laplace transform methods for evolution equations

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LAPLACE TRANSFORM METHODS FOR EVOLUTION EQUATIONS BORIS BA UMER and FRANK NEUBRANDER * Louisiana State University, Baton Rouge, USA

Laplace transform theory of Banach space valued functions and the eld of evolution equations presently o er many research problems which are of general interest in mathematical analysis. Although there is a longstanding relationship between the two elds, it was not until recently that the basic transform principles for Banach space valued functions could be formulated in a satisfactory way. In Section 1 we will give an introduction to some aspects of the mathematical theory of the Banach space valued Laplace transform. In contrast to semigroup methods, where the spectral and regularity assumptions on the characteristic operator are rather restrictive, the Laplace transform method applies to wellposed and illposed linear evolution problems alike. No a priori spectral assumptions have to be made, the characteristic operators might not be closed or closable, might not be densely de ned and could be multivalued. The only assumptions needed are linearity and relative closedness. This class of operators, which includes any sum, product or limit of closed operators, will be discussed in Section 2. Besides new theoretical concepts and a uni ed approach to many aspects of semigroup theory, Laplace transform theory provides analytic tools for many of the basic problems in the theory of evolution equations. We will demonstrate this in Section 3. There we will investigate existence, uniqueness and regularity of solutions of linear evolution equations. The Laplace transform part of this paper builds on results of W. Arendt (1987, 1991), W. Arendt, M. Hieber and F. Neubrander (1994), B. Hennig and F. Neubrander (1993), F. Neubrander (1994, 1994b) and M. Sova (1979, 1980). In applying the Laplace transform methods to evolution equations we follow ideas developed in papers by Yu. I. Lyubich (1966), M. Hieber, A. Holderrieth, and F. Neubrander (1991), B. Baumer and F. Neubrander (1994), and F. Neubrander (1994). * supported by Louisiana Education Quality Support Fund 1

1. THE LAPLACE TRANSFORM The Laplace transform has a long history, dating back to L. Euler's paper `De Constructione Aequationum' from 1737. Since then it has been widely used in mathematics, in particular in ordinary di erential, di erence and functional equations. An informative description of the contributions of mathematicians like Euler, Lagrange, Laplace, Fourier, Poisson, Cauchy, Abel, Liouville, Boole, Riemann, Pincherle, Amaldi, Tricomi, Picard, Mellin, Borel, Heaviside, Bateman, Titchmarsh, Bernstein, Doetsch, Widder and many others can be found in two historical surveys by M. Deakin (1981, 1982). The article `Studio della trasformazione di Laplace e della sua inversa dal punto di vista dei funzionali analitici' by the Italian mathematician Sylvia Martis in Biddau (1933) provides an interesting overall account of the theory prior to D.V. Widder's and G. Doetsch's contributions. D.V. Widder's books `The Laplace Transform' (1946) and `An Introduction to Transform Theory' (1971) as well as G. Doetsch's `Theorie und Anwendung der Laplace-Transformation' (1937) and his monumental three volumes of the `Handbuch der Laplace-Transformation' (1950, 1955, 1956) remain useful and modern works. They are still among the best introductions to the subject. A rst look at Laplace transform theory for functions with values in a Banach space X is contained in E. Hille's monograph `Functional Analysis and Semi-Groups' from 1948. In trying to extend the results of the classical numerical theory of the Laplace transform Z1 e?tf (t) dt ( > 0) (L) r() = 0 to Banach space valued functions, E. Hille remarks on several occasions that this can be done if X is re exive, but not in general. In fact, it was shown by S. Zaidman (1960) (see also W. Arendt (1987) or Theorem 1.6 below) that some important results of classical Laplace transform theory extend to a Banach space X if and only if X has the RadonNikodym property (such as re exive spaces). This \negative" result led to the development of special Laplace transform theories on arbitrary Banach spaces for functions with additional algebraic properties; the most prominent one being the theory of C0 -semigroups. There, the link between the generator A and the semigroup T is given via the Laplace transform Z1 ? 1 e?tT (t)x dt (x 2 X ): (I ? A) x = 0 The crucial algebraic property which made it possible to extend classical transform results to this abstract setting is the semigroup law T (t + s) = T (t)T (s), (t; s  0). In the revised version of E. Hille's monograph from 1957, E. Hille and R.S. Phillips comment as follows on the algebraic methods used to solve evolution equations: 2

\Thus in keeping in spirit of the times the algebraic tools now play a major role and are introduced early in the book; they lead to a more satisfactory operational calculus and spectral theory... On the other hand, the Laplace-Stieltjes transform methods, used by Hille for such purposes, have not been replaced but rather supplemented by the new tools."

In more recent monographs on semigroup theory the role of the Laplace transform is reduced to a footnote at best and almost every proof is based on the semigroup property T (t + s) = T (t)T (s). The major disadvantage of the \algebraic approach" to linear evolution equations becomes obvious if one compares the mathematical theories associated with them (for example, semigroup theories, cosine families, the theory of integro-di erential equations, etc.). It is striking how similar the results and techniques are. Still, without a Laplace transform theory for arbitrary Banach space valued functions at hand, every type of linear evolution equation required its own theory because the algebraic properties of the operator families involved change. In recent years, in search of a general analytic principle behind all these theories, the Laplace transform has been reconsidered for that reason. The rst major result in this direction is due to M. Sova (1979). His analytic Laplace representation theorem (see Theorem 1.16 below) is behind every generation result for analytic solution families of linear evolution equations. The real breakthrough came in 1987 with W. Arendt's paper on `Vector valued Laplace transforms and Cauchy problems'. He showed that Widder's Theorem, one of the main theoretical results of the classical theory, extends to arbitrary Banach spaces if the Laplace transform is replaced by the Laplace-Stieltjes transform (LS )

r() =

Z1 0

e?t dF (t) = lim !1

Z 0

e?t dF (t)

( > 0)

where F 2 Lip0 ([0; 1); X ); i.e., F (0) = 0 and kF kLip := supt;s0 kF (tj)t??sFj(s)k < 1: In order to see how the Laplace-Stieltjes transform relates to the Laplace transR t form, let f 2 L1 ([0; 1); X ). Then F (t) := 0 f (s) ds is in Lip0 ([0; 1);RX ). In this case the transform reduces to the Laplace transform; i.e., 01 e?t dF (t) = R 1 eLaplace-Stieltjes ?tf (t) dt: We recall that a space X is said to have the Radon-Nikodym property if 0 the fundamental theorem of calculus holds; i.e., if any absolutely continuous function is the antiderivative of a Bochner integrable function. Hence, if a space X has the RadonNikodym property, then the Laplace-Stieltjes transform reduces to the Laplace transform. Otherwise it is a proper generalization. As it turned out, it is the generalization needed in dealing with Banach space valued functions. 3

Clearly, the Laplace transform can also be considered for f 2 Lp ((0; 1); X ) and the Laplace-Stieltjes transform for functions F of bounded p-variation or semivariation (see, for example, W. Arendt (1993), J. Pruss (1993), P. Vieten and L. Weis (1993), and L. Weis (1993)). However, we will restrict the discussion of L and LS to the case where f 2 L1 ([0; 1); X ) and F 2 Lip0 ([0; 1); X ). These special cases are relatively easy to deal with and have immediate and important applications to evolution equations. The key to Laplace transform theory is the Riesz-Stieltjes representation of bounded linear operators from L1 (0; 1) into X . The proof of this well known observation is taken from F. Neubrander (1994). We will denote by [0;t] the characteristic function of the interval [0; t].

Theorem 1.1 (Riesz-Stieltjes Representation). There exists an isometric iso1 morphism 0 ([0; 1); X ) ! L(L (0; 1); X ) given by RS (F ) = T where Tg := R 1 g(t) dF (Rt)Sfor: Lip all continuous functions g 2 L1 (0; 1) and T[0;t] = F (t), (t  0). 0 PROOF. Let F 2 Lip0 ([0; 1); X ) and let D be the linear span of the functions with compact support which are either continuous or of bounded variation. For such functions R 1 we can de ne an operator TF on the subspace D by TF g := 0 g(s) dF (s) and it is easy to see thatRfor such g we have that kTF gk  kF kLip kgk1 . It follows from F (0) = 0 that TF [0;t] = 0t dF (s) = F (t) for all t  0. Since D is dense in L1 (0; 1), there exists a unique extension of TF in L(L1 (0; 1); X ), denoted by the same symbol. Clearly, kTF k  kF kLip and, if TF = 0, then TF [0;t] = F (t) = 0 for all t  0. This shows that the linear mapping RS : F ! TF is one-to-one. Moreover, if g is a continuous function in L1 (0; 1), then g = limt!1 g[0;t] in L1 (0; 1): Thus,

TF g = tlim !1 TF (g[0;t] ) = tlim !1

Zt 0

g(s) dF (s) =

Z1 0

g(s) dF (s):

To prove that RS is onto, let T 2 L(L1 (0; 1); X ). De ne F (t) := T[0;t] : Then kF (t) ? F (s)k = kT[s;t] kR kT kk[s;t] k1 = kT kjt ? sj: So F 2 Lip0 ([0; 1); X ) ; kF kLip  kT k; and T[0;t] = F (t) = 0t dF (s) = TF [0;t] : Since the characteristic functions f[0;t] ; t  0g are total in L1 (0; 1), we conclude that T = TF and kTF k = kF kLip for all F 2 Lip0 ([0; 1); X ):

}

4

The Riesz-Stieltjes Representation Theorem is crucial for the following reason. Considering the Laplace-Stieltjes transform LS we want to know how properties of F and its transform r a ect each other. Observe that r() = TF e? and F (t) = TF [0;t] for all ; t > 0, where e? denotes the exponential function t ! e?t. Thus, if one knows the function F , then the operator TF is determined on the set of characteristic functions, which is total in L1 (0; 1). Hence TF , and in particular TF e? = r(), is completely determined. Conversely, any information on r() for  > 0 translates into information on TF on the set of exponential functions, which is also total in L1 (0; 1) (see below). Thus, r determines the properties of TF and, in particular, of TF [0;t] = F (t); (t  0).

