Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

Подождите немного. Документ загружается.

278 D. Guidetti

is a Banach space. If β ∈ (0, 1) an equivalent norm can be obtained replacing in

(1.1) f (.+h)−2f +f(.−h)

)

with f (.+h)−f

)

(see [12], Theorem 4).

Now we deﬁne the space N

)incaseβ ≤ 0.

Deﬁnition 1.3. Let β ∈ R, β ≤ 0, p ∈ [1, +∞]. We set

):=



f =

|s|≤|[β]|

∂

: f

∈ N

{β}

)



equipped with the norm

f

β,p,R

:= inf



|s|≤|[β]|

f



{β},p,R

: f =

|s|≤|[β]|

∂

∈ N

{β}

)



. (1.3)

Deﬁnition 1.4. Let n ∈ N, p ∈ [1, ∞], β ∈ R andletΩbeanopensubsetofR

We set

(Ω) := {f

|Ω

: f ∈ N

)}. (1.4)

If g ∈ N

(Ω), we set

g

β,p,Ω

:= inf{f

β,p,R

: f

|Ω

= g}. (1.5)

We shall write C

(Ω) in alternative to N

∞

(Ω).

Remark 1.5. If β ∈ R

\Z, C

(Ω) coincides with the space of elements in C

[β]

(Ω)

whose derivatives of order [β]areH¨older continuous of exponent {β}.

For the sake of simplicity, here we shall consider only the case that the bound-

ary ∂Ω of Ω is “smooth”, that is, ∂ΩisaC

∞

submanifold of R

, and Ω lies on

one side of ∂Ω, although this restrictive condition could be considerably relaxed.

The spaces N

(Ω) are locally invariant with respect to compositions with

smooth (that is, C

∞

) diﬀeomorphisms (see [14], 2.10). This implies that we can

deﬁne by local charts the spaces N

(S)ifS is a smooth submanifold of R

.The

following fact holds (see [6], Theorem 1.19):

Theorem 1.6. Assume 1 ≤ p ≤ +∞, β>m+ p

−1

,withm ∈ N

. Then there

exists T ∈L(N

(Ω),

j=0

β−j−1/p

(∂Ω)), such that, if u ∈ N

(Ω) ∩ C

(Ω),

Tu=(u

|∂Ω

,...,

∂

∂ν

Nikol’skij spaces arise in real interpolation theory. If θ ∈ (0, 1) and m

∈

,withm

,wehave

(Ω),W

(Ω))

θ,∞

= N

(1−θ)m

+θm

(Ω), (1.6)

with equivalent norms (see [1], 7.32). In general, if β

,β

∈ R and θ ∈ (0, 1),

(Ω),N

(Ω))

θ,∞

= N

(1−θ)β

+θβ

(Ω), (1.7)

again with equivalent norms (see [14], 3.3.6). Finally, if β

, β

∈ R, m ∈ N

<m<β

,andm =(1−θ)β

+ θβ

,withθ ∈ (0, 1), ∀f ∈ N

(Ω),

f

m,p

(Ω)

≤ Cf

1−θ

,p,Ω

f

,p,Ω

, (1.8)

On Linear Elliptic and Parabolic etc. 279

with C>0, independent of f. This is a consequence of the fact that

(Ω),N

(Ω))

θ,1

= B

p,1

(Ω) → W

m,p

(Ω),

(see [14, 3.3.6]), where B

p,1

(Ω) is a Besov space.

2. Elliptic problems depending on a parameter with

nonhomogeneous boundary conditions

Let now Ω be a bounded open subset of R

, with smooth boundary ∂Ω. We

consider the partial linear operator of order 2m (m ∈ N)

A(x, ∂

|s|≤2m

(x)∂

, (2.1)

with smooth coeﬃcients in Ω. We indicate with A

(x, ∂

) its principal part

|s|=2m

(x)∂

.Fork =1,...,m, we introduce partial linear operators

(x, ∂

|s|≤μ

(x)∂

, (2.2)

again with smooth coeﬃcients in Ω. We assume the following:

(H1) A(x, ∂

) is strongly elliptic, in the sense that there exists c>0, such that,

∀x ∈ Ω, ∀ξ ∈ R

, (−1)

Re(A

(x, ξ)) ≤−c|ξ|

(H2) 0 ≤ μ

<μ

< ···<μ

≤ 2m − 1; moreover, if we indicate with B

(x, ∂

)

the principal part of B

(x, ∂

), B

(x, ν(x)) =0for each k =1,...,m.

