Analytic semigroup explained

In mathematics, an analytic semigroup is particular kind of strongly continuous semigroup. Analytic semigroups are used in the solution of partial differential equations; compared to strongly continuous semigroups, analytic semigroups provide better regularity of solutions to initial value problems, better results concerning perturbations of the infinitesimal generator, and a relationship between the type of the semigroup and the spectrum of the infinitesimal generator.

Definition

Let Γ(t) = exp(At) be a strongly continuous one-parameter semigroup on a Banach space (X, ||·||) with infinitesimal generator A. Γ is said to be an analytic semigroup if

• for some 0 < θ < π/&hairsp;2, the continuous linear operator exp(At) : X → X can be extended to t ∈ Δ_θ&hairsp;,

\Delta _{\theta} = \{ 0 \} \cup \{ t \in C : | arg (t) | < \theta \},

and the usual semigroup conditions hold for s, t ∈ Δ_θ&hairsp;: exp(A0) = id, exp(A(t + s)) = exp(At)&thinsp;exp(As), and, for each x ∈ X, exp(At)x is continuous in t;

• and, for all t ∈ Δ_θ \ , exp(At) is analytic in t in the sense of the uniform operator topology.

Characterization

The infinitesimal generators of analytic semigroups have the following characterization:

R _{λ(A)} of the operator A we have\| R _{λ} (A) \| \leq (C)/(| λ- \omega |)

for Re(λ) > ω. Such operators are called sectorial. If this is the case, then the resolvent set actually contains a sector of the form

\left\{ λ \in C : | arg ( λ - \omega) | < (\pi)/(2) + \delta \right\}

for some δ > 0, and an analogous resolvent estimate holds in this sector. Moreover, the semigroup is represented by

\exp (At) = (1 { 2)/(\pi) i} \int _{\gamma} e ^{λ} ( λ id - A ) ^{-1} d λ ,

where γ is any curve from e^−iθ∞ to e^+iθ∞ such that γ lies entirely in the sector

\ { λ \in C : | arg ( λ - \omega) | \leq \theta \ },

with π/&hairsp;2 < θ < π/&hairsp;2 + δ.

References

• Book: Renardy , Michael . Rogers, Robert C.. An introduction to partial differential equations. Texts in Applied Mathematics 13. Second. Springer-Verlag. New York. 2004. xiv+434. 0-387-00444-0. 2028503.