Special Session 1: Recent Advances in Brezis-Nirenberg Problem

Helical vortex structures with compactly supported cross-sectional vorticity for the 3D incompressible Euler equations

Averkios Averkiou University of Bath England Co-Author(s):    Monica Musso

Abstract: In this talk, we first revisit the vortex filament conjecture for three-dimensional incompressible Euler flows with helical symmetry and no swirl. By adapting gluing methods, we obtain the first construction of a smooth helical vortex filament in the whole space $\mathbb{R}^3$ whose cross-sectional vorticity remains compactly supported in $\mathbb{R}^2$ for all times. Building on this construction, we then investigate the leapfrogging phenomenon of helical vortex filaments, in which several interacting vortex helices with a common symmetry axis alternately overtake one another while preserving their coherent structure. This is joint work with Monica Musso.

Classification results, rigidity theorems and semilinear pdes on Riemannian manifolds: a P-Function approach

Giulio Ciraolo University of Milan Italy Co-Author(s):

Abstract: We consider solutions to critical and sub-critical semilinear elliptic equations on complete, noncompact Riemannian manifolds and study their classification as well as the effect of their presence on the underlying manifold. When the Ricci curvature is non-negative, we prove both the classification of positive solutions to the critical equation and the rigidity for the ambient manifold. The same results are established for solutions to the Liouville equation on Riemannian surfaces. Our results are obtained via an appropriate P-function whose constancy implies the classification of both the solutions and the underlying manifold. The analysis carried out on the P-function also makes it possible to classify non-negative solutions for subcritical equations on manifolds enjoying a Sobolev inequality and satisfying an integrability condition on the negative part of the Ricci curvature. We will also present results on weighted Riemannian manifolds.

Blow-up and stability for quadratic derivative nonlinear wave equations

Oliver Gough University of Bath England Co-Author(s):

Abstract: This talk concerns finite-time blow-up in nonlinear wave equations with quadratic derivative nonlinearities. A central question is to understand the blow-up profile and mechanism in such equations, and whether the corresponding singular solutions continue to describe the dynamics under small perturbations. I will discuss recent work on these questions.

Sharp quantitative stability estimates for the Brezis-Nirenberg problem

Seunghyeok Kim Hanyang University Korea Co-Author(s):    Haixia Chen, Juncheng Wei

Abstract: We study quantitative stability for the classical Brezis-Nirenberg problem associated with the critical Sobolev embedding $H_0^1(\Omega) \hookrightarrow L^{\frac{2n}{n-2}}(\Omega)$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n \geq 3$). To the best of our knowledge, this is the first quantitative stability result for the Sobolev inequality on bounded domains. A key discovery of our work is the emergence of unexpected stability exponents, arising from an intricate interaction among the nonnegative solution $u_0$, the lower-order term $\lambda u$ in the Brezis-Nirenberg equation, bubble formation, and the boundary effect of the domain. A main difficulty is to quantify this boundary effect, which makes the problem fundamentally different from the Euclidean setting and from the case of smooth closed manifolds. Our proof refines and streamlines several arguments from the existing literature, while also resolving new analytical difficulties specific to the bounded domains. This talk is based on joint work with Haixia Chen (Hanyang University, Central China Normal University) and Juncheng Wei (The Chinese University of Hong Kong).

The Brezis-Peletier conjecture for one and several bubbles: blow-up analysis in dimension three

Tobias K\"onig Goethe University Frankfurt Germany Co-Author(s):    Rupert Frank, Hynek Kovarik, Paul Laurain

Abstract: My talk is concerned with the following question: How do sequences of positive solutions $u_\varepsilon$ to the Brezis-Nirenberg-type equation $-\Delta u_\varepsilon + (a + \varepsilon V) u_\varepsilon = u_\varepsilon^\frac{N+2}{N-2}$ with Dirichlet boundary conditions on a bounded domain $\Omega \subset \mathbb R^N$ blow up? Brezis and Peletier (1989) conjectured that the blow-up location and speed of single-peak sequences should be universally characterized by the geometry of the domain, with explicit formulas. This was confirmed shortly afterwards by Rey (1989) and Han (1991) for the case of dimension $N \geq 4$, which is non-critical in the sense of the Brezis-Nirenberg. In critical dimension $N=3$, an additional cancellation phenomenon makes the analysis substantially more challenging. In recent work with R. Frank (Munich) and H. Kovarik (Brescia) we overcome this difficulty and prove the Brezis-Peletier conjecture for single-peak sequences in dimension three. Subsequently with P. Laurain (Paris/Champs-sur-Marne) we develop a multi-peak analogue of the Brezis-Peletier conjecture, thus providing a complete picture of blow-up in the Brezis-Nirenberg problem at arbitrary levels of energy. In my talk I will present an overview of these results and the main proof ideas.

Multi-Bubble Analysis for the Brezis-Nirenberg Problem

Paul Laurain University Gustave Eiffel France Co-Author(s):    Tobias Konig

Abstract: In their celebrated paper, Brezis and Nirenberg showed that the behavior of the Yamabe-type equation, can vary greatly depending on the data of the problem. In particular, there exists a notion of critical potential $a$ below which no positive solution exists. We are interested in the behavior of solutions as the potential approaches the critical potential. In particular, we will see how the location and the blow-up rate of these solutions are constrained by the Robin function of the operator $\Delta + a$ This is joint work with Tobias Konig.

Least-energy solutions for the Brezis-Nirenberg problem in dimension 3 in the non-coercive case

Bruno Premoselli Universite Libre de Bruxelles Belgium Co-Author(s):    Hussein Cheikh Ali

Abstract: We consider in this talk the celebrated Brezis-Nirenberg equation in the non-coercive case $\lambda > \Lambda_1$, where $\Lambda_1$ is the first eigenvalue of the Laplacian on a bounded open set of $\mathbb{R}^n$. We prove in dimension 3 the existence of least-energy sign-changing solutions under a positive mass condition. This is the first general existence result for the Brezis-Nirenberg problem in dimension 3 in the non-coercive case. We introduce for this a new non-smooth variational problem, inspired from eigenvalue-optimisation problems in conformal geometry and we show that its minimisers, when they exist, provide least-energy solutions of the Brezis-Nirenberg problem. This is joint work with H. Cheikh Ali (Universite de Lille).

Some new nodal solutions to the Yamabe equation

Liming Sun Academy of mathematics and systems science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Yuanli Li, Angela Pistoia

Abstract: We find several new nodal solutions to the Yamabe equation that are not Kelvin invariant.

Normalized solutions to Sobolev critical Schr\odinger equations

Gianmaria Verzini Politecnico di Milano Italy Co-Author(s):    Dario Pierotti, Junwei Yu

Abstract: We study the existence and multiplicity of positive solutions with prescribed $L^2$-norm for the (stationary) nonlinear Schr\odinger equation with Sobolev critical power nonlinearity. In the free case on the full space, the associated energy functional has a mountain pass geometry on the $L^2$-sphere, which boils down, in higher dimension, to the existence of a mountain pass solution. We consider this problem, either in bounded domains (i.e., the normalized Brezis-Nirenberg problem) or in presence of a potential, wondering (i) whether a local minimum solution appears, thus providing an orbitally stable family of solitons, and (ii) if the existence of a mountain pass solution persists. This talk is based on joint works with Dario Pierotti and Junwei Yu (Politecnico di Milano).