Lemma 1.2. Let n (n 2 IN) be a sequence of distinct complex numbers with Ren 

> 0 for some > 0. If (1:1)

N X n=1

? 1j ! 1 1 ? jjn + 1j n

as N ! 1, then the exponential functions e?n are total in L1 (0; 1). PROOF. a) Let < n0 2 IN. Clearly, the monomials t ! n1 tn=n (n 2 IN0 ) are total in C [0; 1] and thus in L1 (0; 1). Since  : L1 (0; 1) ! L1 (0; 1) de ned by (g)(t) := n0 e?n t g(e?n t ) is an isometric isomorphism which maps the monomials onto the set E of exponential functions e?n (n  n0 ), it follows that E is total in L1 (0; 1). b) Let H be the closure of the linear span of the functions e?n (n 2 IN) in L1 (0; 1). If H contains the exponential functions e?n (n  n0 ), the claim follows from a). Assume that there exists m  n0 such that e?m 62 H. By the Hahn-Banach theorem there exists  2 L1 (0; 1) suchR that (H)  0 and (e?m) 6= 0. Let 0 < < . The function  ! () := (e?) = 01 e?t(t) dt is analytic and bounded for Re  and the function  : z ! 11+?zz + is a conformal map between the unit disk and the halfplane fRe > g. De ne h(z ) := ((z )) and n := nn ?? ?+11 . Then h is analytic and bounded on the unit ?1j = 1 ? jn j disk and h(n ) = (n ) = (e?n ) = 0 for all n 2 IN. Let an := 1 ? jjnn ?? +1 j j  ? 1 j n and bn := 1 ? jn +1j . Then an = jn + 1j  jn ? + 1j2 ? jn ? ? 1j2  jn + 1j + jn ? 1j bn jn ? + 1j jn ? + 1j + jn ? ? 1j jn + 1j2 ? jn ? 1j2 jn + 1j + jn ? 1j : 1j  Ren ?  = j j?n + + 1j Re j ? + 1j + j ? ? 1j 0

0

0

0

n

n

n

n

5

Since jn + 1j  jn ?  1j and jn ? 1j  jn + 1j ? 2 it follows that an  Ren ?  jn + 1j + jn ? 1j bn Ren jn ? + 1j + jn ? ? 1j n ?  2jn + 1j ? 2  Re Ren 2jn + 1j  (1 ? )(1 ? 1 +1 ) > 0:

P

P

P

Since Nn=1 bn ! 1 as N ! 1, we obtain that Nn=1 an = Nn=1 1 ? jn j ! 1 as N ! 1. Thus h() = 0 for jj < 1 (see, for example, W. Rudin (1987), Theorem 15.23). But this implies that () = 0 for Re > contradicting (m) 6= 0. Thus H = L1 (0; 1).

}

Typical examples of sequences satisfying (1.1) are equidistant sequences n = a + nb (a; b > 0) or convergent sequences n ! ; (Re > 0). Examples of sequences not satisfying (1.1) are given by n = n ; ( > 1) and n = 1 + in. Combining Lemma 1.2 with the Riesz-Stieltjes Representation 1.1 we obtain immediately the following uniqueness theorem for the Laplace-Stieltjes transform (see Yu-Cheng Shen (1947) or G. Doetsch (1950), Satz 2.9.6).

Corollary 1.3 (Uniqueness). Let F 2 Lip0 ([0; 1); X ). If R01 e?nt dF (t) = 0 for a sequence of complex numbers n satisfying (1.1), then F  0. Next we determine the range of the Laplace-Stieltjes transform. Let F 2 Lip0 ([0; 1); X ). R 1 ? t Then r() = 0 e dF (t) is analytic and (1:2)

r(n) () =

Z1 0

e?t(?t)ndF (t)

for n 2 IN0 and Re > 0. For a proof, see for example, G. Doetsch (1950), Satz 3.2.1 or W. Arendt, M. Hieber and F. Neubrander (1995). To obtain an estimate for the Taylor coecients of r let x 2 X  and de ne fx (t) := hF (t); x i. Then fx 2 Lip0 ([0R ; 1)) and kfx kLip  kF kLip kx k: It follows that fx is di erentiable a.e. and fx (t) = 0t fx0  (s) ds for all t  0. Clearly, kfx0  (t)k  kF kLip kx k a.e. and, by (1:2), Z1 1 1 k +1 ( k )  k +1 h k! r (); x i =  k! e?t(?t)k fx0  (t) dt: 0 6

R

Since n+1 01 e?t tnn! dt = 1 ( > 0, n 2 IN0 ) we obtain jhk+1 k1! r(k) (); x ij  kF kLip kx k for all x 2 X  and  > 0. Thus (1:3)

krkW := sup sup kk+1 k1! r(k)()k  kF kLip : k2IN0 >0

Let CW1 ((0; 1); X ) := fr 2 C 1 ((0; 1); X ) : krkW < 1g. Then the Widder space CW1 ((0; 1); X ) is a Banach space and we obtain the following crucial result due to D.V. Widder (1936) for the numerical case and due to W. Arendt (1987) for the vector-valued case. The following proof is a modernized version of Widder's classical proof. It is taken from F. Neubrander (1994).

Widder's Theorem 1.4. The Laplace-Stieltjes transform LS : Lip0 ([0; 1); X ) ! CW1 ((0; 1); X ) is an isometric isomorphism. PROOF. It follows from (1:3) that LS maps Lip0 ([0; 1); X ) into CW1 ((0; 1); X ) and that kLS (F )kW = krkW  kF kLip . It follows from Corollary 1.3 that LS is one-to-one. It remains to be shown that LS is onto. Let r 2 CW1 ((0; 1); X ). De ne Tk 2 L(L1 (0; 1); X ) by  k k+1  k  Z1 1 k r(k) t dt: f (t)(?1) k! t Tk f := 0 Then kTk k  krkW for all k 2 IN0 and

Tk e? = (?1)k k1!

Z1 0

e?t( kt )k+1 r(k) ( kt ) dt:

Using change of variables and integration by parts one obtains (with some work; for details, see D.V. Widder (1946) or B. Hennig and F. Neubrander (1993)),

Tk e? = u

Z 1 1  k k+1 0

k! u

where h(t) := 1t r( 1t ) and 1=u := : The functions

e

?kt k u t h(t) dt

 k k+1 ?kt 1 e u tk (t  0) k (t) := k! u 7

are \approximate identities"; i.e., they are positive, their L1 -norm is one and for all  > 0 R and all open intervals I  [0; 1) containing u we have t=2I k (t) dt <  for all suciently large k. Since h is continuous in u > 0 and khk1  krkW , it follows from (1:4)

k

Z1 0

k (t)h(t) dt ? h(u)k = k

Z1 0

k (t) (h(t) ? h(u)) dtk

 2khk1

Z

R that 01 k (t)h(t) dt ! h(u) as k ! 1. Hence,

t=2I

k (t) dt + sup kh(t) ? h(u)k t2I

Tk e? ?! uh(u) = r( u1 ) = r() as k ! 1. By Lemma 1.2, the set of exponential functions fe? :  > 0g is total in L1 (0; 1). Thus, by the Theorem of Banach-Steinhaus, there exists a bounded operator T 2 L(L1 (0; 1); X ) with kT k  krkW such that Tk f ! Tf for all f 2 L1 (0; 1). In particular, r() = klim T e = Te? : !1 k ?

It follows from the Riesz-Stieltjes Representation Theorem 1.1R that there exists F 2 Lip0 ([0; 1); X ) with kF kLip = kT k  krkW such that Tg = 01 g(t) dF (t) for all continuous g 2 L1 (0; 1): Hence, for all  > 0;

r() = Te? =

Z1 0

e?t dF (t):

}

Closely related to Widder's Theorem is the following inversion of the Laplace transform which has been widely used in applications to di erential equations.