(H3) ∀θ

∈ [−π/2,π/2], ∀x ∈ ∂Ω, ∀(ξ



,r) ∈ (T

(∂Ω) ×[0, +∞))\{0, 0)}, the o.d.e.

problem

⎧

⎪

⎨

⎪

⎩

(x, iξ



+ ν(x)∂

) − r

iθ

)v(t)=0,t≥ 0,

(x, iξ



+ ν(x)∂

)v(0) = 0,k=1,...,m,

v bounded in R

has only the trivial solution.

The following theorem summarizes in its statement Lemmas 2.12 and 2.14

(using also Theorem 2.17) in [6]:

Theorem 2.1. Let Ω be a bounded open subset in R

, with smooth boundary ∂Ω,and

let A(x, ∂

) and, for k =1,...,m, B

(x, ∂

),...,B

(x, ∂

) operators satisfying

the conditions (H1)–(H3). We assume that β ∈ R, 1 ≤ p ≤∞, μ + p

−1

−2m<β,

with μ := max

1≤k≤m

. We consider the problem



λu(x) − A(x, ∂

)u(x)=0,x∈ Ω,



,∂

)u(x



) −g



)=0 x



∈ ∂Ω,k=1,...,m,

(2.3)

with λ ∈ C, Re(λ) ≥ 0,fork =1,...,m, g

∈ N

2m+β−μ

(Ω).Then:

280 D. Guidetti

(I) there exist r

and C positive, such that, if |λ|≥r

, (2.3) has a unique solution

u belonging to N

2m+β

(Ω).

(II) The following estimate holds:

|λ|

u

(Ω)

+ |λ|u

β,p,Ω

+ u

2m+β,p,Ω

≤ C

k=1

(g



2m+β−μ

,p,Ω

+ |λ|

2m+β−μ

g



(Ω)

(2.4)

(III) If, moreover, 0 < {β} <p

−1

, the following estimate holds:

|λ|

u

(Ω)

+ |λ|u

β,p,Ω

+ u

2m+β,p,Ω

≤ C

k=1

(g



2m+β−μ

,p,Ω

+ |λ|

2m+[β]−μ

g



{β},p,Ω

(2.5)

Remark 2.2. In the statements of Lemmas 2.12 and 2.14 in [6], concerning the

formula corresponding to (2.4), there appear summands of the form |λ|

g



l,p

(Ω)

and |λ|

g



l,p,Ω

,withl intermediate between, respectively, 2m + β − μ

and 0,

and 2m + β −μ

and {β}. They can be all skipped, using (1.8) and (1.7).

We want to generalize estimate (2.5). More precisely, we want to prove the following

Theorem 2.3. Assume that the assumptions of Theorem 2.1 are satisﬁed. Let λ ∈ C,

with Re(λ) ≥ 0 and |λ|≥r

,withr

as in the statement of Theorem 2.1.Further,

let θ ∈ (0, 1/p).Then,thereexistsC>0, independent of λ and g

(1 ≤ k ≤ m),

such that

|λ|

u

(Ω)

+ |λ|u

β,p,Ω

+ u

2m+β,p,Ω

≤ C



k=1

g



2m+β−μ

,p,Ω

+ |λ|

2m+β−μ

−θ

g



θ,p,Ω



(2.6)

To this aim, we need the following lemmas:

Lemma 2.4. Assume 1 ≤ p<+∞, 0 <β≤ θ<

, g ∈ N

) and vanishes if

>r.Then:

(I) there exists C>0, independent of g, such that, ∀r ∈ R

g

)

≤ Cr

g

θ,p,R

;

(II) there exists C>0, independent of g, such that, ∀r ∈ R

g

β,p,R

≤ Cr

θ−β

g

θ,p,R

;

Proof. Concerning (I), see Lemma 2.13 in [6].

