Winter/spring term 2020:

17 February, 14.30h-17.00h, Louvain-la-Neuve (Chemin du Cyclotron 2), salle B.203-5

14.30h-15.30h: Cristiana de Filippis (Oxford), Manifold constrained non-uniformly elliptic problems

Abstract: I will cover some new regularity results for manifold constrained non-uniformly elliptic problems. Precisely, for manifold-valued minima of energies $$\mathcal E(w,\Omega):=\int_\Omega k(x) |Dw|^{p(x)} d x\qquad\text{ and } \qquad \mathcal P(w,\Omega):=\int_\Omega \left[|Dw|^p+a(x)|Dw|^q \right]d x$$ I shall present sharp interior and up to the boundary partial regularity results, analysis and dimension reduction of the singular set. (pdf)

15.30h-16.00h: Coffee break

16.00h-17.00h: Patrick Dondl (Freiburg), A phase-field approximation of the perimeter under a connectedness constraint

Abstract: We develop a phase-field approximation of the perimeter functional in the plane under a connectedness constraint. Our approximation is based on the classical Modica-Mortola functional and a diffuse quantitative version of path-connectedness. We prove convergence of the approximating energies in the sense of Gamma-convergence and present numerical results and applications to image segmentation and Ohta-Kawasaki functionals modeling charged droplets. Join work with M. Novaga (Pisa), B. Wirth (Münster), and S. Wojtowytsch (Princeton).

2 March, 14.30h-17.00h, Louvain-la-Neuve (Chemin du Cyclotron 2), salle CYCL07

14.30h-15.30h: Vito Crismale (Paris), Existence and approximation of equilibrium configurations for epitaxially strained crystalline films

Abstract: In the realization of electronic device structures, thin layers of highly strained composites such as InGaAs/GaAs or SiGe/Si are deposited onto a substrate with different elastic stiffness. This process is said epitaxial growth of the film on the substrate. It is governed by the competition of the bulk elastic energy of the composite (due to the elastic misfit with the substrate) and its surface energy (localized on the free surface). I will present a work with M. Friedrich (University of Münster), in which we prove existence and approximation of equilibrium configurations for the corresponding energy introduced by Bonnetier and Chambolle in 2002, in any dimension and without any a priori regularity assumption on the displacement of the thin film. The analysis extends the available 2-dimensional results.

15.30h-16.00h: Coffee break

16.00h-17.00h: Michał Łasica (Warsaw), Existence of 1-harmonic map flow

Abstract: The talk will concern the functional of total variation of maps valued in a connected, complete Riemannian manifold. I will report on recent progress on the existence of steepest descent curves of this functional with respect to the $$L^2$$ distance. The results are joint work with L. Giacomelli (Rome) and S. Moll (Valencia).

Fall/winter term 2019:

3 December: Journée FNRS, 10.00h-17.00h, Université Libre de Bruxelles, Campus de la Plaine, Bâtiment NO, 9th floor

10.00h-11.00h: Frédéric Robert (Nancy), Impact of localization of the Hardy potential on the stability of Pohozaev obstructions

Abstract: The Pohozaev obstruction yields a sufficient condition on the potential for the nonexistence of solutions to some nonlinear elliptic PDE on a star-shaped domain. This condition (C) is not stable under perturbation of the potential. Unlike large dimensions, Druet-Laurain have proved that the Pohozaev obstruction is stable in small dimension. In this talk, I will discuss this issue on PDEs involving Hardy-type potential. The stability of the obstruction depends both on the dimension and on the localization of the Hardy weight.

11.00h-11.30h: Coffee break

11.30h-12.30h: Henrik Schumacher (RWTH Aachen), Gradient Flows for the Möbius Energy

Abstract: Aiming at optimizing the shape of closed embedded curves within prescribed isotopy classes, we use a gradient-based approach to approximate stationary points of the Möbius energy. The gradients are computed with respect to certain fractional-order Sobolev scalar products that are adapted to the Möbius energy. In contrast to $$L^2$$-gradient flows, the resulting flows are ordinary differential equations on an infinite-dimensional manifold of embedded curves. In the fully discrete setting, this allows us to completely decouple the time step size from the spatial discretization, resulting in a very robust optimization algorithm that is orders of magnitude faster than following the discrete $$L^2$$-gradient flow.

12.30h-14.30h: Lunch break

14.30h-15.30h: Angela Pistoia (Rome), Elliptic systems with critical growth

Abstract: I will present some results concerning the existence of nodal solutions to the Yamabe equation on the sphere and their connections with the existence of positive solutions to competitive elliptic systems with critical growth in the whole space.

15.30h-16.00h: Coffee break

16.00h-17.00h: Flaviana Iurlano (Paris), Concentration versus oscillation effects in brittle damage

Abstract: This talk is concerned with an asymptotic analysis, in the sense of Γ-convergence, of a sequence of variational models of brittle damage in the context of linearized elasticity. The study is performed as the damaged zone concentrates into a set of zero volume and, at the same time and to the same order ε, the stiffness of the damaged material becomes small. Three main features make the analysis highly nontrivial: at ε fixed, minimizing sequences of each brittle damage model oscillate and develop microstructures; as ε→0, concentration and saturation of damage are favoured; and the competition of these phenomena translates into a degeneration of the growth of the elastic energy, which passes from being quadratic (at ε fixed) to being of linear-growth type (in the limit). Consequently, homogenization effects interact with singularity formation in a nontrivial way, which requires new methods of analysis. We explicitly identify the Γ-limit in two and three dimensions for isotropic Hooke tensors. The expression of the limit effective energy turns out to be of Hencky plasticity type.