Theorem 1.5 (Post-WidderRInversion). Let f 2 L1loc([0; 1); X ). Assume that there exists ! 2 IR such that r() = 01 e?tf (t) dt exists for all  > !. Then for all continuity points t > 0 of f ,

k1 f (t) = klim ( ? 1) !1 k!

8

 k k+1 t

r(k) ( kt ):

PROOF. Since we will make no use of this result in the applications below, we will verify it only for bounded continuous functions f . For a proof of the general case see, for example, G. Doetsch (1950), Satz 8.2.1 or W. Arendt, M. Hieber and F. Neubrander (1995). If f is bounded and continuous, then it follows from (1.4) that

 k k+1 Z 1 ?ks 1 f (t) = klim e t sk f (s)ds: !1 k! t 0

}

Now the statement follows from (1:2).

If F : [a; b] ! X is Lipschitz continuous and g : [a; b] ! C Rcontinuous, then g and F b g(t) dF (t) = F (b)g(b) ? are Riemann-Stieltjes integrable with respect to each other and a R F (a)g(a) ? ab F (t) dg(t): Thus, for F 2 Lip0 ([0; 1); X ) and  > 0 the Laplace-Stieltjes transform equals the Laplace-Carson transform, or -multiplied Laplace transform; i.e.,

r() =

Z1 0

e?t dF (t) = 

Z1 0

e?tF (t) dt:

Combining this with the Post-Widder inversion formula we obtain that the inverse LaplaceStieltjes transform L?S 1 : CW1 ((0; 1); X ) ! Lip0 ([0; 1); X ) is given by k1 F (t) = klim ( ? 1) !1 k!

 k k+1  r() (k) t



=k=t

= klim !1

k X j =0

(?1)j 1

 k j

j! t

r(j )

k

t :

Studying whether or not Widder's original proof of the surjectivity of LS extends to Banach space valued functions, E. Hille (1948) remarks that Widder's \ ... argument seems to hold for any abstract space in which bounded sets are weakly compact, but it is not clear in the present writing whether or not his conditions are always sucient for the existence of the representation."

This statement was made precise by S. Zaidman (1960) and W. Arendt (1987). The result is the following characterization of Banach spaces with the Radon-Nikodym property.

9

Theorem 1.6 (Riesz Representation). Let X be a Banach space. The following are

equivalent. R (i) The antiderivative operator I : f ! F , F (t) := 0t f (s) ds is an isometric isomorphism between L1 ([0; 1); X ) and Lip0 ([0; 1); X ). R (ii) The Riesz operator R : f ! Tf , Tf g := 01 g(t)f (t) dt is an isometric isomorphism between L1 ([0; 1); X ) and L(L1 (0; 1); X ). R (iii) The Laplace transform L : f ! r, r() := 01 e?t f (t) dt is an isometric isomorphism between L1 ([0; 1); X ) and CW1 ((0; 1); X ).

R

PROOF. As shown above, the Riesz-Stieltjes operator RS : F ! TF , TF g = 01 g(t)dF (t) is an isometric isomorphism between Lip0 ([0; 1); X ) and L(L1 (0; 1); X ) and the LaplaceR Stieltjes transform LS : F ! r, r() = 01 e?t dF (t) is an isometric isomorphism between Lip0 ([0; 1); X ) and CW1 ((0; 1); X ). Now the statement follows from the fact that L = LS  I and R = RS  I on L1 ([0; 1); X ). }

Let X = L1 [0; 1) and f (t) := [0;t] (t  0). Since kf (t) ? f (s)k = 1 for all t 6= s, the range of f does not contain a countable dense set. Thus, by Pettis' Theorem (see E. Hille and R.S. Phillips (1957), Theorem 3.5.3), f is not measurable. However, f is Riemann integrable. In fact,

F (t) := (R)

Zt 0

f (s) ds = jlim j!0

n X i=1

[0;i ] (si ? si?1 ) = (t ? )[0;t]

for all t  0. It is easy to see that F is Lipschitz continuous. However, although F is Lipschitz continuous and an inde nite Riemann integral, F is nowhere di erentiable. Again, it is easy to check that h1 (F (t + h) ? F (t)) does not converge in X as h ! 0 for any t  0. Since F 2 C0 [0; 1)  L1 [0; 1), it follows that C0 and L1 -spaces do not have the Radon-Nikodym property. Since L1 [0; 1)  L(Lp [0; 1)) for 1  p  1 it follows that L(Lp [0; 1)) does not have the Radon-Nikodym property. In fact, there seems to be no known in nite dimensional Banach space X for which L(X ) has the Radon-Nikodym property. 10

The function F : [0; 1) ! c0 , F (t) := ( n1 sin(nt))n2IN is another example of a Lipschitz continuous function which is nowhere di erentiable. Thus c0 and any Banach space containing it does not have the Radon-Nikodym property. Examples of spaces with the Radon-Nikodym property are re exive Banach spaces and spaces with a boundedly complete Schauder basis (like `1 ). For a proof of this statement and a thorough discussion of the Radon-Nikodym property, see J. Diestel and J.J. Uhl (1977). As we will see in Section 3, the inversion formulas for the Laplace and LaplaceStieltjes transforms play a fundamental role in applications. Compared to the Post-Widder inversion, it is remarkable that in the following inversion formula (due to G. Doetsch (1950), Satz 8.1.1) only the values of r(n) for large n 2 IN are needed and that the convergence is uniform for all t  0.

1.7 (Phragmen-Doetsch Inversion). Let F 2 Lip0 ([0; 1); X ) and r() = RTheorem 1 e?t dF (t). Then 0 1 (?1)j +1 X tnj r(nj )k  c krkW e kF (t) ? j! n j =1

for all t  0 and n 2 IN, where c  1:0159:::. PROOF. By the Riesz-Stieltjes Representation Theorem 1.1 and Widder's Theorem R 1 1 1.3, there exists T 2 L(L (0; 1); X ) such that r() = 0 e?t dF (t) = Te? ( > 0), T[0;t] = F (t) (t  0) and kT k = krkW = kF kLip . Thus 1 (?1)j +1 1 (?1)j +1 X X tnj tnj e?nj k1 : kF (t) ? e r ( nj ) k  k T k k  e [0;t] ? j =1 j ! j =1 j !

P

j+1

(?1) tnj De ne pn;t := 1 ? e?ente?n  = 1 j =1 j ! e e?nj : Then ( )

k(0;t] ? pn;tk1 =

Zt 0

jpn;t (s) ? 1j ds +

Z1 t

jpn;t(s)j ds =



Zt 0

e?en(t?s) ds +

Z1 t

n(t?s)

1 ? e?e



ds

Ze 1 Z 1 1 ? e?u Z 1 1 ? e?u Z11 1 1 1 ? u ? u =n u e du + n 0 u du  n 1 u e du + 0 u du 1 R for all t  0 and n 2 IN. Now the claim follows from the fact that 11 u1 e?u du + R 1 1?e?u du = ?2Ei(?1) +  1:0159::: where Ei(z) is the exponential integral and u 0

is Euler's constant; see N.N. Lebedev (1972), 3.1. } nt

11

The Phragmen-Doetsch inversion of the Laplace-Stieltjes transform shows that any r 2 R CW1 ((0; 1); X ) (and thus also the unique F 2 Lip0 ([0; 1); X ) with r() = 01 e?t dF (t) ( > 0)) is uniquely determined by the sequence xn := r(n) (n 2 IN). We ask now the converse question: i.e. to characterize those sequences (xn )n2IN in a Banach space X for which there exists r 2 CW1 ((0; 1); X ) (or equivalently F 2 Lip0 ([0; 1); X )) such that

xn = r(n) =

Z1 0

e?t dF (t) =

Z1 0

tn dH (t)

(where H (t) := ?F (? ln t)). This special \Hausdor moment problem" can be solved, among others, by adapting H. Hahn's (1927) method to solve the moment problem (see, for example, IV.5 in K. Yosida (1971) or Theorem 2.7.6 in E. Hille and R.S. Phillips (1957)).

