Andrea Bisterzo (Università degli Studi di Milano-Bicocca): “Weak maximum principle for elliptic operators in unbounded Riemannian domains and an application to a symmetry problem”
Abstract: The necessity of a maximum principle arises naturally when one is interested in studying qualitative properties of solutions to partial differential equations. Typically, ensuring the validity of such principles requires additional assumptions on the underlying space or on the differential operator. This talk will focus on proving a weak maximum principle for second-order elliptic operators acting on unbounded Riemannian domains, with the condition that the spectrum of the operator is positive. In the second part of the talk, we will see how this maximum principle can be applied to derive a symmetry result for stable solutions to semilinear PDEs in isoparametric Riemannian domains.

Denis Bonheure (Université Libre de Bruxelles): “Asymptotics of the solution to the Stokes problem near multiple collisions”

Abstract: It is known since the seminal works of Cox on the lubrication approximation exploiting the disparity between two length scales that the knowledge of the asymptotics of Stokes solutions allows to understand the dynamics near contacts of particles immersed in a fluid. The lubrication theory in the 3D case remains rigorously unjustified except for very peculiar symmetric geometries. I will present the general framework in both 2D and 3D including multiple isolated contacts. We only assume non degeneracy of the contacts (which entails a condition on the curvatures). The talk is based on a joint work with Edoardo Bocchi (Milano) and Matthieu Hillairet (Montpellier)

Lorenzo Carletti (Université Libre de Bruxelles): “The optimal constant for the critical Sobolev inequality of higher order on manifolds”
Abstract: Let $(M,g)$ be a compact Riemannian manifold, and $k\geq 1$ an integer, we assume that $\dim M > 2k$. In the Euclidean setting, there is an optimal constant $K_0(n,k)$ such that $\big(\int_{\mathbb{R}^n} |u|^{2^\sharp}\big)^{2/2^\sharp} \leq K_0^2 \int_{\mathbb{R}^n} |\nabla^k u|^2$ for all $u \in C_c^\infty(\mathbb{R}^n)$, where $2^\sharp = \frac{2n}{n-2k}$. This constant remains optimal for the critical Sobolev embedding $H^{k}(M) \subset L^{2^\sharp}(M)$ : We show that there exists $B_0$ such that $\|u\|_{L^{2^\sharp}(M)}^2 \leq K_0^2 \int_M |\nabla^k u|^2 + B_0 \|u\|_{H^{k-1}(M)}^2$ for all $u \in H^{k}(M)$. In this talk, I will present the main ideas of the proof. This leads to studying the pointwise blow-up of a sequence of positive functions $(u_\alpha)_{\alpha\geq 1}$ satisfying $(\Delta_g + \alpha)^k u_\alpha = u_\alpha^{2^\sharp-1}$ in $M$, for all $\alpha$.

Francesca De Marchis (Universita Roma I): “Asymptotic behavior of solutions of problems involving power nonlinearities”
Abstract: We discuss, highlighting analogies and differences, the asymptotic behavior both of solutions to the classical Lane Emden equation both to a linear elliptic equation subject to a Neumann boundary condition involving a power nonlinearity. Based on joint papers with M. Grossi, H. Fourti, I. Ianni and F. Pacella.

Tobias Lamm (Karlsruher Institut für Technologie):”Parabolic equations with rough initial data”
Abstract: In an ongoing joint project with Herbert Koch (Bonn) we study the local resp. global well-posedness of parabolic systems with rough initial data. Examples include the mean curvature flow for Lipschitz data, the harmonic map heat flow with BMO initial data and various other interesting examples. I will sketch the main ideas involved in our argument.

Paul Laurain (Université Gustave Eiffel): “Multi-Bubble Analysis for the Brezis-Nirenberg Problem”
Abstract: In a seminal article, Brezis and Nirenberg highlighted that the behavior of the Yamabe-type equation $$ \Delta u + a u = u^{2^*-1}, $$ where $2^*$ is the Sobolev critical exponent, can vary significantly depending on the problem’s data. In particular, there exists a notion of critical potential, below which no positive solution exists. Our interest lies in the behavior of solutions as the potential approaches this critical potential. Specifically, we will examine how the location and rate of blow-up of these solutions are constrained by the Robin function of the operator $\Delta +a$ . This work is in collaboration with Tobias König.

Nicolas Marque (Université de Lorraine): “Dark matter mass and Q-curvature”
Abstract: Dark matter is introduced to correct discrepancies between observations and Relativistic predictions. Another option is to study other gravitational theories of higher order, induced by quadratic Lagrangian, which satisfy many properties similar to General Relativity. One can then apply the same procedure that, in relativistic setting yields the ADM mass. Applied to quadratic Lagrangian yields a fourth order mass, linked to physical and mathematical phenomena, and in particular to a fundamental higher order geometric quantity : the Q-curvature.