Concerning (II), from (I) and the fact that

)=(L

),N

))

β/θ,∞

we obtain

g

β,p,R

≤ C

g

1−β/θ

)

g

β/θ

θ,p,R

≤ C

g

θ,p,R

)

1−β/θ

g

β/θ

θ,p,R

. 

On Linear Elliptic and Parabolic etc. 281

Lemma 2.5. Assume 1 ≤ p<+∞, β ∈ (0, 1], θ ∈ (0,

∧β),andlet{χ

}

0<r≤1

afamilyofC

functions in [0, +∞), such that

(a) χ

(t)=0if t ≥ r ∀r ∈ (0, 1];

(b) |χ

(t)| + r|χ



(t)|≤A, ∀r ∈ (0, 1], ∀t ≥ 0,forsomeA ∈ R

Then:

(I) there exists C ∈ R

, depending only on θ, p, A, such that, ∀g ∈ N

∀r ∈ (0, 1],ifg



)=g(x



)χ

g



θ,p,R

≤ Cg

θ,p,R

;

(II) assume, moreover, that the functions χ

are of class C

,andr

|χ



(t)|≤

A, ∀r ∈ (0, 1], ∀t ≥ 0.Then,∀g ∈ N

), ∀r ∈ (0, 1],

g



β,p,R

≤ C(g

β,p,R

+ r

θ−β

g

θ,p,R

Proof. (I) is essentially proved in [5], Lemma 1.6 (see also [6], Lemma 2.13).

Concerning (II), we shall use the following equivalent norm in N

)(see

for this [8], Ch. VII, Theorem 2.1):

g



β,p,R

. := g

)

j=1

sup

t>0

−β

g(. +2te

) − 2g(. + te

)+g

)

Now, let g ∈ N

). Obviously,

g



)

n−1

j=1

sup

t>0

−β

g

(. +2te

) −2g

(. + te

)+g



)

≤ Cg



β,p,R

with C depending only on A.Moreover,

g

(. +2te

) −2g

(. + te

)+g



)

≤(χ

+2t) −χ

+ t))(g(. +2te

) −g(. + te

))

)

+ (χ

+2t) − 2χ

+ t)+χ

))g(. + te

))

)

+ (χ

+ t) − χ

))(g(. +2te

) −g(. + te

))

)

+ χ

)(g(. +2te

) − 2g(. + te

)+g)

)

:= I

+ I

(2.7)

First of all,

−β

≤ Cg



β,p,R

. (2.8)

Next,

−β

≤ Cχ



β−θ

([0,+∞[)

g

θ,p,R

≤ Cr

θ−β

g

θ,p,R

(2.9)

owing to well-known interpolatory inequalities, which are consequences of (a)–(b).

−β

can be estimated similarly. Concerning t

−β

, we start by observing that

(χ

+2t) −2χ

+ t)+χ

))g(x + te

)=0

if x

>r. So, from Lemma 2.4, we have

−β

≤ Cr

−β

(χ

+2t) −2χ

+ t)+χ

))g(. + te

)

θ,p,R

282 D. Guidetti

Assume ﬁrst that t ≥ r. Then, by (I)

−β

(χ

+2t) −2χ

+ t)+χ

))g(. + te

)

θ,p,R

= r

−β

χ

)g(. + te

)

θ,p,R

≤ Cr

−β

g

θ,p,R

≤ Cr

θ−β

g

θ,p,R

Finally, assume 0 <t<r. We deﬁne

r,t

(s):=rt

−1

(χ

(s +2t) − 2χ

(s + t)+χ

(s)).

Then, ∀r, t,

r,t

(s)| + r|c



r,t

(s)|≤4A.

From (I) we deduce

−β

= r

−1

1−β

c

r,t

)g(. + te

)

)

≤ Cr

θ−1

1−β

c

r,t

)g(. + te

)

θ,p,R

≤ Cr

θ−1

1−β

g

θ,p,R

, ≤ Cr

θ−β

g

θ,p,R

Collecting together the previous estimates, we obtain the conclusion. 