Theorem 1.8 (Moment Theorem). Let M > 0 and xn 2 X (n 2 IN). The following

are equivalent. (i) There exists T 2 L(L1 (0; 1); X ) with kT k  M and xn = T (e?Rn) for all n 2 IN. (ii) There exists F 2 Lip0 ([0; 1); X ) with kF kLip  M and xn = 01 e?nt dF (t) for all n 2 IN. (iii) There exists r 2 CW1 ((0; 1); X ) with krkW  M and xn = r(n) for all n 2 IN. P P (iv) k ni=1 i xi k  M k ni=1 i e?ikL for all n 2 IN and i 2 C. 1

PROOF. The equivalence of (i),(ii) and (iii) follows from the Riesz-Stieltjes represenP tation and Widder's Theorem. The implication (i))(iv) follows from k ni=1 i xi k  kT k k Pni=1 ie?ikL . P P To prove (iv))(i), let z = ni=1 i e?i 2 spanfe?i ; i 2 INg =: E . If z = ni=1 i e?i = Pm  e , then k Pn  x ? Pm  x k = k Pl x k  M k Pl e k = j ?j j =1 i=1 i i j =1 j j k=1 k k k=1 k ?k L P P n m M k i=1 i e?i ? j =1 j e?j kL = 0. Thus we can de ne on E an operator T by T (z ) := Pn  x , where z = Pn  e . Now (i) follows from kT (z)k  M kzk and the density L i=1 i i i=1 i ?i of E . } 1

1

1

1

We remark that any r 2 CW1 ((0; 1); X ) has a Laplace-Stieltjes representation and thus an analytic extension for Re > 0 which we denote by the same symbol. Widder's growth conditions \sup>0 kk+1 k1! r(k) ()k  M " translate via Cauchy's Integral Theorem to the following complex conditions (see also M. Sova (1980)). 12

Theorem 1.9 (Complex Widder Conditions). Let M > 0 and let r : fRe > 0g ! X be an analytic function with lim!1 kr()k = 0 and supRe> kr()k < 1 for all > 0. The following are equivalent. (i) r 2 CW1 ((0; 1); X ); with krkW  M: (ii) For some, or equivalently, for all k0 2 IN0 ,



1 ( k ) k +1

sup  k! r ()

 M for all k  k0 : >0

(iii) For some, or equivalently, for all k0 2 IN0 ,

1 Z 1 r( + it)

dt  M for all k  k0 : sup sup

2

>0 s>0 ?1 (1 ? ist)k+2

PROOF. Clearly, (i) implies (ii). First we show that (ii) implies (i). This follows from the observation that sup kk +1 k1 ! r(k ) ()k  M for some k0 2 IN implies that

0

>0

0

0

sup kk+1 k1! r(k) ()k  M >0

for all 0  k  k0 ? 1. In order to prove this statement de ne

k+1 Z 1 ( ? 1) rk () := k! ( ? )k r(k ) () d ( > 0):  These integrals exist absolutely and 0



Z 1 ( ? )k M k 0!

k +1

d = M (k0 k??kk? 1)! : krk ()k  k!    0

0

Since rk(k ?)1 = r(k ) it follows that (rk ?1 ? r)(k ) = 0. Thus rk ?1 ? r is a polynomial with lim!1 rk ?1 () ? r() = 0. Hence r = rk ?1 and 0

0

0

0

0

0

0

0

k! kr(k)()k = krk ?k?1()k  M k +1  0

for all k < k0 . Next we show the equivalence of (ii) and (iii). Let  > > 0; n 2 IN and let ?n be the counterclockwise oriented path consisting of the line segments ?n;1 := + i[?n; n]; ?n;2 := 13

+ n + i[?n; n]; ?n;3 := [ ; + n] + in; ?n;4 := [ ; + n] ? in. By Cauchy's Integral Theorem we obtain r(k+1) () =

r(z ) dz (k 2 IN ): (k + 1)! Z 0 2i ?n (z ? )k+2

For n ! 1 the boundedness of r for Re  implies that

r(k+1) () = ? (k + 1)!

Z

2i

Hence ( ? )k+2

r(z ) dz:

+iIR (z ? )k+2

r(z ) dz 1 r(k+1) () = (?1)k+1 ( ? )k+2 Z (k + 1)! 2i

+iIR (z ? )k+2 Z k+2 1 r( + it) dt: = (?1)k+1 ( ?2) ?1 ( + it ? )k+2 k+1 Z 1 r( + it) ( ? 1) dt; = 2 ?1 (1 ? ist)k+2

1 r (k+1) ()k  where s := ?1 . Thus (ii) implies (iii) and (iii) implies that k( ? )k+2 (k+1)! M for all > 0 and all k  k0 . Taking ! 0 we obtain that (iii) implies (ii). }

The following inversion theorem uses the fact that any r 2 CW1 ((w; 1); X ) is a Laplace-Stieltjes transform and therefore has an analytic extension to fRe > 0g which will be denoted by the same symbol.

Theorem 1.10 (Complex Inversion). The inverse Laplace-Stieltjes transform L?S 1 : CW1 ((0; 1); X ) ! Lip0 ([0; 1); X ) is given by

Z 1+in

t r() d; e LS?1 (r)(t) = F (t) = nlim !1 1?in 

where the limit is uniform on compact intervals.

14

PROOF. By the Riesz-Stieltjes Representation Theorem 1.1 and Widder's Theorem 1.3, R there exist F 2 Lip0 ([0; 1); X ) and T 2 L(L1 (0; 1); X ) such that r() = 01 e?t dF (t) = Te? ( > 0) and T[0;t] = F (t) (t  0). Then

Z 1+in e? Z 1+in r() t et  dk1 : e  dk  kT kk[0;t] ? kF (t) ? 1?in 1?in R Now it is an exercise to show that the functions 1+in et e? d converge towards the 1?in



characteristic function [0;t] in L1 (0; 1) as n ! 1 uniformly (in t) on compact intervals (see F. Neubrander (1994b)). }

Other consequences of the Riesz-Stieltjes Representation Theorem are Laplace transform versions of Trotter-type approximation theorems of semigroup theory (for further theorems of this type, see B. Hennig and F. Neubrander (1993) and F. Neubrander (1994a)).

Theorem 1.11 (Approximation Theorem). Let Fn 2 Lip0 ([0; 1); X ) with kFnkLip  R 1 M for all n 2 IN and rn () = 0 e?t dFn (t), ( > 0). The following are equivalent. (i) There exist a; b > 0 such that limn!1 rn (a + kb) exists for all k 2 IN0 . (ii) There exists F 2 Lip0 ([0; 1); X ) with kF kLip  M such that limn!1 rn () = R 1 e?t dF (t) uniformly (in  > 0) on compact sets.

0

(iii) limn!1 Fn (t) exists for all t  0. (iv) There exists F 2 Lip0 ([0; 1); X ) with kF kLip  M such that Fn(t) ! F (t) uniformly (in t  0) on compact sets. PROOF. By the Riesz-Stieltjes Representation Theorem there exist Tn 2 L(L1 (0; 1); X ) with kTn k  M such that Tn e? = rn () and Tn [0;t] = Fn (t) for all n 2 IN, all t  0 and all  > 0. Each of the statements imply that the operators Tn converge on a total subset of L1 (0; 1). By the theorem of Banach-Steinhaus there exists T 2 L(L1 (0; 1); X ) such that Tn ! T as n ! 1. Let b > 0. Then the set of characteristic functions b := f[0;t] : 0  t  bg and the set of exponential functions Eb := fe? : 1b    bg are compact in L1 (0; 1). Hence, the uniformly bounded sequence Tn converges uniformly on b and Eb (H.H. Schaefer (1980), Theorem III.4.5). Now the statement follows from Theorem 1.1. } 15

Applying vector-valued Laplace-Stieltjes transform theory to evolution equations, it is desirable to allow for Lipschitz functions with arbitrary exponential growth. This can be done by using the following shifting procedure.

Remark 1.12 (Laplace Transform of Functions with Exponential Growth). For w 2 IR let Lipw ([0; 1); XR ) be the space of all functions G : [0; 1) ! X with G(0) = 0 and kG(t) ? G(s)k  M st ewr dr for all 0  s  t and some constant M . Then Lipw ([0; 1); X ) is a Banach space with norm

kGkLip(w) := inf fM : kG(t) ? G(s)k  M

R

Zt s

ewr dr for all 0  s  tg;

and Iw de ned by Iw G(t) := 0t e?ws dG(s) is an isometric isomorphism between the spaces Lipw ([0; 1); X ) and Lip0 ([0; 1); X ). For w 2 IR let CW1 ((w; 1); X ) be the Banach space of all functions r 2 C 1 ((w; 1); X ) with norm krkW;w := sup sup k k1! ( ? w)k+1r(k)()k < 1: k2IN0 >w

The shift Sw r() := r( ? w) is an isometric isomorphism between CW1 ((0; 1); X ) and CW1 ((w; 1); X ). Hence, the Laplace-Stieltjes transform Sw LS Iw is an isometric isomorphism between Lipw ([0; 1); X ) and CW1 ((w; 1); X ) and all theorems mentioned so far in this section can be rephrased for these spaces. }

In general, it is often impossible to verify the Widder growth conditions sup k k1! r(k) ()k  ( ?M!)k+1 (k 2 IN0 )

>!

or the equivalent complex conditions (Theorem 1.10), whereas the growth of r in the whole complex halfplane can be estimated. In these cases one can use the following representation theorem. 16

Theorem 1.13 (Complex Representation Theorem). Let q : fRe > !  0g ! X be analytic and supRe>! kq()k < 1. Then, for all b > 0 there exists f 2 C ([0; 1); X ) R 1 f ( t ) b ? !t with supt>0 ke tb k < 1 such that q() =  0 e?tf (t) dt for Re > !.

