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Sobolev imbeddings and integrated semigroups. (English) Zbl 0762.47013
Semigroup theory and evolution equations, Proc. 2nd Int. Conf., Delft/Neth. 1989, Lect. Notes Pure Appl. Math. 135, 29-40 (1991).
In the reviewing paper, the author considers the Cauchy problem $u'(t) = Au(t), \quad t \geq 0, \quad u(0) = x,$ with Schrödinger operator $$A = i\Delta_ p$$, $$\Delta_ p$$ being the Laplacian on $$L^ p(\Omega )$$, where $$\Omega \subset \mathbb{R} ^ n$$ is a bounded set.
To this aim, he deals with the concept of $$k$$-times integrated semigroup, see [the author, Israel J. Math. 59, 327-352 (1987; Zbl 0637.44001), and F. Neubrander, Pac. J. Math. 135, 111-155 (1988; Zbl 0675.47030)]. Let the operator $$A$$ be an operator on a Banach space $$E$$ , and let $$k \in \mathbb{N}$$. One says that $$A$$ generates a $$k$$-times integrable semigroup $$S$$, if there exists $$w \in \mathbb{R}$$ and a (uniquely determined) strongly continuous function $$S: [0, \infty ) \rightarrow \mathcal L (E)$$ satisfying $$\sup_{t\geq 0} | e^{-wt} S (t)| < \infty$$ , such that $$( w,\infty ) \subset \rho (A)$$ (the resolvent set), and $R (\lambda , A) := (\lambda - A )^{-1} = \lambda \int_{0}^{\infty } e^{-\lambda t}S(t) \text d t \text {\;\;\;for all\;\;\;} \lambda > 0 .$ (In the case that $$A$$ generates a $$C_ 0$$ - semigroup $$T$$, one can write $S(T) = \int_{0}^{t} [(t-s)^{k-1}/(k-1)!]T(S) \text{d} S ;$ and the operator which generates the $$k$$-times integrated semigroup, generates, for all $$l \geq k ,$$ the $$k$$-times integrated semigroup as well. Hence, the best possible $$k$$ is an expression for the regularity of the Cauchy problem defined by $$A$$.)
Using the technique of interpolation of semigroups and Sobolev imbedding theorems, the author shows that for $$\Omega \in C^ 2 (\Omega )$$ $$i\Delta$$ (with Dirichlet or Neumann boundary conditions) generates a $$k$$-times integrable semigroup on $$L^{\infty }(\Omega )$$ ($$1 < p <\infty$$) if $$k \geq \frac{N}{2}| \frac{1}{2}-\frac{1}{p} |$$ and in $$C_ 0(\Omega)$$, $$C(\overline \Omega)$$, $$L^ \infty (\Omega )$$, $$L^ 1(\Omega )$$ if $$k > \frac{N}{4}$$.
The method still works if the Laplacian is replaced by a strictly elliptic operator with smooth coefficients.
For the entire collection see [Zbl 0741.00048].
Reviewer: P.Doktor (Praha)

MSC:
 47D06 One-parameter semigroups and linear evolution equations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems