Related papers: On De Giorgi Conjecture in Dimension $N \geq 9$

We prove the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s >= 3; this is new for s > 3, and the cases s<3 have also been previously established. More precisely, we establish that if f : [N] -> [-1,1] is a function with || f…

Let $(M,g)$ be an $n$-dimensional asymptotically flat Riemannian manifold with nonnegative scalar curvature that admits a noncompact area-minimizing hypersurface $\Sigma \subset M$. In the case where $n = 3$, O. Chodosh and the first-named…

For any unitary conformal field theory in two dimensions with the central charge $c$, we prove that, if there is a nontrivial primary operator whose conformal dimension $\Delta$ vanishes in some limit on the conformal manifold, the…

Given a family of hypergraphs $\mathcal{H}$, we say that a hypergraph $\Gamma$ is $\mathcal{H}$-universal if it contains every $H \in \mathcal{H}$ as a subgraph. For $D, r \in \mathbb{N}$, we construct an $r$-uniform hypergraph with…

In models with a low quantum gravity scale, one might expect that all operators consistent with gauge symmetries are present in the low-energy effective theory. If this is the case, some mechanism must be present to adequately suppress…

We provide a classification result for positive solutions to $\Delta u = u^{-\gamma}$ in the half space, under zero Dirichlet boundary condition.

In the projective space $\mathrm{PG}(N,q)$ over the Galois field of order $q$, $N\ge3$, an iterative step-by-step construction of complete caps by adding a new point on every step is considered. It is proved that uncovered points are evenly…

Let $G$ be a graph, and let $u$, $v$, and $w$ be vertices of $G$. If the distance between $u$ and $w$ does not equal the distance between $v$ and $w$, then $w$ is said to resolve $u$ and $v$. The metric dimension of $G$, denoted $\beta(G)$,…

In 2014, Keevash proved the existence of $(n,q,r)$-Steiner systems (equivalently $K_q^r$-decompositions of $K_n^r$) for all large enough $n$ satisfying the necessary divisibility conditions. In 2021, Glock, K\"uhn, and Osthus proposed a…

We give an alternative proof of the Schoen--Simon--Yau curvature estimates and associated Bernstein-type theorems (1975), and extend the original result by including the case of $6$-dimensional (stable minimal) immersions. The key step is…

Let $u_1$ and $u_2$ be two different positive smooth solutions of the equation $\Delta u + n (n - 2) u^{{n + 2}\over {n - 2}} = 0$ in $R^n (n \ge 3).$ By a result of Gidas, Ni and Nirenberg, $u_1$ and $u_2$ are radially symmetric above the…

We examine the fourth order problem $\Delta^2 u = \lambda f(u) $ in $ \Omega$ with $ \Delta u = u =0 $ on $ \partial \Omega$, where $ \lambda > 0$ is a parameter, $ \Omega$ is a bounded domain in $ R^N$ and where $f$ is one of the following…

For all $N \geq 9$, we find smooth entire epigraphs in $\R^N$, namely smooth domains of the form $\Omega : = \{x\in \R^N\ / \ x_N > F (x_1,\ldots, x_{N-1})\}$, which are not half-spaces and in which a problem of the form $\Delta u + f(u) =…

We show that, for any $0<\gamma<1/2$, any $(\alpha,\beta)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrt{\gamma}$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such…

Let $N\geq 2$ and $1 < p < (N+2)/(N-2)_{+}$. Consider the Lane-Emden equation $\Delta u + u^p = 0$ in $\mathbb{R}^N$ and recall the classical Liouville type theorem: if $u$ is a non-negative classical solution of the Lane-Emden equation,…

In this paper we give a proof of Enomoto's conjecture for graphs of sufficiently large order. Enomoto's conjecture states that, if $G$ is a graph of order $n$ with minimum degree $\delta(G)\geq \frac{n}{2}+1$, then for any pair of vertices…

Erd\H{o}s, Gallai, and Tuza posed the following problem: given an $n$-vertex graph $G$, let $\tau_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $\alpha_1(G)$ denote the largest size of a…

We focus on the classification of positive solutions to $(-\Delta)^s u=\frac{x_n^{\alpha}}{u^\gamma}$ in the half space with $\gamma>0$, subject to the Dirichlet condition. We show that when $-2s<\alpha<(\gamma-1)s$, all positive solutions…

We study the following $1$-Yamabe equation on a connected finite graph $$\Delta_1u+g\mathrm{Sgn}(u)=h|u|^{\alpha-1}\mathrm{Sgn}(u),$$ where $\Delta_1$ is the discrete $1$-Laplacian, $\alpha>1$ and $g, h>0$ are known. We show that the above…

In present paper, we prove the existence of solutions $(\lambda, u)\in \R\times H^1(\R^N)$ to the following Schr\"odinger equation $$ \begin{cases} -\Delta u(x)+V(x)u(x)+\lambda u(x)=g(u(x))\quad &\hbox{in}~\R^N\\ 0\leq u(x)\in H^1(\R^N),…