Remark 2.6. An example of family of functions satisfying conditions (a) and (b)

in the statement of Lemma 2.5 can be obtained setting χ

(t):=χ(

), with χ :

[0, +∞) → C suitably smooth.

Now,weareabletoproveTheorem2.3:

Proof of Theorem 2.3. For every x

∈ ∂Ω there exist an open neighbourhood U

of x

, and a smooth diﬀeomorphism Φ : U → Φ(U) ⊆ R

, such that Φ(x

)=0,

Φ(U ∩ ∂Ω) = Φ(U) ∩{(y



, 0) : y



∈ R

n−1

} and Φ(U ∩ Ω) = Φ(U) ∩ R

.AsΩis

bounded, there exist x

,..., x

, with corresponding diﬀeomorphisms Φ

,...,Φ

such that ∂Ω ⊆∪

j=1

. We add an open subset U

of Ω, with U

⊆ Ω, such that

Ω ⊆∪

j=0

.Let{ψ

,...,ψ

} be a smooth partition of unity in a neighbourhood

of Ω, such that the support of ψ

is contained in U

(j =0,...,N). Let (χ

)

0<r≤1

be a family of smooth functions of domain [0, +∞), such that χ

(t)=1if0≤ t ≤

r/2, χ

(t)=0ift ≥ r, |χ

(l)

(t)|≤C

−l

, ∀r ∈ (0, 1], ∀l ∈ N

, ∀t ≥ 0. We put, for

k =1,...,m, r>0 suﬃciently small

(x):=

j=1

(x)χ

(Φ

(x)

(x).

Then, applying Theorem 2.1 (II), we have ∀r ∈ (0,r

|λ|

u

(Ω)

+ |λ|u

β,p,Ω

+ u

2m+β,p,Ω

≤ C

k=1

(g



2m+β−μ

,p,Ω

+ |λ|

2m+β−μ

g



(Ω)

(2.10)

On Linear Elliptic and Parabolic etc. 283

with C>0, independent of λ, g

(1 ≤ k ≤ m)andr ∈ (0,r

]. We have

g



(Ω)

≤

j=1

ψ

(Φ

(x)



(Ω)

≤ C

j=1

χ

)(ψ

◦ Φ

−1

)

)

We set g

∗

:= ψ

◦ Φ

−1

. Then, by Lemmas 2.4 and 2.5,

χ

∗



)

≤ C

χ

∗



θ,p,R

, ≤ C

g

∗



θ,p,R

, ≤ C

g



θ,p,Ω

(2.11)

Moreover,

g



2m+β−μ

,p,Ω

≤

j=1

ψ

(Φ

(x)



2m+β−μ

,p,Ω

≤ C

j=1

χ

)(ψ

◦ Φ

−1

)

2m+β−μ

,p,R

We have

χ

∗



2m+β−μ

,p,R

≤ C

+|s

|≤2m+[β]−μ

χ

)

)∂

∗



{β},p,R

. (2.12)

Assume ﬁrst that {β}≤θ. Then, if |s

|≥1, we have, employing Lemma 2.5,

χ

)

)∂

∗



{β},p,R

≤ r

−s

r

)

)∂

∗



{β},p,R

≤ Cr

−s

∂

∗



{β},p,R

≤ Cr

−s

g

∗



|+{β},p,R

≤ Cr

|−(2m+[β]−μ

)

g



|+{β},p,Ω

. (2.13)

If s

=0,

χ

)

∗



{β},p,R

≤ Cr

θ−{β}

χ

)

∗



θ,p,R

≤ Cr

θ−s

−{β}

g

∗



θ,p,R

≤ Cr

θ−(2m+β−μ

)

g



θ,p,Ω

. (2.14)

Instead, we assume that θ<{β}. Again applying Lemma 2.5, we have

χ

)

)∂

∗



{β},p,R

= r

−s

r

)

)∂

∗



{β},p,R

≤ Cr

−s

(∂

∗



{β},p,R

+ r

θ−{β}

∂

∗



θ,p,R

) (2.15)