R

+in q() 1 lim PROOF. For  > ! we de ne f (t) := 2i n!1 ?in et b d: Observe that the integral is absolutely convergent and that it converges uniformly for t 2 [0; m] (m 2 IN). Hence f 2 C ([0; 1); X ). Let ?;R be the path  + i(?1; ?R] [  + Rei[?  ;  ] [  + i[R; 1). It follows from Cauchy's integral theorem that the de nition of f is independent of  and that Z Z ?R 1 q (  ) 1 t e(+ir)t q((++irir)b) dr e b d = 2 f (t) = 2i ?1 ?;R  Z Z1 i i )t q( + Re ) i 1 1 (  + Re (+ir)t q ( + ir ) dr e Re d + + 2  e i b ( + Re ) 2 R ( + ir)b ? 2

2

2

2

for all R > 0: Hence, for all t  0,

Z 1 Z1 1 M M t kf (t)k   e e(+R cos )t d b +1 dr + 2 b  r R R ?  Z 1 et + M 1 et eRt cos d : M b Rb  Rb 0 2

2

2

Choosing R = 1t ; we obtain for all  > ! that kf (t)k  Ctb et for some C > 0. Hence, kf (t)k  Ctb e!t : For Re > ! we choose ! <  < Re and let ?R be the path consisting of the halfcircle  + Re?i[?  ;  ] and the line segment  + i[?R; R]. Then the residue theorem and Cauchy's theorem imply that 2

Z1 0

2

e?t f (t) dt =

Z1

Z +i1 q() et b d dt 2i

e?t 1

?i1 Z  + i 1 1 1 q() d = 21i ?i1Z  ?  b 1 1 1 0

= Rlim !1 2i = 1b q():

17

?R

 ?  b q() d

}

The following is an immediate consequence of the Complex Representation Theorem 1.13. We say that a function q is polynomially bounded for  2 if there exists a polynomial p such that kq()k  p(jj) for  2 . Similarly, q is said to be exponentially bounded if there exists ! 2 IR and M > 0 such that kq()k  Me!jj for all  2 .

Corollary 1.14. Let q be a function with values in X . The following are equivalent. (i) q is analytic and polynomially bounded for some half-plane fRe > !g. (ii) There R exists b  0 and an exponentially bounded f 2 C ([0; 1); X ) such that q() = b 01 e?tf (t) dt for some half-line  > !0 .

As another consequence of Theorem 1.13 we obtain the following representation result due to J. Pruss (1993).

Corollary 1.15. Let q : fRe > 0g ! X be analytic. If there exists M > 0 such that kq()k  M and k2 q0 ()k RM for Re > 0, then there exists a bounded function f 2 C ((0; 1); X ) such that q() = 01 e?tf (t) dt for Re > 0. PROOF. It follows from the Complex Representation Theorem 1.13 that there are functions fi 2 C ([0; 1); X ), (i 2 f0; 1g) and C > 0 such that kfi (t)k  Ct for t > 0,

q() =  for Re > 0. Hence

q0 () =

Z1 0

Z1 0

e?tf0 (t) dt; and q0 () = 

e?tf0 (t) dt ? 

Integrating by parts yields

Z1



0

e?t

hZ t 0

Z1 0

Z1 0

e?ttf0 (t) dt =

i

f0 (s) ds ? tf0 (t) dt = 

Z1 0

e?tf1 (t) dt

Z1

e?t

0

Zt 0

e?t f1 (t) dt:

f1 (s) ds dt:

R

R

Since the Laplace transform is one-to-one, it follows that tf0 (t) = 0t f0 (s) ds ? 0t f1 (s) ds: Thus f0 2 C 1 ((0; 1); X ) and tf00 (t) = ?f1 (t): Therefore kf00 (t)k  C for all t > 0 and

q() =  if Re > 0.

Z1 0

e?tf0 (t) dt =

18

Z1 0

e?tf00 (t) dt

}

We conclude this section with M. Sova's characterization of those functions q : (!; 1) ! X which are Laplace transforms of analytic, exponentially bounded functions de ned on a sector. For a proof see M. Sova (1979), F. Neubrander (1989, 1994b), W. Arendt (1991) or W. Arendt, M. Hieber, F. Neubrander (1995). For 0 <   we denote by  the open sector  := frei : r > 0; ? < < g and by  its closure.

Theorem 1.16 (Analytic Representation Theorem). Let 0 <  2 ; ! 2 IR and q : (!; 1) ! X . The following are equivalent. (i) There exists an analytic function f :  ! X such that supz2 ke?!z f (z )k < 1 for R all 0 < < and q() = 01 e?tf (t) dt for all  > !: (ii) The function q is analytic in the sector ! + +  and sup2!+ for all 0 < < : 2



+2

k( ? !)q()k < 1

Moreover, for all 0 < < there exists a constant C > 0 such that

kzk f (k)(z)k  C e!Rez (j!jjzj + 1)k

(1:5) for all z 2  .

}

Let f and q be as in the previous theorem and assume that that k( ? !)q()k  M for all  2 ! +  +  . Let !0 < !. It is easy to see that if q is analytic for all Re > !0 , then there exists 0 < < such that q is analytic on !0 +  +  and k( ? !0 )q()k  M~ . Thus, by (1:5), for all k 2 IN0 , there exists a constant C > 0 such that 2

2

kf (k)(t)k  Ce!0t (j!0 j + 1t )k for all t > 0: This shows that the exponential growth (at in nity) of the function f and its derivatives is determined by the abscissa of analyticity of its Laplace transform q, i.e., inf f! 2 IR : q extends analytically to fRe > !gg.

19

2. EVOLUTION EQUATIONS, CAUCHY PROBLEMS AND CLOSED OPERATORS To simplify the demonstration of the Laplace transform method, we think of a linear evolution equation as a system of equations which can be written as an abstract Cauchy problem (ACP )

u0 (t) 2 Au(t) ; u(0) = x ; 0  t < T (T  1)

where A is a linear, but in general unbounded, not closed, not densely de ned and even multivalued operator on some Banach space X . It is clear from applications to partial di erential equations that A is in general unbounded with possibly nondense domain. As a typical example of a problem where A might be not closed, consider an inhomogeneous initial boundary value problem for an integro-di erential equation; i.e.,

w0 (t) = A0 w(t) +

Zt 0

b(t ? s)Bw(s) ds + f (t); w(0) = w0 ; Cw(t) = g(t):

Here A0 ; B are operators on a Banach space X0 ; C has its domain in X0 and maps in a Banach space Y ; w0 2 X0 , f 2 F  L1loc ([0; 1); X0 ), and g 2 G  L1loc ([0; 1); Y ) (where F and G are Banach function spaces). This equation can be written as an (ACP ) on X = X0  F  G with initial value x = (w0 ; f; g), and

0 A  01 0 0 @ A = b()B DF 0 A ; 0

0

DG

where DF ; DG denote the rst derivative operators on F and G , and

D(A) = f(x0 ; f; g) : x0 2 D(A0 ) \ D(B ) \ D(C ) ; f 2 D(DF ) ; g 2 D(DG ) ; Cx0 = g(0)g: As for many other abstract systems of equations (i.e., if A is an operator matrix on products of Banach spaces), the operator A might not be closed. And even if the operator A is closable, we might not want to switch to its closure Ac , because much of the basic information on the underlying evolution problem is contained in the domain of A and would be lost by considering the larger and often dicult to describe domain of the closure Ac . 20

Illposed problems connected with evolution equations (backwards equations, observation and inverse problems) lead to abstract Cauchy problems where the operator A has \bad" spectral properties (like the pointspectrum p(A) covering a right or upper halfplane). For example, consider the heat equation wt (r; t) = wrr (r; t) in an in nitely long bar. Suppose we observe the temperature and ux of heat at the point r = 0 over one time unit and that the initial temperature distribution is unknown. We would like to estimate from our observations the initial temperature distribution w(r; 0) for 0  r < . This leads to wt (r; t) = wrr (r; t) ; w(0; t) = f (t) ; wr (0; t) = g(t) ; t 2 [0; 1] ; 0 < r < ; where f; g 2 C [0; 1]. Interchanging t and r we obtain the initial value problem

wr (t; r) = wtt (t; r) ; w(0; r) = f (r) ; wt (0; r) = g(r) ; r 2 [0; 1] ; 0  t < ; where f; g 2 C [0; 1], and where we ask for the value of w(t; 0) for 0  t < . Clearly, this initial value problem  0 canI be written as an (ACP ) if we choose X := C [0; 1]  C [0; 1], x := (f; g), and A := D 0 , where D denotes the rst derivative on C [0; 1]. Since the point spectrum of D covers the whole complex plane, the same holds for A. For further examples of illposed Cauchy problems see, for example, S. Agmon and L. Nirenberg (1963). As an example of a linear evolution equation which leads to a multivalued linear operator A, consider the degenerate Cauchy problem (DCP )