7 October, 14.30h-17.00h, room B.203 at Louvain-la-Neuve (Chemin du Cyclotron 2)

14.30h-15.30h: Manuel del Pino (Bath), Singularity formation by loss of compactness in nonlinear diffusions

Abstract: A fundamental question in nonlinear evolution equations is the analysis of solutions which develop singularities (blow-up) in finite time or as time goes to infinity. We review recent results on the construction of solutions to nonlinear parabolic PDE which exhibit this kind of behavior in the form of "bubbling". This means solutions that at main order look like asymptotically singular time-dependent scalings of a fixed finite energy entire steady state. We mainly focus on the energy critical heat equation, the classical two-dimensional harmonic map flow into the sphere, and the Keller-Segel system of chemotaxis.

15.30h-16.00h: Coffee break

16.00h-17.00h: Lisa Beck (Augsburg), On the minimization of convex, variational integrals of linear growth

Abstract: We study the minimization of functionals of the form $$\int_\Omega f(Du) \, dx$$ with a convex integrand $$f$$ of linear growth (such as the area integrand), among all functions in the Sobolev space $$W^{1,1}$$ with prescribed boundary values. Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and might in fact fail, as it is well-known already for the non-parametric minimal surface problem. In such cases, the functional is extended suitably to the space of functions of bounded variation via relaxation, and for the relaxed functional one can in turn guarantee the existence of minimizers. However, in contrast to the original minimization problem, these so-called generalized minimizer might in principle have interior jump discontinuities or not attain the prescribed boundary values. After a short introduction to the problem I want to discuss what is known about the regularity of generalized minimizers. In particular, I will review several results which were obtained in the last years in cooperation with Miroslav Bulicek (Prague), Franz Gmeineder (Bonn), Erika Maringova (Prague), and Thomas Schmidt (Hamburg).

21 October, 14.30h-17.00h, Université Libre de Bruxelles

14.30h-15.30h: Jean-Baptiste Castéras (Helsinki), Nonlinear Helmholtz equation in the hyperbolic space

Abstract: We will be interested in the impact of the geometry in the solvability of the nonlinear Helmholtz equation. We will show in particular that, in the hyperbolic space, this equation admits solution for a larger class of nonlinearity than in the Euclidian space. We also obtain counter-examples to the Strichartz estimates in the hyperbolic space. Joint work with Rainer Mandel.

15.30h-16.00h: Coffee break

16.00h-17.00h: Katarzyna Mazowiecka (UCLouvain), On the size of the singular set of minimizing harmonic maps

Abstract: Minimizing harmonic maps (i.e., minimizers of the Dirichlet integral) with prescribed boundary conditions are known to be smooth outside a singular set of codimension 3. I will consider mappings from an n-dimensional domain with values in the two dimensional sphere. I will present an extension of Almgren and Lieb's linear law on the bound of the singular set. Next, I will investigate how the singular set is affected by small perturbations of the prescribed boundary map and present a stability theorem, which is an extension of Hart and Lin's result. I will also discuss possible extensions to different target manifolds and the optimality of our assumptions. This is joint work with Michał Miśkiewicz and Armin Schikorra.

4 November, 14.30h-17.00h, Université Libre de Bruxelles

14.30h-15.30h: Alexis Michelat (ETH/Paris), Morse Index of Branched Willmore Spheres

Abstract: In several variational problems (critical catenoid for free boundary minimal surfaces, Willmore problem...), one expects to characterise the eventual solutions by their Morse indices, which is the number of negative directions at a critical point of the associated Lagrangian. We will explain in the special case of Willmore surfaces (critical points of the integral of mean-curvature squared) how one can reduce the computation of the Morse index to the computation index of some associated finite-dimensional matrix.

15.30h-16.00h: Coffee break

16.00h-17.00h: Céline Grandmont (Paris), Some existence results for fluid-beam interaction problems

Abstract: We are intersted in an unsteady nonlinear fluid-structure interaction problem that can be viewed as a toy model to describe blood flow in large arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier-Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation with or without damping. The fluid and the structure systems are coupled via interface conditions prescribing the continuity of the velocities at the fluid?structure interface and the action-reaction principle. The aim is to investigated existence of strong or weak solutions with possible contact between the structure and the bottom of the fluid cavity, depending on the considered elastic model. We will present existence of local in time strong solutions (joint work with M. Hillairet and J. Lequeurre), existence of global in time strong solution in the case of a damped beam (joint work with M. Hillairet), and existence of weak solution beyong contact" in the case where the structure is described by a beam equation (joint work with J.-J. Casanova and M. Hillairet).

18 November, 14.30h-17.00h, Louvain-la-Neuve, CYCL05

14.30h-15.30h: Julien Brasseur (Paris), On the restriction property in low regularity function spaces

Abstract : In this talk, I will be interested in the "restriction property" on low regularity function spaces. For example: given a function in $$H^1(\mathbf{R}^2)$$, one can show that its restrictions to almost every lines still belong to $$H^1$$ : we then say that $$H^1$$ has the restriction property. This type of property plays a important role in lifting theory and, by extension, in some reaction-diffusion problems of Ginzburg-Landau type. We will present new results which exhibit a large class of Besov spaces for which, surprisingly, this property is not satisfied. We will also present an optimal (or close to being optimal) characterization of the aforementionned loss of regularity in terms of spaces of so-called "generalized smoothness" recently introduced by D. Edmunds and H. Triebel in connection with fractal geometry.

15.30h-16.00h: Coffee break

16.00h-17.00h: Franz Gmeineder (Bonn), Regularity for the Dirichlet problems on BV and BD

Abstract: In this talk I give an overview of recent results on the regularity of generalised minima for variational problems on BV and BD. This comprises both higher Sobolev regularity in the convex as well as partial Hölder continuity in the quasiconvex case. The underlying results are, in parts, joint work with J. Kristensen (Oxford).