≤ C(r

|+{β}−(2m+β−μ

)

g



|+{β},p,Ω

+ r

|+θ−(2m+β−μ

)

g



|+θ,p,Ω

Observe now that the last expressions in (2.13), (2.14), (2.15) can be all majorized

with

C(g



2m+β−μ

,p,Ω

+ r

θ−(2m+β−μ

)

g



θ,p,Ω

). (2.16)

In fact, if θ<ρ<2m + β − μ

, by (1.7),

ρ−(2m+β−μ

)

g



ρ,p,Ω

≤ C(r

θ−(2m+β−μ

)

g



θ,p,Ω

)

2m+β−μ

−ρ

2m+β−μ

−θ

g



ρ−θ

2m+β−μ

−θ

,∞

2m+β−μ

,p,Ω

≤ C(g



2m+β−μ

,p,Ω

+ r

θ−(2m+β−μ

)

g



θ,p,Ω

, ).

284 D. Guidetti

So, from (2.10), we obtain, ∀r ∈ (0,r

|λ|

u

(Ω)

+ |λ|u

β,p,Ω

+ u

2m+β,p,Ω

(2.17)

≤ C

k=1

(g



2m+β−μ

,p,Ω

+ r

θ−(2m+β−μ

)

g



θ,p,Ω

+ |λ|

2m+β−μ

g



θ,p,Ω

Choosing r = |λ|

−1/(2m)

, we obtain the conclusion. 

As a consequence, we obtain the following reﬁnement of Propositions 2.15–

2.16 in [6]:

Theorem 2.7. Let Ω be a bounded open subset in R

, with smooth boundary ∂Ω,and

let A(x, ∂

) and, for k =1,...,m, B

(x, ∂

),...,B

(x, ∂

) operators satisfying

the conditions (H1)–(H3). We assume that β ∈ R, μ + p

−1

− 2m<β<ν+ p

−1

with μ := max

1≤k≤m

, ν := min

1≤k≤m

.Letθ ∈ (0,p

−1

) be such that ν+θ ≥ β.

We consider the problem



λu(x) − A(x, ∂

)u(x)=f(x),x∈ Ω,



,∂

)u(x



) − g



)=0 x



∈ ∂Ω,k=1,...,m,

(2.18)

with λ ∈ C, Re(λ) ≥ 0, f ∈ N

(Ω),fork =1,...,m g

∈ N

2m+β−μ

(Ω).

Then, there exist r

and C positive, such that, if |λ|≥r

, (2.18) has a unique

solution u belonging to N

2m+β

(Ω). Moreover, the following estimate holds:

|λ|u

β,p,Ω

+ u

2m+β,p,Ω

≤ C(f

β,p,Ω

k=1

g



2m+β−μ

,p,Ω

+ |λ|

2m+β−μ

−θ

g



θ,p,Ω

(2.19)

Proof. Consider a strongly elliptic operator A(x, ∂) with coeﬃcients which are

smooth and bounded with all their derivatives in R

, such that the restriction of

A(x, ∂) to Ω coincides with A(x, ∂) (see the proof of Proposition 2.15 in [6] for the

construction of A(x, ∂)). We ﬁx f



∈ N

), such that f



|Ω

= f and

f





β,p,R

≤ Cf

β,p,Ω

. (2.20)

Then, if |λ| is large enough and Re(λ) ≥ 0, owing to Proposition 2.5 in [6], there

exists a unique u



∈ N

2m+β

), such that

λu



− A(x, ∂)u



= f



in R

(2.21)

and

|λ|u





β,p,R

+ u





2m+β,p,R

≤ Cf





β,p,R

. (2.22)

This implies that, ∀ρ ∈ [β,2m + β],

u





ρ,p,R

≤ C(ρ)|λ|

ρ−(2m+β)

f





β,p,R

. (2.23)

On Linear Elliptic and Parabolic etc. 285

We indicate with v the restriction of u



to Ω. Now we consider the problem



λz(x) − A(x, ∂

)z(x)=0,x∈ Ω,



,∂

)z(x



) −(g



) − B



,∂

)v(x



)) = 0,x



∈ ∂Ω,k=1,...,m,

(2.24)

By Theorem 2.3, if |λ| is suﬃciently large, (2.24) has a unique solution z in

2m+β

(Ω). Clearly, u := v + z is the unique solution of (2.18) in N

2m+β

(Ω).

