相关论文: Zero energy critical points of functionals dependi…

It is widely believed that the energy functional $E_p:(\mathbb{S}^2)^n \rightarrow \mathbb{R}$ $$ E_p = \sum_{i,j=1 \atop i \neq j}^{n} \frac{1}{\|x_i-x_j\|^p}$$ has a number of critical points, $\nabla E(x) = 0$, that grows exponentially…

Existence and regularity of minimizers in elliptic free boundary problems have been extensively studied in the literature. We initiate the corresponding study of higher critical points by considering a superlinear free boundary problem…

We study critical and stationary, i.e. critical with respect to both inner and outer variations, points of polyconvex functionals of the form $f(X) = g(\det(X))$, for $X \in \mathbb{R}^{2\times 2}$. In particular, we show that critical…

Given a closed Riemann surface $(\Sigma,g)$ and any positive smooth weight, we use a minmax scheme together with compactness, quantization results and with sharp energy estimates to prove the existence of positive critical points of the…

Recently the first author studied the bifurcation of critical points of families of functionals on a Hilbert space, which are parametrised by a compact and orientable manifold having a non-vanishing first integral cohomology group. We…

This paper focuses on the analysis of a free energy functional, that models a dilute suspension of magnetic nanoparticles in a two-dimensional nematic well. The {\it first part} of the article is devoted to the asymptotic analysis of global…

We prove that every continuous function on a separable infinite-dimensional Hilbert space X can be uniformly approximated by smooth functions with no critical points. This kind of result can be regarded as a sort of very strong approximate…

In this paper we study the functional $\SW_{\lambda_1,\lambda_2}$, which is the the sum of the Willmore energy, $\lambda_1$-weighted surface area, and $\lambda_2$-weighted volume, for surfaces immersed in $\R^3$. This coincides with the…

We study an energy functional on the universal spinor bundle over a closed $n$-dimensional spin manifold $M$. The critical points of this functional, which is modelled on the total torsion functional of $G_2$-structures in seven dimensions,…

We consider a nonlinear problem $F(\lambda,u)=0$ on infinite-dimensional Banach spaces that correspond to the steady-state bifurcation case. In the literature, it is found again a bifurcation point of the approximate problem…

We show that every linear functional on the Dirichlet space that is non-zero on nowhere-vanishing functions is necessarily a multiple of a point evaluation. Continuity of the functional is not assumed. As an application, we obtain a…

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…

On the 2-dimensional unit disk $B_1$ we study the Moser-Trudinger functional $$E(u)=\int_{B_1}(e^{u^2}-1)dx, u\in H^1_0(B_1)$$ and its restrictions to $M_\Lambda:=\{u \in H^1_0(B_1):\|u\|^2_{H^1_0}=\Lambda\}$ for $\Lambda>0$. We prove that…

Given a bounded Lipschitz domain $\omega\subset\mathbb{R}^{d-1}$ and a lower semicontinuous function $W:\mathbb{R}^N\to\mathbb{R}_+\cup\{+\infty\}$ that vanishes on a finite set and that is bounded from below by a positive constant at…

A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new even for isometries of Banach spaces as well as for non-locally compact…

This paper concerns the existence of critical points for solutions to second order elliptic equations of the form $\nabla\cdot \sigma(x)\nabla u=0$ posed on a bounded domain $X$ with prescribed boundary conditions. In spatial dimension…

We consider the optimization problem associated with fitting two-layer ReLU networks with $k$ hidden neurons, where labels are assumed to be generated by a (teacher) neural network. We leverage the rich symmetry exhibited by such models to…

In this paper, we are concerned with the following type of fractional problems: $$ \begin{cases}\dis (-\Delta)^{s} u-\mu\frac{u}{|x|^{2s}}-\lambda u=|u|^{2^*_{s}-2}u+f(x,u), &\text{in} \Omega,\ \ \, u=0\,&\text{in} \R^N\backslash\Omega…

We prove that the number of critical points of a Li-Tam Green's function on a complete open Riemannian surface of finite type admits a topological upper bound, given by the first Betti number of the surface. In higher dimensions, we show…

We use some tools from nonlinear analysis to study two examples of singular stochastic elliptic PDEs that cannot be solved by the contraction principle or the Schauder fixed point theorem. Let $\xi$ stand for a spatial white noise on a…