Bu0 (t) = A0 u(t) ; u(0) = x;

where A0 ; B are linear operators with domains in a Banach space X and ranges in a Banach space Y . Clearly, (DCP) can be written as an abstract Cauchy problem u0 (t) 2 Au(t) ; u(0) = x on the space X , where A := B ?1 A0  X  X is given by f(x; y) : x 2 D(A0 ); y 2 D(B ); A0 x = Byg. Then A is a possibly multivalued linear operator (i.e., A(x + y) = A(x) + A(y) for x; y 2 D(A);  2 C), whose graph is, in general, not closed in X  X . It is one of the major advantages of the Laplace transform method that it applies to all of the problems above. With it we can study abstract Cauchy problems (ACP ) for a wide variety of operators A, including sums, products and limits of closed operators which are not necessarily closed or closable, operators which are not densely de ned, and those where the point spectrum p(A) covers C. 21

For simplicity, we will include in the following survey only the case of single valued operators and refer to C. Knuckles and F. Neubrander (1994) for the multivalued case. For other related approaches to illposed Cauchy problems see I. Cioranescu and G. Lumer (1994), R. deLaubenfels (1994), and G. Lumer (1994). In order to demonstrate the Laplace transform method for evolution equations, it is convenient to consider relatively closed operators (see S. Agmon and L. Nirenberg (1963), S.R. Caradus (1973), and B. Baumer and F. Neubrander (1994)). This class of linear operators is quite large. It contains basically all linear operators appearing in applications and imposes just enough continuity on the operator A such that A commutes with the Bochner and Stieltjes integral for suciently regular functions. If a Banach space Z is continuously embedded in a Banach space X , then we will use the notation Z ,! X .

De nition 2.1. A linear operator A is called relatively closed in a Banach space X if there exists an auxiliary Banach space XA such that D(A)  XA ,! X and A is closed in XA  X . A relatively closed operator will also be called (XA ,! X )-closed. Examples 2.2 (Sums and Products). An important class of examples of relatively

closed operators are sums and products of closed operators A; B . In general, the sum S := A + B with domain D(S ) = D(A) \ D(B ) and the composition C := BA with domain D(C ) = fx 2 D(A) : Ax 2 D(B )g will not be closed or closable in X . However, both S and C are relatively closed if we choose as XS and XC the Banach space [D(A)] with the graph norm kxkA := kxk + kAxk. We remark that S and C can be relatively closed even if the operators A; B are not closed themselves. As example take a pair of jointly closed operators A; B on a Banach space X ; i.e., D(A) \ D(B ) 3 xn ! x, and Axn ! y1 , and Bxn ! y2 implies that x 2 D(A)\D(B ), Ax = y1 , and Bx = y2 (see also N. Sauer (1982)). Then XS := [D(A)\D(B )] with norm kxkXS := kxk + kAxk + kBxk is a Banach space and D(S )  XS ,! X . It is easy to see that the sum S is (XS ,! X )-closed. }

22

As an example of an operator which is not closable in X, but relatively closed, we consider the following composition of closed operators. Let A be the rst derivative on X = C [0; 1] with maximal domain and let B be the bounded operator Bf := f (0)g, where 0 6= g 2 X . As seen above, the composition Cf := BAf = f 0 (0)g with domain D(C ) = D(A) = C 1 [0; 1] is relatively closed. However, since there is a sequence fn 2 D(C ) with fn ! 0 and fn0 (0) = 1, it follows from Cfn = fn0 (0)g = g 6= 0 that C is not closable; i.e., the closure of the graph of C in X  X which is given by X  C g is not the graph of a single-valued operator. Because the multivalued closure of C does not contain any information about the original operator, and because closedness is absolutely necessary for most operations, it is necessary to consider the graph as a subset of XA  X . In this example one might consider the domain D(Cmax ) := ff 2 X : f 0 (0) existsg instead of C 1 [0; 1]. De ne bounded operators on X by At f := f (t)?t f (0) g. Then, for each f 2 D(Cmax ) one has that At f ! Cmax f as t ! 0. Because Cmax is the pointwise limit of closed operators, it follows from the next theorem that it is relatively closed (for proofs of the following theorems see B. Baumer and F. Neubrander (1994)).

Theorem 2.3. For all t 2 I := (0; 1] let At be a (XAt ,! X )-closed operator. Suppose there exists a Banach space Y such that Y ,! XAt for all t 2 I . Then Ax := limt!0 At x T with D(A) = fx 2 t2I D(At ) \ Y : limt!0 At x exists g is (XA ,! X )-closed, where T XA = fx 2 t2I D(At ) \ Y : kxkXA := kxkY + supt2I kAt xk < 1g. Theorem 2.4. For all n 2 IN let An be a (Xn ,! X )-closed operator. Suppose there P 1 exists a Banach space Y such that Y ,! Xn for all n 2 IN. Then Ax := n=0 An x T P A x existsg is (X ,! X )-closed, where with D(A) = fx 2 n2IN D(An ) \ Y : 1 A n=0 n T P n XA = fx 2 n2IN D(An ) \ Y : kxkXA = kxkY + supn2IN k i=1 Ai xk < 1g. Theorem 2.5. For all n 2 IN let An be a (Xn ,! X )-closed operator. Then for all n 2 IN, Cn x := An    A0 x with D(Cn ) = fx 2 D(A0 ) : Ak    A0 x 2 D(Ak+1 ) for all 0  k  n ? 1g is (XCN ,! X )-closed, where XCn := fx 2 D(A0 ) : Ak    A0 x 2 D(Ak+1 ) for all 0  k  n ? 2; An?1    A0 x 2 Xn and kxkCn := kxkX + sup0kn?1 kAk    A0 xkXk < 1g. If there exists a Banach space Y such that Y ,! XCn for all n 2 IN, then Cx := T limn!1 Cn x with D(C ) := fx 2 n2IN D(Cn ) \ Y : limn!1 Cn x exists g is (XC ,! X )T closed, where XC := fx 2 n2IN D(Cn ) \ Y : kxkXC := kxkY + supn2IN kCn xk < 1g. 0

23

+1

Clearly, most linear operators that appear in applications can be decomposed into sums, products and/or limits of relatively closed operators. In order to nd linear operators which are not relatively closed one has to consider \small" domains. For example, consider the identity operator I on X = C [0; 1] with the polynomials P as its domain. Clearly, I is closable. Assume it were relatively closed. Then there exists a Banach space XI such that the graph G = f(p; p) : p 2 Pg of I is closed in XI  X . Thus G is a complete metric space of rst category, which is a contradiction to Baire's theorem. It follows that the operator I with domain P is not relatively closed. The following is a classical result of functional analysis due to E. Hille. For a proof, see E. Hille and R.S. Phillips (1957), Theorem 3.7.12.

Proposition 2.6. Let A be a (XA ,! X )-closed operator. Assume that u() maps an interval I into DR(A). If u() : I ! XRA and Au() : RI ! X are (improperly) Bochner integrable, then I u(t) dt 2 D(A) and I Au(t) dt = A I u(t)dt . 3. THE LAPLACE TRANSFORM METHOD Let A be a relatively closed operator on a Banach space X , x 2 X and 0 < T  1. In this section we will study the abstract Cauchy problem (ACP ) u0 (t) = Au(t); u(0) = x; 0  t < T in terms of the asymptotic characteristic equation (CE ) (kI ? A)yk = x ? ak (k0  k 2 IN); where ak is a sequence with exponential decay ?T in X ; i.e., lim supk!1 k1 ln kak k  ?T . If T = 1, we set ak = 0 for all k 2 IN. To obtain necessary and sucient conditions for the existence of solutions, we include the integral equations Zt n (ACPn ) v(t) = A v(s) ds + tn! x ; 0  t < T 0 in our considerations (n 2 IN0 ). Formally, the (n + 1)-times integrated Cauchy problem (ACPn ) can be obtained by integrating (ACP ) (n + 1)-times from 0 to t. Then a solution v() of (ACPn ) can be thought of as the n-th normalized antiderivative Z t (t ? s)n?1 [ n ] u (t) := u(s) ds 0 (n ? 1)! of a solution u() of (ACP ). 24

For v 2 C ([0; T ]); X ) let vT be the 0-continuation of v to IR+ ; i.e., vT := v on [0; T ] and vT := 0 on (T; 1). For v 2 L1loc ([0; 1); X ) we denote by absX (v) the in mum of all a 2 IR for which the Laplace transform

v^() :=

Z1 0

e?t v(t) dt = rlim !1

Zr 0

e?tv(t) dt

of v exists for Re > a. If !  0, then absX (v)  ! if and only if !X (v[1] )  !, where the exponential growth bound !X (v[1] ) is de ned to be the in mum of the numbers !0 2 IR for which there exists a constant M such that kv[1] (t)kX  Me!0 t for all t  0 (see G. Doetsch (1950), Satz 7 [2.2]). Notice that absX (vT ) = !X (vT ) = ?1 if T < 1. We denote by IN! the set of all integers larger than !. The following lemma is the foundation of the Laplace transform method. It shows that the Laplace transform allows us to switch freely between the Cauchy problem and the asymptotic characteristic equation for any relatively closed operator A (see B. Baumer and F. Neubrander (1994)). Notice that the following lemma contains also a uniqueness result.

