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Research Seminar

Research Seminar (Continuing Class)

Talks in summer 2025

The talks are taking place in seminar room 2.066.

13.05.2025, 14:00 Uhr Franz Gmeineder (Konstanz)

A panoramic view on quasiconvexity and non-uniform ellipticity
By Morrey's and Ball's foundational works, quasiconvexity is the central notion of the vectorial Calculus of Variations; yet, little is known in the realm of energy functionals with non-uniformly elliptic behaviour. Such functionals arise e.g. in the description of cavitation, but the systematic study of such minimizers and their regularity properties had remained open until recently. As a key point and essentially optimal substitute of the De Giorgi-Nash-Moser theory, we outline how partial gradient Hölder continuity can be established in the critical degenerate elliptic regime.
Partially based on joint work with Jan Kristensen (Oxford).

20.05.2025, 14:00 Uhr Patrick Tolksdorf (Karlsruhe)

An introduction to Shen's L^p extrapolation theorem
On the whole space, regularity estimates to elliptic PDE, such as estimates on second derivatives of solutions to the Laplace equation follow by classical Calderón-Zygmund theory. Corresponding regularity properties can be transported to smooth enough domains via localization. In rough situations, however, analogous L^p-estimates on second derivatives fail dramatically - even if the gradient is estimated instead and if p is too large.
Shen's L^p-extrapolation theorem is a valuable substitute for classical Calderón-Zygmund theory und unfolds its strength for PDE in rough situations. It allows to derive L^p-estimates on solutions for specific values of p and therefore opens the door for more refined regularity statements. In this talk, we will introduce to this L^p-extrapolation theorem of Shen and show how to apply it in the context of elliptic PDE in order to derive optimal regularity estimates such as the higher integrability property of the gradient of solutions to elliptic systems in divergence form with L^{\infty}-coefficients due to N. Meyers from 1963.

27.05.2025, 14:00 Uhr Maximilian Ruff (Karlsruhe)

Error analysis of splitting methods for 3D semilinear wave equations with finite-energy solutions
We study splitting schemes for the time integration of the 3D energy-(sub)critical semilinear wave equation on the full space and the torus under the finite-energy condition. In the case of a cubic nonlinearity, we show that a filtered Strang splitting converges with almost second order in $L^2$ and almost first order in $H^1$. If the nonlinearity has a quartic form instead, we show an analogous convergence result with an order reduced by 1/2. For the energy-critical quintic nonlinearity, we show first-order convergence in $L^2$ for the filtered Lie and Strang splittings. Our approach relies on continuous- and discrete-time Strichartz estimates. We also make use of the integration and summation by parts formulas to exploit cancellations in the error terms. Moreover, in the torus case, error bounds for a full discretization using the Fourier pseudo-spectral method in space are given. Finally, we discuss a numerical example indicating the sharpness of our theoretical results.

05.06.2025, 9:45 Uhr, SR 2.067 El Maati Ouhabaz (Bordeaux)

Commutator estimates and Poisson bounds for Dirichlet-to-Neumann operators with variable coefficients
The Dirichlet-to-Neumann operator appears in many mathematical problems and modelling such as electrical impedance tomography, Calderòn's inverse problem, spectral theory,... Its analysis has attracted attention in recent years. We consider the Dirichlet-to-Neumann operator $N$ associated with a general elliptic operator
\[ A u = - \sum_{k,l=1}^d \partial_k (c_{kl}\, \partial_l u) + \sum_{k=1}^d \Big( c_k\, \partial_k u - \partial_k (b_k\, u) \Big) +c_0\, u \]
with possibly complex coefficients. We study three problems:
1) Boundedness on $C^\nu$ and on $L_p$ of the commutator $[N, M_g]$, where $M_g$ denotes the multiplication operator by a smooth function $g$.
2) Hölder and $L_p$-bounds for the harmonic lifting associated with $A$.
3) Poisson bounds for the heat kernel of $N$.
We solve these problems in the case where the coefficients are Hölder continuous and the underlying domain is bounded and of class $C^{1+\kappa}$ for some $\kappa > 0$. For the Poisson bounds we assume in addition that the coefficients are real-valued.
The talk is based on a joint work with T. ter Elst (Univ. Auckland).

08.07.2025, 14:00 Uhr Piero D'Ancona (Rom)

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22.07.2025, 14:00 Uhr Richard Nutt (Karlsruhe)

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You can find previous talks in the archive of the research seminar.