Special Session 73: Nonlinear elliptic and parabolic equations and related functional inequalities

New solutions for the Lane-Emden problem in planar domains

Luca Battaglia Universita degli Studi Roma Tre Italy Co-Author(s):    Isabella Ianni, Angela Pistoia

Abstract: We consider the Lane-Emden problem on a smooth bounded planar domain. We find nodal concentrating solutions, for large values of p, where both the positive and negative part blow up, the latter with a non-radial profile. Up to our knowledge, this is the first result concerning concentration with such a pattern for planar elliptic problems.

Nonlinearities and singularities

Marta Calanchi Universit\`a degli Studi di Milano Italy Co-Author(s):

Abstract: We discuss some applications of singularity theory to nonlinear differential equations. In particular, we consider an elliptic equation with a nonlinearity which interacts with the k-th eigenvalue, and show that only fold and cusp singularities occur.

Fine bounds for best constants in subcritical Sobolev`s embeddings and applications

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

Abstract: We establish upper and lower estimates for the optimal embedding constants related to the classical Sobolev embeddings. In particular we derive fine bounds for best constants of the fractional subcritical Sobolev embeddings $$ W_{0}^{s,p}\left(\Omega\right)\hookrightarrow L^{q}\left(\Omega\right), $$ where $N\geq1$, $0

Singularity formation for the Keller-Segel system in the plane

Manuel del Pino England Co-Author(s):

Abstract: The classical model for chemotaxis is the planar Keller-Segel system $$ u_t = \Delta u - \nabla\cdot ( u\nabla v ), \quad v(\cdot, t) = \frac 1{2\pi} \log 1{|\cdot |} * u(\cdot ,t) . $$ in $\R^2\times (0,\infty)$. A blow-up of finite mass solutions is expected to occur by aggregation, a concentration of bubbling type, common to many geometric flows. We build with precise profiles solutions in the critical-mass case $8\pi$, where blow-up takes place in infinite time. We establish the stability of the phenomenon detected under arbitrary mass-preserving small perturbations and present new constructions in the finite time blow-up scenario. The results presented are in collaboration with Federico Buseghin, Juan Davila, Jean Dolbeault, Monica Musso, and Juncheng Wei.

Recent developments in the study of nonrelativistic limit of nonlinear Dirac equations

Qi Guo Renmin University of China Peoples Rep of China Co-Author(s):    Pan Chen, Xiaojing Dong, Yanheng Ding, Huayang Wang, Yuanyang Yu

Abstract: This talk presents a summary of recent findings on nonrelativistic limit of nonlinear Dirac equations. The discussion is divided into two main parts. Firstly, we discuss the ground states of nonlinear Dirac equations without constraints, focusing on the nonrelativistic limit of these ground states and the convergence behavior in cases with different potentials. Secondly, we delve into the nonrelativistic limit of normalized solutions of nonlinear Dirac equations. Specifically, we demonstrate that the first two components of the normalized ground states converge to the normalized ground states of the corresponding nonlinear Schr\odinger equations and the last two components converges to trivial states. This talk aims to provide an overview of the pertinent findings in this domain.

Relationship between maximizers of maximization problems and ground state solutions of semilinear elliptic equations

Masato Hashizume Osaka University Japan Co-Author(s):    Masato Hashizume

Abstract: We consider maximization problems in $\mathbb{R}^2$ with the Sobolev norm constraints and with the Dirichlet norm constraints. Typical maximization problems are the Sobolev inequalities and the Trudinger-Moser inequalities, and the existence and non-existence of maximizers for these variational problems have been studied so far. In this talk we focus on properties of maximizers for the maximization problems. We show that maximizers of the maximization problems are ground state solutions of corresponding elliptic equations. Moreover, we also discuss other connections between maximizers of maximization problems and ground state solutions of corresponding elliptic equations.

Existence of solutions to a semilinear heat equation in uniformly local weak Zygmund type spaces

Norisuke Ioku Tohoku University Japan Co-Author(s):    Kazuhiro Ishige, Tatsuki Kawakami

Abstract: We consider an optimal sufficient condition for the existence of solutions to the critical semilinear heat equation by introducing uniformly local weak Zygmund type spaces. Differences between standard (weak) Zygmund spaces and the space introduced here will be explained by focusing several properties for the heat semigroup in these spaces.

On the asymptotic behavior of noncompact orbits for dynamical systems

Michinori Ishiwata Osaka university Japan Co-Author(s):

Abstract: In this talk, we are concerning the asymptotic behavior of noncompact orbits for dynamical systems. First we introduce a semilinear parabolic problem defined onentire spatial domain as a typical example and give a typical behavior which comes from the noncompactness of the spatial domain. The possibility to develop a general dynamical system theory allowing the noncompactness of the orbit (i.e., a generalization of the LaSalle principle) will be also discussed. The method is based on the profile decomposition.