Moreover, employing (2.23),

|λ|u

β,p,Ω

+ u

2m+β,p,Ω

≤|λ|u





β,p,R

+ u





2m+β,p,R

+ |λ|z

β,p,Ω

+ z

2m+β,p,Ω

≤ C[f





β,p,R

k=1

(g

−B

(., ∂)v

2m+β−μ

,p,Ω

+ |λ|

(2m+β−μ

)−θ

g

−B

(., ∂)v

θ,p,Ω

)]

≤ C[f

β,p,Ω

k=1

(g



2m+β−μ

,p,Ω

+ |λ|

2m+β−μ

−θ

g



θ,p,Ω

)

k=1

(B

(., ∂)v

2m+β−μ

,p,Ω

+ |λ|

2m+β−μ

−θ

B

(., ∂)v

θ,p,Ω

)]

≤ C[f

β,p,Ω

k=1

(g



2m+β−μ

,p,Ω

+ |λ|

2m+β−μ

−θ

g



θ,p,Ω

)

+ v

2m+β,p,Ω

k=1

|λ|

2m+β−μ

−θ

v

+θ,p,Ω

]

≤ C[f

β,p,Ω

k=1

(g



2m+β−μ

,p,Ω

+ |λ|

2m+β−μ

−θ

g



θ,p,Ω

)]. 

We conclude the section with the following simple result of perturbation:

Theorem 2.8. Assume that the assumptions of Theorem 2.7 are satisﬁed. Let β



∈

R, β



<β, P ∈L(N

2m+β



(Ω),N

(Ω)) and let λ ∈ C. We consider the problem



λu(x) − A(x, ∂

)u(x) − Pu(x)=f (x),x∈ Ω,



,∂

)u(x



) − g



)=0,x



∈ ∂Ω,k=1,...,m.

(2.25)

Then there exist θ

∈ (π/2,π), r

,C ∈ R

, such that, if |λ|≥r

, |Arg(λ)|≤θ

f ∈ N

(Ω),fork =1,...,m g

∈ N

2m+β−μ

(Ω), (2.18) has a unique solution u

belonging to N

2m+β

(Ω).Moreover,anestimatelike(2.19) holds.

Proof. Obviously, we may assume that 2m + β



≥ β. By the well-known continu-

ation method, it suﬃces to prove an a priori estimate. So, let u ∈ N

2m+β

(Ω) be a

286 D. Guidetti

solution of (2.25). We ﬁx μ ∈ C, such that |μ| = |λ|,andRe(μ) ≥ 0. Then (2.25)

canbewrittenintheequivalentform



μu(x) −A(x, ∂

)u(x)=f(x)+(μ − λ)u(x)+Pu(x),x∈ Ω,



,∂

)u(x



) − g



)=0 x



∈ ∂Ω,k=1,...,m,

(2.26)

so that, if |λ| is suﬃciently large, we obtain from Theorem 2.7 the estimate

|λ|u

β,p,Ω

+ |λ|

β−β



u

2m+β



,p,Ω

+ u

2m+β,p,Ω

≤ C

(f

β,p,Ω

k=1

g



2m+β−μ

,p,Ω

+ |λ|

2m+β−μ

−θ

g



θ,p,Ω

+ u

2m+β



,p,Ω

+ |λ − μ|u

β,p,Ω

(2.27)

If θ

−

is suﬃciently small, it is possible to choose μ in such a way that C

|λ−μ|

|λ|

≤

.If|λ| is so large that C

|λ|



−β

≤

, from (2.27) we immediately deduce (3.10).



