξ)andu

(x) deﬁned

in D×[0,T)×R×R

and in D, respectively, ﬁnd a function u(x, t)inD×[0,T)

such that

⎧

⎨

⎩



∂

∂t

− A



u(x, t)=f (x, t, u, grad u)inD × (0,T),

Lu(x



,t):=μ(x



)

∂u

∂n

+ γ(x



)u =0 on∂D × [0,T),

u(x, 0) = u

(x)inD.

(1.7)

Here:

(4) μ ∈ C

∞

(∂D)andμ(x



) ≥ 0on∂D.

(5) γ ∈ C

∞

(∂D)andγ(x



) ≤ 0on∂D.

(6) n =(n

,...,n

) is the unit interior normal to the boundary ∂D (see

Figure 1.1).

Recall that the operator A

is a unbounded linear operator from L

(D)

into itself given by the following formulas:

(a) The domain of deﬁnition D(A

)ofA

is the space

D(A



u ∈ H

2,p

(D)=W

2,p

(D):Lu(x



)=μ(x



)

∂u

∂n

+ γ(x



)u =0



(b) A

u = Au, u ∈D(A

By using the operator A

, we can formulate problem (1.7) in terms of the

abstract Cauchy problem in the Banach space L

(D) as follows:



= A

u(t)+F (t, u(t)) , 0 <t<T,

t=0

= u

(1.8)

Here u(t)=u(·,t)andF (t, u(t)) = f (·,t,u(t), grad u(t)) are functions deﬁned

on the interval [0,T), taking values in the space L

(D).

First, we consider the case N<p<∞:

Theorem 10.1. Let N<p<∞, and assume that conditions (A) and (B)

are satisﬁed:

(A) μ(x



) ≥ 0 on ∂D.

(B) γ(x



) < 0 on M = {x



∈ D : μ(x



)=0}.

If the nonlinear term f(x, t, u, ξ) is a locally Lipschitz continuous function

with respect to all its variables (x, t, u, ξ) ∈ D×[0,T)×R×R

with the possible

exception of the x variables, then, for every function u

of D(A

), problem

(1.8) has a unique local solution u ∈ C ([0,T

]; L

(D)) ∩ C

((0,T

); L

(D))

where T

= T

(p, u

) is a positive constant.

10.2 Fractional Powers and Imbedding Theorems 163

Here C ([0,T]; L

(D)) denotes the space of continuous functions on the closed

interval [0,T] taking values in L

(D), and C

((0,T); L

(D)) denotes the

space of continuously diﬀerentiable functions on the open interval (0,T)taking

values in L

(D), respectively.

In the case 1 <p<N, the domain D(A

) is large compared with the case

N<p<∞. Hence we must impose some growth conditions on the nonlinear

term f(x, t, u, ξ):

Theorem 10.2. Let N/2 <p<N, and assume that conditions (A) and (B)

are satisﬁed. Furthermore, we assume that there exist a non-negative contin-

uous function ρ(t, r) on R × R and a constant 1 ≤ γ<N/(N −p) such that

the following four conditions are satisﬁed:

(a) |f(x, t, u, ξ)|≤ρ(t, |u|)(1 + |ξ|

(b) |f(x, t, u, ξ) − f (x, s, u, ξ)|≤ρ(t, |u|)(1+|ξ|

) |t − s|.



1+|ξ|

γ−1

+ |η|

γ−1



|ξ − η|.

(d) |f(x, t, u, ξ) − f (x, t, v, ξ)|≤ρ(t, |u|+ |v|)(1+|ξ|

) |u − v|.

Then, for every function u

of D(A

), problem (1.8) has a unique local

solution u ∈ C ([0,T

]; L

(D)) ∩ C

((0,T

); L

(D)) where T

= T

(p, u

) is

a positive constant.

Theorems 10.1 and 10.2 prove Theorem 1.5, and are a generalization of

Pazy [Pa, Section 8.4, Theorems 4.4 and 4.5] to the degenerate case.

10.2 Fractional Powers and Imbedding Theorems

First, we study the imbedding properties of the domains of the fractional

powers (−A

)

(0 <α<1) into Sobolev spaces of L

type. By virtue of

Theorem 2.8, this allows us to solve, by successive approximations, problem

(1.8), proving Theorems 10.1 and 10.2.

By Theorem 7.1, we may assume that the operator A

satisﬁes condition

(2.21) in Subsection 2.1.2 (see Figure 7.1):

(1) The resolvent set of A

contains the region Σ as in Figure 10.1:

(2) There exists a positive constant M such that the resolvent (A

− λI)

−1

satisﬁes the estimate



− λI)

−1



≤

(1 + |λ|)

for all λ ∈ Σ. (10.1)

By using estimate (10.1), we can deﬁne the fractional powers (−A

)

for

0 <α<1onthespaceL

(D) as follows (cf. formula (2.23)):

(−A

)

−α

= −

sin απ



∞

−α

− sI)

−1

ds,

164 10 Application to Semilinear Initial-Boundary Value Problems

Fig. 10.1.

and

(−A

)

=theinverseof(−A

)

−α

We recall that (−A

)

is a closed operator with domain D((−A

)

) ⊃D(A

In this section we study the imbedding characteristics of D((−A

)

), which

will make these spaces so useful in the study of semilinear parabolic diﬀerential

equations.

