Analytical semigroup

An analytic semigroup , sometimes called a holomorphic semigroup , is a family of bounded linear operators of a real or complex Banach space in itself, where is a complex-valued sector and an angle. Analytical semigroups are a special form of the strongly continuous semigroups , which are used in analysis to prove the existence and uniqueness of solutions to partial differential equations such as the heat conduction equation . ${\displaystyle \left(T(z)\right)_{z\in \Sigma _{\delta }}}$ ${\displaystyle X}$ ${\displaystyle \Sigma _{\delta }:=\{\lambda \in \mathbb {C} \setminus \{0\}:|\operatorname {arg} \lambda |<\delta \}\cup \{0\}}$ ${\displaystyle \delta \in (0,\pi /2]}$

The investigation of the analytical semigroups is particularly interesting because of their smoothing properties: For example, the solution of the assigned Cauchy problem is always infinitely differentiable and, for positive ones, is always in the domain of the generator instead of just at the end of the domain as with the strongly continuous semigroups. ${\displaystyle t}$ ${\displaystyle t}$

definition

A family is called an analytic semigroup if the following applies to an angle : ${\displaystyle T=\left(T(z)\right)_{z\in \Sigma _{\delta }}\subset {\mathcal {L}}(X)}$ ${\displaystyle \delta \in (0,\pi /2]}$

• ${\displaystyle T(0)=I}$.
• ${\displaystyle T(z_{1}+z_{2})=T(z_{1})T(z_{2})}$for everyone . ${\displaystyle z_{1},z_{2}\in \Sigma _{\delta }}$
• the mapping is on analytical. ${\displaystyle z\mapsto T(z)}$ ${\displaystyle \Sigma _{\delta }}$
• the mapping is on for strongly steady. ${\displaystyle z\mapsto T(z)}$ ${\displaystyle \Sigma _{\delta '}\cup \{0\}}$ ${\displaystyle \delta '\in (0,\delta )}$

Additionally, if for every in is limited, limited analytical semigroup called (but: limited strongly continuous semi-group analysis, in general, is not limited analytical semigroup). ${\displaystyle \|T(z)\|}$ ${\displaystyle \delta '\in (0,\delta )}$ ${\displaystyle \Sigma _{\delta '}}$ ${\displaystyle \left(T(z)\right)_{z\in \Sigma _{\delta }}}$

Infinitesimal producer

Analogous to strongly continuous semigroups, one considers the operator with ${\displaystyle A}$

${\displaystyle Ax:=\lim _{\Sigma _{\delta }\ni z\rightarrow 0}{\frac {T(z)x-x}{z}}}$

and

${\displaystyle D(A):=\left\{x\in X:\lim _{\Sigma _{\delta }\ni z\rightarrow 0}{\frac {T(z)x-x}{z}}{\text{ existiert}}\right\}}$.

The operator is called (infinitesimal) generator and is densely defined and closed .

properties

The spectrum of a producer ${\displaystyle A}$

• Creates an analytic semigroup , then ${\displaystyle A}$ ${\displaystyle T}$
  • exist and with for everyone . If the half group is limited, you can choose . ${\displaystyle M\geq 1}$ ${\displaystyle \omega \geq 0}$ ${\displaystyle \|T(z)\|\leq Me^{\omega \mathrm {Re} \,z}}$ ${\displaystyle z\in \Sigma _{\delta }}$ ${\displaystyle \omega =0}$
  • exists such that creates a bounded analytic semigroup. ${\displaystyle \omega >0}$ ${\displaystyle A-\omega }$
  • applies to everyone . ${\displaystyle T(t)X\subset D(A)}$ ${\displaystyle t>0}$
  • the inverse Laplace transform of the resolvent agrees with the semigroup, i.e. for and a suitable path in . ${\displaystyle T(t)={\frac {1}{2\pi i}}\int _{\gamma }e^{\lambda t}R(\lambda ,A)\mathrm {d} \lambda }$ ${\displaystyle t\geq 0}$ ${\displaystyle \gamma }$ ${\displaystyle \mathbb {C} }$
• If it creates a bounded analytic semigroup , then the resolvent set contains the sector for all . ${\displaystyle A}$ ${\displaystyle T}$ ${\displaystyle \rho (A)}$ ${\displaystyle \Sigma _{\pi /2+\delta '}}$ ${\displaystyle \delta '\in (0,\delta )}$
• ${\displaystyle A}$generates a bounded analytic semigroup if and only if a strongly continuous semigroup generates with for all and (real characterization). ${\displaystyle A}$ ${\displaystyle T}$ ${\displaystyle \operatorname {rg} (T(t))\subset D(A)}$ ${\displaystyle t>0}$ ${\displaystyle \sup _{t>0}\|tAT(t)\|<\infty }$

Examples

• If it creates a strongly continuous semigroup, then it creates an analytic semigroup with an angle . ${\displaystyle A}$ ${\displaystyle A^{2}}$ ${\displaystyle \pi /2}$
• If there is a region with a Dirichlet-regular boundary ( e.g. Lipschitz boundary or smooth boundary), the Laplace operator with Dirichlet boundary condition , i.e. H. , a bounded analytic semigroup. ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle D(\Delta ):=W^{2,p}(\Omega )\cap W_{0}^{1,p}(\Omega )}$

The Cauchy problem

If a bounded analytic semigroup creates the abstract Cauchy problem ${\displaystyle A}$ ${\displaystyle (T(t))_{t\geq 0}}$

${\displaystyle \left\{{\begin{array}{lll}u'(t)&=&A(u(t))+f(t){\text{ für alle }}t>0\\u(0)&=&u_{0}\end{array}}\right.}$

for the initial value and a Hölder continuous function through the function ${\displaystyle u_{0}\in X}$ ${\displaystyle f\in C^{\alpha }([0,\infty );X)}$

${\displaystyle u(t):=T(t)u_{0}+\int _{0}^{t}T(t-s)f(s){\rm {d}}s}$

solved.

literature

• Klaus-Jochen Engel, Rainer Nagel : One-parameter semigroups for linear evolution equations. Springer, New York NY 2000, ISBN 0-387-98463-1 ( Graduate Texts in Mathematics 194).
• Tosio Kato : Perturbation Theory for Linear Operators. Corrected printing of the 2nd edition. Springer, Berlin 1980, ISBN 0-387-07558-5 ( The basic teachings of the mathematical sciences in individual representations 132), (Reprint. Springer-Verlag, Berlin et al. 1995, ISBN 3-540-58661-X ( Classics in mathematics )).
• Ammon Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin et al. 1983, ISBN 3-540-90845-5 ( Applied Mathematical Sciences 44).