Lemma 3.1 (Fundamental Lemma). Let A be (XA ,! X )-closed, x 2 X , n 2 IN0 , 0 < T  1 and !  0. Let v 2 C ([0; T ]; X ) with v[1] 2 C ([0; T ]; XA ), absX (vT )  ! and absXA (vT[1] )  !. The following are equivalent. R R all t 2 [0; T ]. (i) 0t v(s) ds 2 D(A) and v(t) = A 0t v(s) ds + tnn! x for  P ?1 (T )i x if Re > !. (ii) v^T () 2 D(A) and (I ? A)nv^T () = x ? e?T n v(T ) + ni=0 i! (iii) v^T (k) 2 D(A) for all k 2 IN! and there exists a sequence ak of exponential decay ?T such that (kI ? A)knv^T (k) = x ? ak for all k 2 IN! . PROOF. We outline the proof given in B. Baumer and F. Neubrander (1994). Assume (i) holds. If Re > !, then (ii) follows from

v^T () =

ZT 0

e?tv(t) dt =

ZT 0

e?tAv[1] (t) dt +

ZT 0

n e?t tn! x dt

?1 (T )i e?T nX

= ? 1 e?T v(T ) + 1 Av^T () + n1+1 x ? n+1 Clearly, (ii) implies (iii). 25

i=0

i! x:

and

Assume (iii) holds. Let !0 > !. Then vT[2] 2 Lip!0 ([0; 1); X ), vT[3] 2 Lip!0 ([0; 1); XA )

y() := n v^T () = n+1

Z1

0 j +1 (?1) tkj 1 j =1 j ! e (kj )n+2 akj

P1

e?tdv[2] (t) = n+2 T

Z1 0

e?tdvT[3] (t):

Since limk!1 = 0 and Ay(k) = ky(k)?x+ak for all suciently large k 2 IN it follows from the (XA ,! X )-closedness of A that v[3] (t) 2 D(A) and 1 (?1)j +1 X tkj Ay(kj ) [3] e Av (t) = klim !1 j =1 j ! (kj )n+2  y(kj )  1 (?1)j +1 X x a kj tkj = klim e !1 j =1 j ! (kj )n+1 ? (kj )n+2 + (kj )n+2 n+2 = vT[2] (t) ? (nt + 2)! x Now (i) follows from the (XA ,! X )-closedness of A. } As the Fundamental Lemma 3.1 shows, a solution of (ACP ) or (ACPn ) can only exist on [0; T ] if the initial value x is in the T -approximate range of the operators I ? A ; i.e., there exists a sequence ak 2 X of exponential decay ?T and a sequence yk such that yk 2 D(A) and (3:1) (kI ? A)yk = x ? ak for all suciently large k 2 IN. If there exists a function y which is analytic and polynomially bounded in a right half-plane such that y(k) = yk for all suciently large k 2 IN, then (3:1) is also sucient for the existence of a solution of (ACPn ) for some n 2 IN0 (see B. Baumer and F. Neubrander (1994)).

Theorem 3.2 (Existence and Uniqueness). Let A be a (XA ,! X )-closed operator, x 2 X , and 0 < T  1. The following are equivalent. [1] (i) There exists n 2 IN 0 and v 2 C ([0; T ]; X ) with v 2 C ([0; T ]; XA ), absX (vT ) < 1, R absXA (vT[1] ) < 1, 0t v(s) ds 2 D(A) and Zt n t v(t) = A v(s) ds + n! x for all t 2 [0; T ]: 0 (ii) There exist a sequence ak in X of exponential decay ?T and, for some ! 2 IR, an analytic, polynomially bounded y : fRe > !g ! XA such that y(k) 2 D(A) and (kI ? A)y(k) = x ? ak for all suciently large k 2 IN. 26

Now let T = 1. It follows from (3:1) that (3:2)

x2

\

Re>!

Im (I ? A)

is a necessary range condition for the existence of a global, Laplace transformable solution of (ACPn) for some n 2 IN0 . Assume now that (3:2) holds; i.e., there exists y() 2 D(A) such that (I ? A)y() = x for Re > !. Then, by the Fundamental Lemma 3.1, (ACPn ) has a solution for some n 2 IN0 if and only if 1n y() = v^() for some v 2 C ([0; 1); X ) with v[1] 2 C ([0; 1); XA ), absX (v) < 1 and absXA (v[1] ) < 1. Given a local resolvent  ! y() solving the characteristic equation (I ? A)y() = x for some x 2 X , any of the Laplace Representation Theorems 1.4., 1.8 and 1.9 will lead to a Hille-Yosida type existence result. For example, Widder's Theorem 1.4 leads to the following local version of the Hille Yosida theorem.

Corollary 3.3 (Local Hille-Yosida Theorem). Let A be (XA ,! X )-closed, x 2 X , n 2 IN and !  0. The following are equivalent. (i) There exists a solution v 2 Lip! ([0; 1); X ) of (ACPn ) with v[1] 2 Lip! ([0; 1); XA ). (ii) There exists y 2 C 1 ((!; 1); XA ) and M > 0 such that (a) (I ? A)y() = x for all  2 IN ,  > !, ?  (b) sup>! k k1! ( ? !)(k+1) ? n1? y() (k) kX  M for all k 2 IN0 , (c) sup>! k k1! ( ? !)(k+1) 1n y() (k) kXA  M for all k 2 IN0 . 1

Moreover, if XA = X , then condition (c) can be dropped. PROOF. Assume (i) holds. For Re > ! de ne y() := n v^(). Then n1? y() = v^() and 1n y() = v^[1] () for Re > !. The statement (a) in (ii) follows from the Fundamental Lemma 3.1 and the statements (b) and (c) in (ii) follow from Widder's Theorem 1.4. Assume (ii) holds. It follows from Widder's Theorem that there exist v 2 Lip! ([0; 1); X ) and w 2 Lip! ([0; 1); XA ) such that n1? y() = v^() and 1n y() = w^ () for all  > !. Because 1n y() = v^() = v^[1] () for all  > !, it follows from the Uniqueness Theorem of Laplace transform theory that v[1] = w. Since n (I ? A)^v() = (I ? A)y() = x for all  2 IN,  > !, the statement (i) follows from the Fundamental Lemma 3.1. } 1

1

27

If a closed operator A is !?dissipative, i.e.,

k(I ? A)xk  ( ? !)kxk for all  > ! and for all x 2 D(A), the growth conditions on y in the corollary above are automatically satis ed. In fact, more can be said. It follows from the closedness and !?dissipativity of A that for  > ! and xn 2 D(A) (3:3)

(I ? A)xn ! z implies xn ! x 2 D(A) and (I ? A)x = z:

As mentioned above, it is necessary for the existence of global, Laplace transformable solutions for the abstract Cauchy problem that the initial data x is an element of \ X~ := Im(I ? A): >!

Let X~ 3 xn ! x and  > !. Then there exists yn 2 D(A) such that (I ? A)yn = xn ! x. By (3.3), there exists y 2 D(A) with (I ? A)y = x. Thus X~ is closed in X whenever (3.3) holds. Let A be closed and !?dissipative and let x~ 2 X~ . Then there exists y : (!; 1) ! D(A) such that (I ? A)y() = x~ and ky()k   ?1 ! kx~k for all  > !. It follows from  y() ? y(0 )  ( > !) y(0 ) = (I ? A) 0 ?  that y() 2 D(A) \ X~ for all  > !. Let A~ be the part of A in X~ ; i.e., D(A~) = ~ = Ax. Since (I ? A)(y() ? x) = Ax for all fx 2 D(A) \ X~ : Ax 2 X~ g and Ax x 2 D(A) \ X~ , we obtain that D(A~) = D(A) \ X~ . Furthermore, it follows that y() 2 D(A~) and (I ? A~)y() = x~ for all x~ 2 X~ . Thus I ? A~ is one-to-one (by !?dissipativity), onto X~ , and kR(; A~)~xk = ky(k  ?1 ! kx~k for all x~ 2 X~ . This, the Fundamental Lemma 3.1 and Widder's Theorem 1.4 imply the following.

