Special Session 7: Recent developments on nonlinear geometric PDEs

Maximum principle for higher order operators and applications to nonlinear PDEs

Daniele Cassani University of Insubria & RISM Italy Co-Author(s):

Abstract: It is well known how the Maximum Principle (MP) in general fails to hold for uniformly elliptic operators of order higher than two, even in smooth convex domains. In D. Cassani and A. Tarsia (2022) it was shown in dimension two and three, by establishing a new Harnack type inequality with nonlocal remainder terms, that the validity of the positivity preserving property can be restored when lower order derivatives are taken into account as a perturbation of the higher order differential operator. We will present further advances in this direction, extending the validity of the MP to any dimension and fairly general domains. Moreover, we will discuss how the presence of inertial terms affects the range of the perturbation parameter, providing a balance between the positivity restoring effect of lower order derivatives and the mass energy. Applications to semilinear biharmonic equations are presented.

Ground States and Nodal Solutions for the Mean Curvature Operator in Minkowski Space

Francesca Colasuonno Universita degli Studi di Torino Italy Co-Author(s):

Abstract: The Dirichlet problem for the mean curvature operator in Minkowski space is considered in a radial domain of $\mathbb R^N$. For this problem, we prove the existence of a ground state and a linking solution in the general case, and the multiplicity of radial nodal solutions in the radially symmetric setting. We also investigate the asymptotic behavior of these solutions as a parameter in the equation tends to infinity.

New ancient solutions to the Yamabe flow on the sphere

Seunghyeok Kim Hanyang University Korea Co-Author(s):    Haixia Chen, Monica Musso

Abstract: The Yamabe flow is a geometric evolution equation that aims to improve the geometry of a manifold by deforming its metric toward one with constant scalar curvature. In this talk, I will introduce the historical background for the Yamabe flow and then discuss recent developments concerning ancient solutions, namely, solutions that exist for all negative times. In the latter part of the talk, I will present joint work with Dr. Haixia Chen (Hanyang University, Central China Normal University) and Prof. Monica Musso (University of Bath) on the existence of non-rotationally symmetric ancient solutions.

Eigenfunctions in domains with small balls removed

Ying Li Central China Normal University&Sapienza University of Rome Peoples Rep of China Co-Author(s):    Laura Abatangelo, Massimo Grossi

Abstract: This work concentrates on the eigenvalues and eigenfunctions of the Dirichlet Laplacian in a bounded domain with a small hole $B(P,\varepsilon)$ removed. We derive pointwise estimates of the $u-$capacity potential $V_{\varepsilon}$ associated with $B(P,\varepsilon)$ in dimensions two and higher. Using these estimates, we obtain a uniform estimate for $u_\varepsilon-u-V_{\varepsilon}$ when the eigenvalue is simple. When the eigenvalue is multiple, we also derive the bifurcation of $\lambda_\varepsilon$, which was previously obtained by Abatangelo, L\`ena and Musolino in 2022 and 2024. Finally, in dimension two, we apply the uniform estimate of $u_\varepsilon$ to study the number of intersection points between the nodal line $\mathcal{N}_\varepsilon$ and $\partial B(P,\varepsilon)$.

Extremising eigenvalues of the GJMS operators in a fixed conformal class

Bruno Premoselli Universite Libre de Bruxelles Belgium Co-Author(s):    Emmanuel Humbert and Romain Petrides

Abstract: Let $(M,g)$ be a closed Riemannian manifold of dimension $n \ge 3$ and $P_g$ be a conformally-covariant operator on $(M,g)$. We consider in this talk two problem at the crossroads of conformal geometry and spectral theory: 1) determining the extremal value that the renormalized eigenvalues of $P_g$ take as $g$ runs through a fixed conformal class and 2) determining whether these extremal values are attained at an extremal metric. Examples of such operators $P_g$ include the famous conformal Laplacian of the Yamabe problem, $P_g = \Delta_g + c_n S_g$, but also its higher-order generalisations of even order. Extremal metrics for these problems, when they exist, are not smooth in general, and yield interesting geometric objects, such as (singular) harmonic maps into large-dimensional spheres or least-energy possibly nodal solutions of prescribed $Q$-curvature type problems. Addressing these questions requires to investigate renormalized eigenvalues functionals that are highly non-smooth. We develop an ad hoc variational theory to study them: we obtain in particular semi-contiuity results and we provide an explicit Euler-Lagrange equation for local extremals. As a consequence we obtain new existence results for extremals by performing a bubble-tree analysis of suitable extremizing sequences. This is joint work with E. Humbert (Universite de Tours) and R. Petrides (Universite Paris Cite) and is based on arXiv:2505.08280.

Miminization of the first eigenvalue of the Dirichlet Laplacian with a small volume obstacle

Gianmaria Verzini Politecnico di Milano Italy Co-Author(s):    Benedetta Noris, Giovanni Siclari

Abstract: We consider the well-known shape optimization problem with spectral cost: minimizing the first eigenvalue of the Dirichlet Laplacian among all subdomains $\Omega$ having prescribed volume and contained in a fixed box $D$; equivalently, we look for the best way to remove a compact set (obstacle) $K\subset\overline{D}$ of Lebesgue measure $|K|=\eps$, $0

Classification of the Struwe decomposition of the Brezis-Nirenberg problem and its application: The one-bubble case

Yuanze Wu School of Mathematics, Yunnan Normal University Peoples Rep of China Co-Author(s):    Rui He, Xiangqing Liu, Juncheng Wei

Abstract: I will report our recent results on the classification of the Struwe decomposition of the Brezis-Nirenberg equation in high dimensions for the one-bubble case. I will also report the application of this refined analysis in constructing nontrivial solutions of the Brezis-Nirenberg equation.

Normalized solutions for the Sobolev critical Schr\{o}dinger equation with trapping potential

Junwei Yu Politecnico di Milano Italy Co-Author(s):

Abstract: In this talk, I would like to present some recent results on the existence and multiplicity of positive solutions in $H^1(\mathbb{R}^N)$, $N\ge3$, with prescribed $L^2$-norm, for the Sobolev critical Schr\{o}dinger equation with trapping potential. Under suitable assumptions on the potential, the associated energy functional exhibits a mountain pass geometry on the $L^2$-sphere, which yields the existence of both a local minimizer and a mountain pass solution. In particular, for small values of the prescribed mass, we prove the existence of two distinct solutions: a ground state and a mountain pass solution.