We let

=thespaceD((−A

)

) endowed with the graph norm

·

of (−A

)

Here

u



u

+ (−A

)

u



1/2

,u∈D((−A

)

) .

Then we have the following three assertions:

(1) The space X

is a Banach space.

(2) The graph norm u

is equivalent to the norm (−A

)

u

(3) If 0 <α<β<1, then we have X

⊂ X

with continuous injection.

Furthermore, since the domain D(A

) is contained in W

2,p

(D), we can

obtain the following imbedding properties of the spaces X

into Sobolev spaces

(cf. [He, Theorem 1.6.1]):

Theorem 10.3. Let 1 <p<∞. Then we have the following continuous

injections:

(i) X

⊂ W

1,q

(D) if

<α<1,

−

2α−1

≤

, 1 <p<N.

(ii) X

⊂ C

(D) if

<α<1, 0 ≤ ν<2α −

, p = N.

10.3 Proof of Theorems 10.1 and 10.2 165

10.3 Proof of Theorems 10.1 and 10.2

This section is devoted to the proof of our local existence and uniqueness

theorems for problem (1.8) (Theorems 10.1 and 10.2). To do this, we verify

that all the conditions of Theorem 2.8 are satisﬁed.

10.3.1 Proof of Theorem 10.1

We verify that all the conditions of Theorem 2.8 are satisﬁed; then

Theorem 10.1 follows from an application of the same theorem.

Since p>N, we can choose a positive constant α such that





<α<1,

so that

1 < 2α −

Then, by part (ii) of Theorem 10.3 with ν := 1, we have

⊂ C

(D)andX

⊂ W

1,p

(D), (10.2)

with continuous injections. Thus we ﬁnd that the function

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

is well deﬁned on [0,T] × X

. Furthermore, since the function f(x, t, u, ξ)is

locally Lipschitz continuous, in view of assertion (10.2) it follows that we have,

for all t, s ∈ [0,t

]andforallu, v ∈ X

with u − u



≤ R, v − u



≤ R

F (t, u) − F (s, v)

≤F (t, u) − F (t, v)

+ F (t, v) − F (s, v)

≤ C

u − v



i=1



∂

∂x

(u − v)



+ |t − s|

≤ C (u − v

1,p

+ |t − s|)

≤ CC



(u − v

+ |t − s|) . (10.3)

Here C = C(t

,R) is a (local) Lipschitz positive constant for the function f ,

and C



is an imbedding (positive) constant for the imbedding: X

⊂ W

1,p

(D).

By inequality (10.3), we obtain that the function F (t, u) is locally Lipschitz

continuous in t and u.

The proof of Theorem 10.1 is complete. 

166 10 Application to Semilinear Initial-Boundary Value Problems

10.3.2 Proof of Theorem 10.2

The proof is similar to that of Theorem 10.1; we verify that all the conditions

of Theorem 2.8 are satisﬁed.

Since N/2 <p<N and 1 ≤ γ<N/(N − p), we can choose a positive

constant α such that

max



γ − 1



<α<1, (10.4)

so that

0 < 2α −

and

−

2α − 1

pγ

≤

Then, by Theorem 10.3 with ν := 0 and q := pγ,wehave

⊂ L

∞

(D)andX

⊂ W

1,pγ

(D), (10.5)

with continuous injections.

We let

F (t, u):=f(x, t, u(x), grad u(x)),t∈ [0,T],u∈ X

Then we have, by condition (a),

F (t, u)

≤ 2

p−1

ρ(t, u

∞

)



(1 + |grad u|

pγ

)dx

≤ 2

p−1

ρ(t, u

∞

)



|D| + u

pγ

1,pγ



where |D| denotes the volume of the domain D. By assertion (10.5), it follows

that the function F (t, u) is well deﬁned on [0,T] × X

for all α satisfying

condition (10.4).

Step 1: First, we verify the local Lipschitz continuity of F (t, u)withre-

spect to the variable t.

By condition (b), it follows that

F (t, u) − F (s, u)



|f(x, t, u(x), grad u(x)) − f(x, s, u(x), grad u(x))|

≤ 2

p−1

ρ(t, u

∞

)

|t − s|



(1 + |grad u|

pγ

)dx

≤ 2

p−1

ρ(t, u

∞

)



|D| + u

pγ

1,pγ



|t − s|

In view of assertion (10.5), this proves that

F (t, u) − F (s, u)

≤ C

(u

) |t − s|, (10.6)

where C

(u

) is a positive constant depending on the norm u

10.3 Proof of Theorems 10.1 and 10.2 167

Step 2: Secondly, we verify the local Lipschitz continuity of F (t, u)with

respect to the variable u.