N-Euclidean Logarithmic Moser-Trudinger-Onofri inequality and some geometrical variants

Gabriele Mancini University of Bari Aldo Moro Italy Co-Author(s):    Natalino Borgia, Silvia Cingolani

Abstract: In this talk, I will provide a brief overview of the history of Onofri`s inequality for the unit sphere, highlighting its connection with Trudinger-Moser type inequalities on Euclidean bounded domains. I will focus on an $N$-dimensional Euclidean version of Onofri`s inequality, proved by del Pino and Dolbeault, for smooth compactly supported functions in $\mathbb{R}^N$, with $N \geq 2$. I will prove that this inequality can be extended to a suitable weighted Sobolev space and, although there is no clear connection with standard Sobolev spaces on $\mathbb{S}^N$ via stereographic projection, I will show that it is equivalent to the logarithmic Moser-Trudinger inequality with sharp constant, obtained by Carleson and Chang for balls in $\mathbb{R}^N$. These results are part of a joint work with N. Borgia and S. Cingolani.

Uniqueness and minimality in Euler`s elastica problem

Tatsuya Miura Kyoto Univeristy Japan Co-Author(s):    Tatsuya Miura

Abstract: Euler`s elastica problem is the oldest variational model for elastic rods and serves as a classic example of higher-order geometric variational problems. In this talk, I will discuss recent advancements concerning the issues of uniqueness and minimality in the elastica problem.

Delaunay-like compact equilibria in the liquid drop model

Monica Musso University of Bath England Co-Author(s):    M. del Pino, A. Zuniga

Abstract: The liquid drop model was originally introduce by Gamow in 1928 to model atomic nuclei. The model describes the competition between surface tension (which keeps the nuclei together) and Coulomb force (which corresponds to repulsion among the protons). Equilibrium shapes correspond to sets in the 3-dimensional Euclidean space which satisfies an equation that links the mean curvature of the boundary of the set to the Newtonian potential of the set. In this talk I will present the construction of toroidal surfaces, modelled on a family of Delaunay surfaces, with large volume which provide new equilibrium shapes for the liquid drop model.

Concentration and oscillation analysis of semilinear elliptic equations with exponential growth in a disc

Daisuke Naimen Muroran Institute of Technology Japan Co-Author(s):

Abstract: We study blow-up positive solutions of semilinear elliptic equations with supercritical exponential growth. We detect infinite sequences of bubbles by a scaling technique. Then, we observe that the infinite sequences of bubbles cause infinite oscillations, around singular solutions, of blow-up solutions. Thanks to this, we finally arrive at a proof of infinite oscillations of bifurcation diagrams which yield the existence of infinitely many solutions.

Concentrations in Bernoulli`s free boundary problem

Michiaki Onodera Tokyo Institute of Technology Japan Co-Author(s):

Abstract: Bernoulli`s free boundary problem is an overdetermined boundary value problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. I will talk about recent progress on a conjecture of Flucher and Rumpf that asserts the existence of a family of free boundaries concentrating at non-degenerate local minima of the Robin function.

Proof of the Brezis-Gallouet inequality via heat semigroup

Tohru Ozawa Waseda University Japan Co-Author(s):    Yi Huang, Chenmin Sun, Taiki Takeuchi

Abstract: Proof of the Brezis-Gallouet inequality is given by means of the heat semigroup in two space dimensions. This talk is based on my recent joint-work with Yi Huang (Nanjing Normal University), Chenmin Sun (Universit\`{e} Paris-Est Cr\`{e}teil), and Taiki Takeuchi (Kyushu University).

Overdetermined problems in cylinders and related questions

Filomena Pacella University of Roma Sapienza Italy Co-Author(s):    Danilo Gregorin Afonso, Paolo Caldiroli, Alessandro Iacopetti, David Ruiz, Pieralberto Sicbaldi

Abstract: We consider overdetermined semilinear elliptic problems in bounded domains contained in an unbounded cylinder in $\mathbb{R^N}$. The variational formulation of these problems naturally leads to study the stationary points (under a volume constraint) of a corresponding energy functional. In particular, domains which are local minima of the energy are of special interest. We will present several results which show that the domains with the simplest geometry (namely bounded cylinders) are not always the best candidates to minimize the energy. This, in turn, suggests the existence of nontrivial domains for which the overdetermined problem admits a solution which can be obtained by globally minimizing the energy or by a bifurcation analysis.

: Global existence of solutions of a class of system of reaction-diffusion equations on evolving domains.