TheoremT3.4 (Dissipative Operators). Let A be closed and !?dissipative. Then (a) X~ := >! Im(I ? A) is a closed subspace of X . (b) A leaves X~ invariant; i.e., if x 2 D(A) \ X~ then Ax 2 X~ . Let A~ denote the part of A in X~ . Then D(A~) = D(A) \ X~ , (!; 1)  (A~) and kR(; A~)k  ?1 ! ( > !).

In particular, if A is densely de ned, then A~ generates a semigroup on X~ . 28

A similar result can be proved for closed, inverse positive operators on Banach lattices with order continuous norm (see B. Baumer and F. Neubrander (1994)). We conclude this section with the following local analytic generation theorem.

Corollary 3.5 (Analytic Solutions). Let A be (XA ,! X )-closed, 0 <  2 and ! 2 IR. Assume that y() 2 D(A) and (I ? A)y() = x for all  > !. If y : ! +  +  ! XA is analytic and sup2!+  k( ? !)y()kXA < 1 for all 0 < < , then there exists an function u :  ! XA such that supz2 ke?!z u(z )k < 1 for all 0 < < , R t uanalytic 0 (s) ds 2 D (A) and 2

+2

Zt

u(t) = A u(s) ds + x for all t > 0: 0

PROOF. The statement follows from the Fundamental Lemma 3.1 and the Analytic Representation Theorem 1.16. } In this section we have indicated how the Laplace transform method can be used to nd mild or integrated `solutions' of u0 (t) = Au(t); u(0) = x for a particular x 2 X . The results mentioned are only a small sample of those possible. The Laplace transform method can also be used to study approximation problems (with approximation theorems similar to the one of Section 1) as well as the asymptotics of the solutions (with the Abelian and Tauberian theorems of Laplace transform theory, see for example W. Arendt (1991) or W. Arendt and C.J.K. Batty (1988)). We have demonstrated how the method works for the abstract Cauchy problem. However, Laplace transform techniques can also be applied directly to evolution equations without reducing them to abstract Cauchy problems. Although the direct approach to linear evolution equations has some advantages, the methods one uses and the problems one encounters remain, in principle, unchanged. Clearly, the Laplace transform method also allows us to characterize those operators A for which (ACP ) has mild or integrated solutions for all x 2 X (i.e., generators of C0 semigroups or integrated semigroups). A more detailed introduction to Laplace transform theory, a Laplace transform approach to strongly continuous and integrated semigroups, and applications of these classes of semigroups to partial di erential equations can be found in a forthcoming monograph on Laplace Transforms and Evolution Equations by W. Arendt, M. Hieber and F. Neubrander (1995). 29

REFERENCES 1. S. Agmon and L. Nirenberg (1963). Properties of solutions of ordinary di erential equations in Banach space. Comm. Pure Appl. Math. 16, 121-239. 2. W. Arendt (1987). Vector valued Laplace transforms and Cauchy problems. Israel J. Math. 59, 327-352. 3. W. Arendt (1991). Linear Evolution Equations. Lecture Notes, Universitat Zurich. 4. W. Arendt (1993). Vector valued versions of some representation theorems in real analysis. Preprint. 5. W. Arendt, C.J.K. Batty (1988). Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 306, 837-852. 6. W. Arendt, M. Hieber and F. Neubrander (1995). The Laplace Transform in Banach Spaces and Evolution Equations. Monograph in preparation. 7. B. Baumer and F. Neubrander (1994). Relatively closed operators and abstract differential equations. Preprint. A preliminary version can be found in the LSU Seminar Notes in Functional Analysis and PDEs. Baton Rouge, 1993. 8. S.R. Caradus (1973). Semiclosed operators. Paci c J. Math. 44, 75-79. 9. I. Cioranescu and G. Lumer (1994). Regularization of evolution equations via kernels K (t), K -evolution operators and convoluted semigroups, generation theorems, Preprint. 10. M. Deakin (1981). The development of the Laplace transform 1737-1937, I. Euler to Spitzer, 1737-1880. Arch. Hist. Math., 346-390. 11. M. Deakin (1982). The development of the Laplace transform 1737-1937, II. Poincare to Doetsch, 1780-1937. Arch. Hist. Math., 351-381. 12. R. deLaubenfels (1994). Existence Families, Functional Calculus and Evolution Equations. Lect. Notes Math. 1570, Springer Verlag. 13. J. Diestel, J.J. Uhl (1977). Vector Measures. Amer. Math Soc., Providence, Rhode Island. 14. G. Doetsch (1937). Theorie und Anwendung der Laplace-Transformation. Verlag Julius Springer, Berlin. 15. G. Doetsch (1950). Handbuch der Laplace-Transformation I. Verlag Birkhauser, Basel. 16. G. Doetsch (1955). Handbuch der Laplace-Transformation II. Verlag Birkhauser, Basel. 17. G. Doetsch (1956). Handbuch der Laplace-Transformation III. Verlag Birkhauser, Basel. 30

18. B. Hennig and F. Neubrander (1993). On representations, inversions and approximations of Laplace transforms in Banach spaces. Applicable Analysis 49, 151-170. 19. M. Hieber, A. Holderrieth and F. Neubrander (1992)). Regularized semigroups and systems of linear partial di erential equations. Ann. Scuola Norm. Sup. Pisa 19, 363-379. 20. E. Hille (1948). Functional Analysis and Semi-Groups. Amer. Math Soc., Providence, Rhode Island. 21. E. Hille and R.S. Phillips (1957). Functional Analysis and Semi-Groups. Amer. Math. Soc. Providence, Rhode Island. 22. C. Knuckles and F. Neubrander (1994). Remarks on the Cauchy problem for multivalued linear operators. Partial Di erential Equations, Models in Physics and Biology. G. Lumer, S. Nicaise, B.-W. Schulze (eds.). Mathematical Research 82, Akademie Verlag. 23. N.N. Lebedev (1972). Special Functions and their Applications. Dover Publications, New York. 24. G. Lumer (1994). Evolution equations. Solutions for irregular evolution problems via generalized solutions and generalized initial values. Applications to periodic shock models. Annales Saraviensis 5, 1-102 25. Yu. I. Lyubich (1966). The classical and local Laplace transformation in an abstract Cauchy problem. Uspehi Mat. Nauk 21, 3-51. 26. Sylvia Martis in Biddau (1933). Studio della transformazione di Laplace e della sua inversa dal punto di vista dei funzionali analitici. Rendiconti del circolo matematico di Palermo LVII, 1-70. 27. F. Neubrander (1989). Abstract elliptic operators, analytic interpolation semigroups, and Laplace transforms of analytic functions. Semesterbericht Funktionalanalysis, Universitat Tubingen. 28. F. Neubrander (1994). The Laplace-Stieltjes transform in Banach spaces and abstract Cauchy problems. In: Proceedings of the `3rd International Workshop Conference on Evolution Equations, Control Theory, and Biomathematics', Han-sur-Lesse, Ph. Clement and G. Lumer (eds.), Lecture Notes in Pure and Applied Math. 155 (417431), Marcel Dekker. 29. F. Neubrander (1994b), Laplace transforms and evolution equations. Lecture Notes, Louisiana State University. 30. J. Pruss (1993). Evolutionary Integral Equations and Applications. Verlag Birkhauser, Basel - Boston - Berlin. 31

31. W. Rudin (1987). Real and Complex Analysis (3rd printing). McGraw-Hill Book Co., New York. 32. H.H. Schaefer (1980). Topological Vector Spaces (4th printing). Springer, New York - Heidelberg - Berlin. 33. N. Sauer (1982). Linear evolution equations in two Banach spaces. Proc. Royal Soc. Edinburgh 91A, 287-303. 34. M. Sova (1979). The Laplace transform of analytic vector-valued functions. Casopis pro pestovani matematiky 104, 267-280. 35. M. Sova (1980). Relations between real and complex properties of the Laplace transform. Casopis pro pestovani matematiky 105, 111-119. 36. P. Vieten and L. Weis (1993). Functions of bounded semivariation and classical integral transforms. Preprint. 37. Yu-Cheng Shen (1947). The identical vanishing of the Laplace integral. Duke Math. J. 14, 967-973. 38. L. Weis (1993). The inversion of the Laplace transform in Lp (X )-spaces. In: Proceedings of the Conference on Di erential Equations in Banach Spaces, Bologna 1991; North Holland. 39. D.V. Widder (1936). A classi cation of generating functions. Trans. Amer. Math. Soc. 39, 244-298. 40. D.V. Widder (1946). The Laplace Transform. Princeton University Press, Princeton, New Jersey. 41. D.V. Widder (1971). An Introduction to Transform Theory. Academic Press, New York. 42. S. Zaidman (1960). Sur un theoreme de I. Miyadera concernant la representation des functions vectorielles par des integrales de Laplace. Tohoku Math. J. 12, 47-51.

Boris Baumer, Frank Neubrander, Dept. of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA ; E-mail: [email protected] and [email protected] ; Phone : 504-388-1612 (oce) ; 504-767-3415 (home); Fax: 504-388-4276. 32