Related papers: Qualitative analysis on the critical points of the…

We study the Dirichlet problem for a second order linear elliptic equation in a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, $n \ge 3$, with the drift $\mathbf{b} $ belonging to the critical weak space $L^{n,\infty}(\Omega )$. We…

For a finite set of points $P$ in $R^d$, the function $d_P: R^d \to R^+$ measures Euclidean distance to the set $P$. We study the number of critical points of $d_P$ when $P$ is a Poisson process. In particular, we study the limit behavior…

In this paper, we address the solvability of the critical Lane-Emden system \[\begin{cases} -\Delta u=|v|^{p-1}v &\mbox{in } \Omega_\epsilon,\\ -\Delta v=|u|^{q-1}u &\mbox{in } \Omega_\epsilon,\\ u=v=0 &\mbox{on } \partial \Omega_\epsilon,…

We consider the nonlinear and nonlocal problem $$ A_{1/2}u=|u|^{2^\sharp-2}u\ \text{in \Omega, \quad u=0 \text{on} \partial\Omega $$where $A_{1/2}$ represents the square root of the Laplacian in a bounded domain with zero Dirichlet boundary…

We prove the existence of a positive {\it SOLA (Solutions Obtained as Limits of Approximations)} to the following PDE involving fractional power of Laplacian \begin{equation} \begin{split} (-\Delta)^su&= \frac{1}{u^\gamma}+\lambda…

We investigate the complexity of the Hamiltonian in the pure $p$-spin spin glass model accompanied with a polynomial-type potential on $\mathbb{R}^N$. In this Hamiltonian, the Gaussian field is anisotropic, and the potential lacks…

We study higher critical points of the variational functional associated with a free boundary problem related to plasma confinement. Existence and regularity of minimizers in elliptic free boundary problems have already been studied…

The asymptotical behavior of physical quantities, like the order parameter, the replicon and longitudinal masses, is studied around the zero-field spin glass transition point when a small external magnetic field is applied. An effective…

We consider an open connected set $\Omega$ and a smooth potential $U$ which is positive in $\Omega$ and vanishes on $\partial\Omega$. We study the existence of orbits of the mechanical system \[ \ddot{u}=U_x(u), \] that connect different…

On a Riemannian compact manifold, we give existence and multiplicity results for solutions of elliptic PDE by introducing isometry invariances. When the groups we used have finite orbits, we get multiplicity results for equations with the…

Let $p_n$ be the characteristic polynomial of an $n \times n$ random matrix drawn from one of the compact classical matrix groups. We show that the critical points of $p_n$ converge to the uniform distribution on the unit circle as $n$…

In this paper, we prove the existence of nontrivial unbounded domains $\Omega\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the…

We prove a Chern-Lashof type formula computing the expected number of critical points of smooth function on a smooth manifold $M$ randomly chosen from a finite dimensional subspace $V\subset C^\infty(M)$ equipped with a Gaussian probability…

In many areas of applied mathematics and statistics, it is a fundamental problem to find the best representative of a model by optimizing an objective function. This can be done by determining critical points of the objective function…

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^d$. Under certain conditions on the coefficients of $L$, we first establish the existence of a unique Green's…

We study the spectrum of the Laplacian on the hemisphere with Robin boundary conditions. It is found that the eigenvalues fall into small clusters around the Neumann spectrum, and satisfy a Szeg\H{o} type limit theorem. Sharp upper and…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$ ($n\geq 3$) such that $0\in\partial \Omega$. In this memoir, we consider issues of non-existence, existence, and multiplicity of variational solutions in $H_{1,0}^2(\Omega)$ for the…

We establish the existence and the pointwise bound of the fundamental solution for the stationary Stokes system with measurable coefficients in the whole space $\mathbb{R}^d$, $d \ge 3$, under the assumption that weak solutions of the…

Let $n\geq 3$ and let $\Omega \subset \mathbb{R}^n$ be a $\mathcal{C}^1$ bounded domain which is diffeomorphic to a ball. We investigate here the problem of finding critical points of the $n$-energy in the space $\mathcal{I}=\{v\in…

We study Green functions for the pressure of stationary Stokes systems in a (possibly unbounded) domain $\Omega\subset \mathbb{R}^d$, where $d\ge 2$. We construct the Green function when coefficients are merely measurable in one direction…