Jyotshana Prajapat University of Mumbai India Co-Author(s):    Vandana Sharma

Abstract: In this talk, I will discuss the existence and global existence of solutions of system of reaction diffusion equation where the components diffuse inside an evolving domain and react on the surface through mass transport type boundary conditions. Furthermore, we also report progress on the case where some components react and diffuse on the boundary of the region in addition to the components reacting and diffusing in the interior of an evolving domain. Interactions for different boundary conditions will be explored.

Critical cases of Boundary Hardy and applcation to Moser-Trudinger inequality

Prosenjit Roy Indian Institute of Technology, Kanpur India Co-Author(s):    Adimurthi, Purbita Jana, Vivek Sahu

Abstract: Boundary Hardy inequality states that if ~$1 < p < \infty$ and $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^d$, then $$\int_{\Omega} \frac{|u(x)|^{p}}{\delta^{p}_{\Omega}(x)} dx \leq C\int_{\Omega} |\nabla u(x) |^{p}dx, \forall \ u \in C^{\infty}_{c}(\Omega),$$ where ~$\delta_\Omega(x)$ is the distance function from $\partial\Omega$. B. Dyda generalised the above inequality to the fractional setting, which says, for $sp >1$ and $s\in (0,1)$ $$ \int_{\Omega} \frac{|u(x)|^{p}}{\delta_{\Omega}^{sp}(x)} dx \leq C \int_{\Omega} \int_{\Omega} \frac{|u(x)-u(y)|^{p}}{|x-y|^{d+sp}} dxdy, \ \forall \ u \in C^{\infty}_{c}(\Omega). $$ The first and the second inequality is not true for $p=1$ and $sp=1$ respectively. In this talk, I will present the appropriate inequalities for the critical cases: $p=1$ for the first and $sp= 1$ for the second inequality. I will also discuss the case when the weight function ($\delta_\Omega$) in the first inequality is replaced by distance function from a $k-$ dimensional sub manifold of $\Omega$ and some related applications to Moser-Trudinger inequality.

Deficit estimates for an entropic form of Gagliardo-Nirenberg inequalities related to nonlinear diffusion equations and their application

Takeshi Suguro Kumamoto University Japan Co-Author(s):

Abstract: We consider functional inequalities concerning the $p$-Laplace equation. In this talk, we introduce an entropic form of the Gagliardo--Nirenberg inequalities based on the Tsallis entropy, a one-parameter extension of the Boltzmann--Shannon entropy. We obtain deficit estimates for the inequality and consider their application to the uncertainty relation inequality.

Bifurcation into spectral gaps for Schr\odinger equations: from local to non local case

Cristina Tarsi Universit\`a degli Studi di Milano Italy Co-Author(s):

Abstract: In this talk we consider a Schr\odinger-Choquard equation of the type $$ -\Delta u +V(x)u=(I_\alpha\ast |u|^p)|u|^{p-2}u+\lambda u \quad x \in \mathbb R^N $$ where $V$ is a periodic and non costant potential, $I_\alpha$ denotes the Riesz potential and $N\geq 3$. In this case, the spectrum of the self-adjoint operator $-\Delta + V$ in $L^2(\mathbb R^N)$ is purely continuous and may contain gaps. An interesting physical and mathematical issue is establishing the existence of branches of solutions converging towards the trivial solution as $\lambda$ approaches some {\emph{bifurcation point}} of the spectrum. An intriguing situation is the so called {\emph{gap-bifurcation}}, occurring at boundary points of the spectral gaps: we review the main known results and open problems in the local case (in presence of a local perturbation $f(u)$) and we address the non local case.

Some nonlinear heat equations with exponential non-linearity and with singular data in two dimensions

Elide Terraneo University of Milano Italy Co-Author(s):    Y. Fujishima, N. Ioku , and B. Ruf

Abstract: In this talk we deal with a class of nonlinear heat equations in two dimensions. Recently, for some specific nonlinearities with exponential growth of Trudinger-Moser type, Ioku et al and Ibrahim et al establish the existence of a singular stationary solution. Then, they prove that the Cauchy problem, with this singular solution as initial data, admits, at least, two different solutions. Here we consider similar nonuniqueness phenomena for a wider class of nonlinearities in two dimensions.

Recent progress in the study of concentrated helical vortices of 3d incompressible Euler equations

Jie Wan Beijing Institute of Technology Peoples Rep of China Co-Author(s):    Daomin Cao, Rui Li, Guolin Qin

Abstract: In this talk, we first introduce recent progress of the existence of concentrated helical vortices of 3d incompressible Euler equations when the orthogonality condition holds. Then, in case that the orthogonality condition fails, we prove the existence of concentrated helical vortices of 3d incompressible Euler equations in infinite cylinders. As parameter $\varepsilon\to0$, the vorticity concentrates near a helix evolved by the vortex filament equations.