Abstract

The aim of this talk is to present parts of the so-called Functional Analysis Toolbox (FA-ToolBox), a unified and general approach to solve PDEs. Hilbert Complexes are of particular interest.

We shall motivate our concept by discussing the well known and prototypical div-curl-system $$\mathring{\mathrm{curl}}\,E=F,\qquad \mathrm{div}\,E=g,$$ arising, e.g., in electro-magneto statics. Employing techniques from linear functional analysis (FA-ToolBox) we develop a comprehensive (and surprisingly simple) solution theory for static problems of the above type. We will introduce the notion of Hilbert complexes $$H_{0}\xrightarrow{\mathrm{A_{0}}}H_{1}\xrightarrow{\mathrm{A_{1}}}H_{2},$$ of densely defined and closed linear operators $$\mathrm{A_{0}}:D(\mathrm{A_{0}})\subset H_{0}\to H_{1},\qquad \mathrm{A_{1}}:D(\mathrm{A_{1}})\subset H_{1}\to H_{2},$$ satisfying the so-called complex property $$R(\mathrm{A_{0}})\subset N(\mathrm{A_{1}}).$$ The latter electro static system is then generalised to $$\mathrm{A_{1}}x=f,\qquad \mathrm{A_{0}}^{*}x=g.$$ The aim is to provide criteria on the complex such that existence and uniqueness of $x$ can be guaranteed. It will turn out that the crucial property is the compactness of the embedding $$D(\mathrm{A_{1}})\cap D(\mathrm{A_{0}}^{*})\hookrightarrow H_{1},$$ i.e., in classical terms the compactness of $$D(\mathring{\mathrm{curl}})\cap D(\mathrm{div})\hookrightarrow L^{2},$$ the co-called Picard-Weber-Weck selection theorem.

Our general theory is not only applicable to the classical de Rham complex involving grad, curl, and div, but also to other important Hilbert complexes, such as the elasticity complex or the biharmonic complex. Moreover, important results can be proved in this general setting, such as general div-curl-type lemmas and informations about generalised Poincaré/Friedrichs estimates, e.g., for the Maxwell constants.

This talk contains parts of joined work with colleagues from Essen, Linz, and Prag, in particular, with Walter Zulehner (JKU Linz). Some parts are strongly related to the work of Doug Arnold (Minnesota) and Ragnar Winther (Oslo) and their co-authors.

Results of this talk can be found in, e.g. [2, 1, 3, 4, 5, 7, 8, 6].

References

1. S. Bauer, D. Pauly, and M. Schomburg, The Maxwell Compactness Property in Bounded Weak Lipschitz Domains with Mixed Boundary Conditions, SIAM Journal on Mathematical Analysis, 48(4), 2912-2943, 2016
2. D. Pauly, On Maxwell’s and Poincaré’s Constants, Discrete and Continuous Dynamical Systems – Series S, 8(3), 607-618, 2015
3. D. Pauly, A Global div-curl-Lemma for Mixed Boundary Conditions in Weak Lipschitz Domains and a Corresponding Generalized $\mathrm{A_{1}}$-$\mathrm{A_{0}^{*}}$-Lemma in Hilbert Spaces, Analysis (Munich), 39(2), 33-58, 2019
4. D. Pauly, On the Maxwell and Friedrichs/Poincare Constants in ND, Mathematische Zeitschrift, 293(3), 957-987, 2019
5. D. Pauly and J. Valdman, Poincaré-Friedrichs Type Constants for Operators Involving grad, curl, and div: Theory and Numerical Experiments, Computers and Mathematics with Applications, 79, 3027-3067, 2020
6. D. Pauly and M. Waurick, The Index of some Mixed Order Dirac-Type Operators and Generalised Dirichlet-Neumann Tensor Fields, arXiv, 2020, https://arxiv.org/abs/2005.07996
7. D. Pauly and W. Zulehner, The divDiv-Complex and Applications to Biharmonic Equations, Applicable Analysis, 99(9), 1579-1630, 2020
8. D. Pauly and W. Zulehner, The Elasticity Complex: Compact Embeddings and Regular Decompositions, arXiv, 2020, https://arxiv.org/abs/2001.11007