Person: Denk, Robert
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An Introduction to Maximal Regularity for Parabolic Evolution Equations
In this note, we give an introduction to the concept of maximal Lp-regularity as a method to solve nonlinear partial differential equations. We first define maximal regularity for autonomous and non-autonomous problems and describe the connection to Fourier multipliers and R-boundedness. The abstract results are applied to a large class of parabolic systems in the whole space and to general parabolic boundary value problems. For this, both the construction of solution operators for boundary value problems and a characterization of trace spaces of Sobolev spaces are discussed. For the nonlinear equation, we obtain local in time well-posedness in appropriately chosen Sobolev spaces. This manuscript is based on known results and consists of an extended version of lecture notes on this topic.
Lp-theory for a fluid–structure interaction model
We consider a fluid–structure interaction model for an incompressible fluid where the elastic response of the free boundary is given by a damped Kirchhoff plate model. Utilizing the Newton polygon approach, we first prove maximal regularity in Lp-Sobolev spaces for a linearized version. Based on this, we show existence and uniqueness of the strong solution of the nonlinear system for small data.
Convex monotone semigroups on lattices of continuous functions
We consider convex monotone C0-semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a σ-Dedekind complete Banach lattice. Typical examples include the space of all bounded uniformly continuous functions and the space of all continuous functions vanishing at infinity. We show that the domain of the classical generator of a convex semigroup is typically not invariant. Therefore, we propose alternative versions for the domain, such as the monotone domain and the Lipschitz set, for which we prove invariance under the semigroup. As a main result, we obtain the uniqueness of the semigroup in terms of an extended version of the generator. The results are illustrated with several examples related to Hamilton-Jacobi-Bellman equations, including nonlinear versions of the shift semigroup and the heat equation. In particular, we determine their symmetric Lipschitz sets, which are invariant and allow to understand the generators in a weak sense.
Elliptic problems with rough boundary data in generalized Sobolev spaces
We investigate regular elliptic boundary-value problems in boun\-ded domains and show the Fredholm property for the related operators in an extended scale formed by inner product Sobolev spaces (of arbitrary real orders) and corresponding interpolation Hilbert spaces. In particular, we can deal with boundary data with arbitrary low regularity. In addition, we show interpolation properties for the extended scale, embedding results, and global and local a priori estimates for solutions to the problems under investigation. The results are applied to elliptic problems with homogeneous right-hand side and to elliptic problems with rough boundary data in Nikolskii spaces, which allows us to treat some cases of white noise on the boundary.
Convex semigroups on Lp-like spaces
In this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having Lp-spaces in mind as a typical application. We show that the basic results from linear C0-semigroup theory extend to the convex case. We prove that the generator of a convex C0-semigroup is closed and uniquely determines the semigroup whenever the domain is dense. Moreover, the domain of the generator is invariant under the semigroup, a result that leads to the well-posedness of the related Cauchy problem. In a last step, we provide conditions for the existence and strong continuity of semigroup envelopes for families of C0-semigroups. The results are discussed in several examples such as semilinear heat equations and nonlinear integro-differential equations.