Giovanni Molica Bisci (Università Telematica San Raffaele Roma): “Symmetries and flower-shape geometries”
Abstract: Several important problems arising in many research fields, such as physics and
differential geometry, lead to consider semilinear variational elliptic equations defined on
unbounded domains of the Euclidean space and a great deal of work has been devoted
to their study. From the mathematical point of view, probably the main interest relies on
the fact that often the tools of nonlinear functional analysis, based on compactness
arguments, cannot be used, at least in a straightforward way, and some new techniques
have to be developed. In a joint paper with Giuseppe Devillanova (Politecnico di Bari)
and Raffaella Servadei (Urbino Carlo Bo) we introduce a group theoretical scheme,
raised in the study of problems which are invariant with respect to the action of
orthogonal subgroups, to show the existence of multiple solutions distinguished by their
different symmetry properties. Aim of the talk is to present this construction, called
flower-shape geometry, and to show its applications to the study of nonlinear problems
set in strip-like domains.

Tristan Robert (Université de Lorraine): “Stochastic quantization of Liouville conformal field theory”
Abstract : In this talk, I will present some results regarding a nonlinear stochastic heat equation on closed surfaces, with exponential nonlinearity, and forced by a space-time white noise. This model arises as the stochastic quantization equation for a two-dimensional conformal field theory. I will first review the construction of the corresponding Gibbs measure, known as the Liouville quantum gravity measure. I will then discuss the almost sure global well-posedness of the dynamics, and finally the invariance of the measure under the resulting dynamics. This is a joint work with Tadahiro Oh (Edinburgh), Nikolay Tzvetkov (ENS Lyon), and Yuzhao Wang (Birmingham).

Raffaella Servadei (Università degli Studi di Urbino Carlo Bo): “Fractional critical problems with jumping nonlinearities”
Abstract: In this talk we deal with some nonlocal problem driven by the fractional
Laplacian in presence of jumping nonlinearities. Using variational and topological methods, we prove the existence of a nontrivial solution for the problem under consideration. These existence results can be seen as the nonlocal counterpart of the ones obtained in the context of the Laplacian equations. In the nonlocal framework the arguments used in the classical setting have to be refined. Indeed the presence of the fractional Laplacian operator gives rise to some additional difficulties, that we are able to overcome proving new regularity results for weak solutions of nonlocal problems, which are of independent interest.
This is a joint paper with Giovanni Molica Bisci, Kanishka Perera and Caterina Sportelli (Journal des Mathématiques Pures et Appliquées, 2024).

Michael Struwe (ETH Zürich): “The prescribed curvature flow on the disc”
Abstract: For given functions f and j on the disc B and its boundary,
we study the existence of conformal metrics g on B with prescribed Gauss curvature f and
boundary geodesic curvature j. Using the variational characterization of such metrics obtained by Cruz-Blazquez and Ruiz in 2018, we show that there is a canonical negative gradient flow of such metrics, either converging to a solution of the prescribed curvature problem, or blowing up to a spherical cap. In the latter case, similar to our work in 2005
on the prescribed curvature problem on the sphere, we are able to exhibit a 2-dimensional
shadow flow for the center of mass of the evolving metrics from which we obtain existence
results complementing the results recently obtained by Ruiz by degree-theory.

Gabriella Tarantello (Universita degli Studi di Roma II): “On CMC-immersions of surfaces into Hyperbolic 3-manifolds”
Abstract: I shall discuss the moduli space of Constant Mean Curvature (CMC) c-immersions of a closed surface S (orientable and of genus at least 2) into hyperbolic 3-manifolds. Interestingly when |c|<1, such space can be parametrized by elements of  the tangent bundle of the Teichmueller space of S. This is attained by showing existence and uniqueness for a “constraint” version of the Gauss-Cadazzi equations governing the immersion. Namely, that the associated “Donaldson functional” (in Gonsalves-Uhlenbeck (2007)) admits a global minimum as its unique critical point.
Since (CMC) 1-immersion into the hyperbolic space play a relevant role  in hyperbolic geometry (in view of their striking analogies  with minimal immersions into the Euclidian space) I shall discuss the asymptotic behavior of such (CMC) c-immersions  as |c| approaches 1.  We see that it is possible to catch at the limit a “regular “ CMC 1-immersions into an hyperbolic 3-manifold, except in very rare  situations.
For example for genus g=2, we see that “compactness” relates to the image of the Kodaira map on the six Weierstrass points of S.

Pierre-Damien Thizy (Université de Lyon): “Big blow-up sets for solutions of Q-curvature equations.“
Abstract: given a bounded domain of the 2m-dimensional Euclidean space, m>1, Adimurthi, Robert and Struwe observed that, even assuming a volume bound like $\int e^{2mu} dx \leq C$, some sequences of solutions of Q-curvature type equations like $(-\Delta)^m u= Q e^{2m u}$ without boundary conditions can blow-up not only on a discrete set, but also on the zero set of a nonpositive nontrivial polyharmonic function. This is in striking contrast with what happens in dimension 2 corresponding here to m=1. Motivated by a work in progress with Ali Hyder and Luca Martinazzi, we will present this problem and then a strategy to construct such solutions. This construction involves some generalized versions of the Walsh-Lebesgue theorem and some problems about elliptic PDEs with measure data.