Advanced Topics in Analysis

The course will provide an introduction to the theory of C_0-semigroups of bounded linear operators. This theory provides the functional analytic setting for solving initial value problems. Roughly speaking, the idea is to consider abstract equations of the form

u'(t) = Au(t) + f(t), t \geq 0,

u(0) = x_0,

where A is a (possibly unbounded) linear operator acting on a Banach space X, and x0 \in X is the initial condition. For example, the heat equation on R^d is obtained as a special case by choosing X = L^2(R^d) and A = \Delta, the Laplace operator. The function f takes values in X and represents an inhomogeneity. Under very mild conditions on the operator A, in the linear case (when F = 0) the initial value problem is well-posed if and only if there exists a family of bounded operators (S(t)){t\geq 0} on X satisfying

(1) S(0) = I (the identity operator)

(2) S(t1) S(t2) = S(t1 + t2) for all t1, t2 \geq 0

(3) the mapping t --> S(t)x is continuous for all x \in X.

In this case, the unique solution of the initial value problem is given by

u(t) = S(t)x_0, t \geq 0.

Such a family is called a C_0-semigroup of linear operators and A its generator. One may think of the operators (S(t))_{t\ge 0} as the solution operator that maps the initial value to the value of the solution at time t. In case A is a bounded operator, one has S(t) = exp(tA), but the power of semigroup theory lies in the fact that it permits unbounded operators such as the Laplace operator. It thus provides a unified framework for studying a great variety of initial value problems arising in applications. Once existence and uniqueness of solutions are established, the following numerical challenges arise to compute the solution u(t):

• space discretization: Due to its unboundedness, the operator A has to be approximated by a sequence of simpler operators Am generating C0-semigroups (Sm(t)){t\ge 0}, giving rise to approximate solutions um(t)=Sm(t)x0. Convergence of um to u can be reduced to approximation properties of A_m via the Trotter-Kato theorem. This provides an abstract framework for e.g. finite difference or finite element methods.

• time discretization: Computing the exponential even of a bounded operator A_m is equally challenging. The way to overcome this are rational approximations of the exponential z \mapsto e^z such as (1-z)^{-1}, resulting in the implicit Euler scheme (1-t/n A)^{-n}. Chernoff’s theorem gives sufficient conditions for these approximations to converge, and the convergence rate can be quantified.

The aim of the course is to prove the well-posedness theorem just mentioned and study the relationships between the operator A, the associated semigroup of linear operators, and the properties of the solution u. Moreover, the approximation of the solutions in space and time is studied from a functional analytic viewpoint using semigroup theory.

(last modified: 28/05/2025)