Remark 2.9. In case f = 0, the following more general estimate holds: if β



≤ α ≤

2m + β,

u

α,p,Ω

≤ C|λ|

α−β

−1

k=1

(g



2m+β−μ

,p,Ω

+ |λ|

2m+β−μ

−θ

g



θ,p,Ω

). (2.28)

This follows from Theorem 2.8 if β ≤ α ≤ 2m + β.Thecaseβ



≤ α<βfollows

from

u

α,p,Ω

= |λ|

−1

[A(., ∂

)+P ]u

α,p,Ω

≤ C|λ|

−1

u

2m+α,p,Ω

3. Parabolic problems

We start with the autonomous parabolic system

⎧

⎪

⎨

⎪

⎩

u(t, x)=A(x, ∂

)u(t, x)+Pu(t, x)+f(t, x), (t, x) ∈ [0,T] ×Ω,



,∂

)u(t, x



) − g

(t, x



)=0,k=1,...,m, (t, x



) ∈ [0,T] × ∂Ω

u(0,x)=u

(x),x∈ Ω,

(3.1)

under the following assumptions:

(I1) Ω is an open bounded subset of R

, with smooth boundary ∂Ω,

A(x, ∂

|s|≤2m

(x)∂

is a linear partial diﬀerential operator of order 2m (m ∈ N),and,fork =

1,...,m

(x, ∂

|s|≤μ

(x)∂

is a linear partial diﬀerential operator, all with smooth coeﬃcients in Ω.

On Linear Elliptic and Parabolic etc. 287

(I2) The conditions (H1)–(H3) are fulﬁlled.

(I3) β ∈ (μ + p

−1

− 2m, ν + p

−1

),withμ := max

1≤k≤m

, ν := min

1≤k≤m

(I4) P ∈L(N

2m+β



(Ω),N

(Ω)),forsomeβ



<β.

We introduce the following operator in N

(Ω):



D(A):={u ∈ N

2m+β

(Ω) : B

(., ∂

|∂Ω

=0,k =1,...,m},

Au := A(., ∂

)u + Pu.

(3.2)

By Theorem 2.8, A is a sectorial operator (with not dense domain) in N

(Ω). We

use θ

with the same meaning as in the statement of Theorem 2.8. Then we can

construct an analytic semigroup {T (t):t>0} (not strongly continuous in 0) in

a standard way (see [9]): we ﬁx a piecewise C

path γ, describing {λ ∈ C \{0} :

|Arg(λ)| = θ

, |λ|≥r

}∪{λ ∈ C \{0} : |Arg(λ)|≤θ

, |λ| = r

},withr

as in the

statement of Theorem 2.8, θ

∈ (π/2,θ

), γ oriented from ∞e

−iθ

to ∞e

iθ

,and

set, for t>0,

T (t):=

2πi



λt

(λ −A)

−1

dλ. (3.3)

If T ∈ R

,fort ∈ (0,T] T (t) satisﬁes the estimate

T (t)f

β,p,Ω

+ tT (t)f

2m+β,p,Ω

≤ C(T )f 

β,p,Ω

. (3.4)

We consider also the operator

(−1)

(t):=



T (s)ds =

2πi



λt

−1

(λ −A)

−1

dλ. (3.5)

This operator satisﬁes, for T ∈ R

,anestimateoftheform

T

(−1)

(t)f

β,p,Ω

+ tT

(−1)

(t)f

2m+β,p,Ω

≤ C(T )tf 

β,p,Ω

. (3.6)

It is convenient to ﬁx β



∈ (β



,β) ∩ (μ + p

−1

− 2m, ν + p

−1

), and consider the

analogues of A, T (t)andT

(−1)

(t)inN



(Ω). We shall indicate these operators

with A



, T



(t)andT

(−1)



(t) respectively. Clearly they satisfy estimates of the

form (3.4) and (3.6), with β



replacing β.Ofcourse,A, T(t)andT

(−1)

(t)arethe

parts of A



, T



(t)andT

(−1)



(t)inN

(Ω).

We pass to consider nonhomogeneous boundary conditions. Let i ∈{1,...,m}

and take g ∈ N

2m+β−μ

(Ω). We indicate with N

(λ)g the solution of system (2.25),

in case f =0,g

=0ifk = i, g

= g.Ift>0, we set

(t)g := (2πi)

−1



λt

(λ)gdλ. (3.7)

We have:

Lemma 3.1. Let T ∈ R

, k ∈{1,...,m}, g ∈ N

2m+β−μ

(Ω), β



≤ α ≤ 2m + β,

θ ∈ (0,p

−1

),withν + θ ≥ β.Then: