Regularity theory for elliptic and parabolic systems and problems in continuum mechanics

Telč, Czech Republic, 12.–15. 9. 2022


The conference programme is available for download (Last update on 23rd September 2022---actual conference programme.).

Invited lectures

Michael Bildhauer

Properties of smooth solutions to the $\mu$-surface equation

We are interested in a class of equations related to the minimal surface equation but not necessarily governed by the geometric properties of minimal surfaces. More precisely we consider smooth solutions the equation \[ {\rm div}\, \Bigg[\frac{g'\big(|\nabla u|\big)}{|\nabla u|} \nabla u\Bigg] = 0 \, , \] where the function $g$ satisfies a linear growth condition together with some additional reasonable assumptions.\\ In the first part of the talk the situation in some sense is not too far away from the minimal surface case and we recover particular geometric properties known from the minimal surface case. In the second part a theorem of Bers and Finn on the removability of isolated singularities is discussed. As the main tool we study the comparison properties of generalized catenoids. We finally sketch some recent developments.

Gilles A. Francfort

Regularity and rigidity of the stress field in Von Mises plasticity

In this joint work with J. F. Babadjian, inspired from prior work with A. Giacomini and J. J. Marigo, we start an investigation of spatial hyperbolicity in Von Mises elasto-plasticity, the ultimate goal being an adjudication of the uniqueness of the plastic strain. After discussing a specific example where uniqueness, or lack thereof, can be established, I will present partial results, focussing on a 2d simplified model.

Giuseppe Mingione

Hopf, Caccioppoli and Schauder, reloaded

So called Schauder estimates are in fact a contribution, at various stages, of Hopf, Caccioppoli and Schauder, between the end of the 20s and the beginning of the 30s. Later on, they were extended, with various degrees of precision, to nonlinear uniformly elliptic equations. I will present the solution to the longstanding open problems of proving estimates of such kind in the nonuniformly elliptic case and for minima of non-differentiable functionals (again considered in the nonuniformly elliptic case). From joint work with Cristiana De Filippis.

Domenico Mucci

Micro-slip-induced multiplicative plasticity: existence of energy minimizers

To account for material slips at microscopic scale, we take deformation mappings as SBV functions $\varphi$, which are orientation-preserving outside a jump set taken to be two-dimensional and rectifiable. For their distributional derivative $F=D\varphi$ we consider the common multiplicative decomposition $F=F^{e}F^{p}$ into so-called elastic and plastic factors, the latter a measure. Then, we consider a polyconvex energy with respect to $F^{e}$, augmented by the measure $|\mathrm{curl}\ F^{p}|$. For this type of energy we prove existence of minimizers in the space of SBV maps. We avoid self-penetration of matter. Our analysis rests on a representation of the slip system in terms of currents (in the sense of geometric measure theory) with both $\mathbb{Z}^{3}$ and $\mathbb{R}^{3}$ valued multiplicity. The two choices make sense at different spatial scales. They describe separate but not alternative modeling choices. The first one is particularly significant for periodic crystalline materials at a lattice level. The latter covers a more general setting and requires to account for an energy extra term involving the slip boundary size. We include a generalized (and weak) tangency condition; the resulting setting agrees to gliding and cross-slip mechanisms. The possible highly articulate structure of the jump set allows one to consider the resulting setting even as an approximation of climbing mechanisms. This is a joint work with P. M. Mariano (DICEA, University of Firenze, Italy).

Emil Wiedemann

Weak and measure-valued solutions of the Euler equations

Several notions of weak or "very weak" solutions have been suggested for the incompressible and compressible Euler systems, motivated by the lack of a satisfactory well-posedness theory for these equations in turbulent regimes. Surprisingly, the speaker and L. Székelyhidi showed in 2012 that distributional and measure-valued solutions of the incompressible system are in a sense the same, although the latter had been expected to be a much weaker notion. In this talk, we turn to the isentropic compressible Euler system, where the situation is fundamentally different. Our approach borrows and develops several tools from the theory of multiple integrals, such as quasiconvexity, Ball-James rigidity, or Müller-Zhang truncation. Joint work with D. Gallenmüller.

Yasemin Şengül

Stability in nonlinear viscoelasticity via monodromy

We consider nonlinear viscoelasticity of strain-rate type under some assumptions allowing for solid phase transformations. We consider the dynamical problem as an elliptic regularisation of the quasistatic case and formulate the latter as a gradient flow in one space dimension, leading to existence and uniqueness of solutions. By approximating general initial data by those in which the deformation gradient takes only finitely many values, we show that under suitable hypotheses on the stored-energy function the deformation gradient is instantaneously bounded and bounded away from zero, which is an important feature for the stability analysis. In order to prove convergence as time tends to infinity of solutions to a single equilibrium, it seems necessary to impose a nondegeneracy condition on the constitutive equation for the stress. We will investigate this condition and show how in some cases it can be proved using the monodromy group of a holomorphic function.

Lenka Slavíková

Fractional Orlicz-Sobolev embeddings

We establish the optimal Orlicz target space for embeddings of fractional-order Orlicz-Sobolev spaces in the Euclidean space. We also present an improved embedding with an Orlicz-Lorentz target space, which is optimal in the broader class of all rearrangement-invariant spaces. We will consider both spaces of order less than one as well as higher-order spaces. This is a joint work with A. Alberico, A. Cianchi and L. Pick.

Cristiana De Filippis

Quasiconvexity meets nonlinear potential theory

A classical problem in the regularity theory for vector-valued minimizers of multiple integrals consists in proving their smoothness outside a negligible set, cf. Evans (ARMA ’86), Acerbi \& Fusco (ARMA ’87), Duzaar \& Mingione (Ann. IHP-AN ’04), Schmidt (ARMA ’09). In this talk, I will show how to infer sharp partial regularity results for relaxed minimizers of degenerate/singular, nonuniformly elliptic quasiconvex functionals, using tools from nonlinear potential theory. In particular, in the setting of functionals with $(p,q)$-growth - according to the terminology introduced by Marcellini (Ann. IHP-AN ’86; ARMA ‘89) - I will derive optimal local regularity criteria under minimal assumptions on the data. This talk is partly based on joint work with Bianca Stroffolini (University of Naples Federico II).

© 2015–2022 Miroslav Bulíček; Last modified: Fri 23 Sep 2022 03:39:00 PM CEST