To do this, we remark that

F (t, u) − F (t, v)



|f(x, t, u(x), grad u(x)) − f (x, t, v(x), grad v(x))|

≤ 2

p−1



|f(x, t, u(x), grad u(x)) − f (x, t, u(x), grad v(x))|

p−1



|f(x, t, u(x), grad v(x)) − f (x, t, v(x), grad v(x))|

dx. (10.7)

We estimate the two terms on the right-hand side of inequality (10.7):

Step 2-1: By condition (c), it follows that



|f(x, t, u(x), grad u(x)) − f (x, t, u(x), grad v(x))|

≤ 3

p−1

ρ(t, u

∞

)





1+|grad u|

p(γ−1)

+ |grad v|

p(γ−1)



|grad (u − v)|

dx. (10.8)

However, by H¨older’s inequality it follows that



|grad (u − v)|

dx ≤





1dx



(γ−1)/γ





|grad (u − v)|

pγ



1/γ

≤|D|

(γ−1)/γ

u − v

1,pγ

, (10.9)



|grad u|

p(γ−1)

|grad (u − v)|

dx ≤





|grad u|

pγ



(γ−1)/γ





|grad (u − v)|

pγ



1/γ

≤u

p(γ−1)

1,pγ

u − v

1,pγ

(10.10)



|grad v|

p(γ−1)

|grad (u − v)|

dx ≤v

p(γ−1)

1,pγ

u − v

1,pγ

. (10.11)

Thus, by carrying these three inequalities (10.9), (10.10) and (10.11) into

inequality (10.8) we obtain that



|f(x, t, u(x), grad u(x)) − f (x, t, u(x), grad v(x))|

≤ 3

p−1

ρ(t, u

∞

)



|D|

(γ−1)/γ

+ u

p(γ−1)

1,pγ

+ v

p(γ−1)

1,pγ



×u − v

1,pγ

. (10.12)

168 10 Application to Semilinear Initial-Boundary Value Problems

Step 2-2: By condition (d), it follows that



|f(x, t, u(x), grad v(x)) − f (x, t, v(x), grad v(x))|

≤ 2

p−1

ρ(t, u

∞

+ v

∞

)



(1 + |grad v|

pγ

) |u − v|

≤ 2

p−1

ρ(t, u

∞

+ v

∞

)

u − v

∞



(1 + |grad v|

pγ

) dx

≤ 2

p−1

ρ(t, u

∞

+ v

∞

)



|D| + v

pγ

1,pγ



u − v

∞

. (10.13)

Therefore, by combining inequalities (10.7), (10.12) and (10.13) we obtain

that

F (t, u) − F (t, v)

≤ 6

p−1

ρ(t, u

∞

)



|D|

(γ−1)/γ

+ u

p(γ−1)

1,pγ

+ v

p(γ−1)

1,pγ



u − v

1,pγ

p−1

ρ(t, u

∞

+ v

∞

)



|D|+ v

pγ

1,pγ



u − v

∞

In view of assertion (10.5), this proves that

F (t, u) − F (t, v)

≤ C

(u

, v

) u − v

, (10.14)

where C

(u

, v

) is a positive constant depending on the norms u

and v

Summing up, we ﬁnd from inequalities (10.6) and (10.14) that the function

F (t, u) is locally Lipschitz continuous in t and u.

The proof of Theorem 10.2 is now complete. 

Concluding Remarks

This book is devoted to a careful and accessible exposition of the functional

analytic approach to the problem of construction of Markov processes with

Ventcel’ boundary conditions in probability theory. More precisely, we prove

that there exists a Feller semigroup corresponding to such a diﬀusion phe-

nomenon that a Markovian particle moves continuously in the state space

D \M until it “dies” at the time when it reaches the set M where the particle

is deﬁnitely absorbed (see Figure 11.1). Our approach here is distinguished

by the extensive use of the ideas and techniques characteristic of the recent

developments in the theory of pseudo-diﬀerential operators which may be

considered as a modern theory of the classical potential theory.

More generally, it is known (see [BCP], [SU], [Ta2], [We]) that the inﬁnites-

imal generator W of a Feller semigroup {T

}

t≥0

is described analytically by

a Waldenfels operator W and a Ventcel’ boundary condition L,whichwe

formulate precisely.

Let W be a second-order, elliptic integro-diﬀerential operator with real

coeﬃcients such that

Wu(x)=Au(x)+S

u(x)

⎛

⎝



i,j=1

(x)

∂

∂x

(x)+



i=1

(x)

∂u

∂x

(x)+c(x)u(x)

⎞

⎠



s(x, y)

⎡

⎣

u(y) − u(x) −



j=1

− x

)

∂u

∂x

(x)

⎤

⎦

dy.

Here:

(1) a

∈ C

∞

(D), a

(x)=a

(x) for all x ∈ D and 1 ≤ i, j ≤ N,andthere

exists a positive constant a

such that



i,j=1

(x)ξ

≥ a

|ξ|

for all (x, ξ) ∈ Ω ×R

K. Taira, Boundary Value Problems and Markov Processes, 169

Lecture Notes in Mathematics 1499, DOI 10.1007/978-3-642-01677-6

11,

 Springer-Verlag Berlin Heidelberg 2009