In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron–Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations

In this paper we prove a Pohozaev identity for the weighted anisotropic $p$-Laplace operator. As an application of the identity, we deduce the nonexistence of nontrivial solutions of the Dirichlet problem for the weighted anisotropic $p$-Laplacian in star-shaped domains of $\mathbb R^n$. We also provide an upper bound estimate for the first Dirichet eigenvalue of the anisotropic $p$-Laplacian on bounded

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain, and let $\delta$ be the distance to $\partial \Omega$. In this paper, we study positive solutions of the equation $(\star)\ -L_\mu u+ g(|\nabla u|) = 0$ in $\Omega$, where $L_\mu=\Delta + {\mu}/{\delta^2} $, $\mu \in (0,{1}/{4}]$ and $g$ is a continuous, nondecreasing function on ${\mathbb R}_+$. We prove that if $g$ satisfies

In this article, we will show the scattering of the cubic defocusing nonlinear Schrödinger equation on the waveguide $\mathbb{R}^2\times \mathbb{T}$ in $H^1$. We first establish the linear profile decomposition in $H^{1}(\mathbb{R}^2 \times \mathbb{T})$ motivated by the linear profile decomposition of the mass-critical Schrödinger equation in $L^2(\mathbb{R}^2)$. Then by using the solution of the cubic

The defining equation $$(\ast)\qquad \dot \omega_t=-F'(\omega_t)$$ of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation $(\ast)$ into the family of slowed down gradient flow equations: $\dot \omega^{\varepsilon}_t=- \varepsilon F'( \omega ^{ \varepsilon}_t)$, where $\varepsilon > 0$, and (ii) by considering

In this paper, we consider a blow-up solution $u(t)$ (close to the soliton manifold) to the $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5)_x=0$, with finite blow-up time $T < +\infty$. We expect to construct a natural extension of $u(t)$ after the blow-up time. To do this, we consider the solution $u_{\gamma}(t)$ to the saturated $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5-\gamma

In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schrödinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schrödinger–Poisson system $$\left\{\begin{array}{lll} - \Delta u+ u + \rho (x) \phi u = |u|^{p-1} u, &x\in \mathbb{R}^3, \\ -\Delta \phi=\rho(x) u^2, & x\in \mathbb{R}^3, \end{array}\right.$$ under different assumptions

We show that there exists an integrable function on the $n$-sphere $(n \geq 2)$, whose Cesàro $(C, (n − 1)/2)$ means with respect to the spherical harmonic expansion diverge unboundedly almost everywhere. This extends results of Stein (1961) for flat tori and complements the work of Taibleson (1985) for spheres. We also study relations among Cesàro, Riesz, and Bochner–Riesz means.

We study the cohomological equation associated to linear cocycles on semi simple Lie groups $\mathcal G$ over hyperbolic dynamics. We give sufficient conditions for the solution of the cohomological equation of fiber-bunched cocycles to be unique and for the Hölder conjugacy class of the cocycle to coincide with $C^\nu(M,\mathcal G)$. In particular, we prove that there exists an open and dense subset

We describe a Lefschetz fibration of genus one on the disk cotangent bundle of any closed orientable surface $\Sigma$. As a corollary, we obtain an explicit genus one open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of $\Sigma$.

In this paper we study a Bloch–Torrey regularization of the Rosensweig system for ferrofluids. The scope of this paper is twofold. First of all, we investigate the existence and uniqueness of Leray–Hopf solutions of this model in the whole space $\mathbb R^2$. In the second part of this paper we investigate both the long-time behavior of weak solutions and the propagation of Sobolev regularities in

It is known that the compactness of the commutators of point-wise multiplication with bilinear homogeneous Calderón–Zygmund operators acting on product of Lebesgue spaces is characterized by the multiplying function being in the space CMO. This space is the closure in BMO of its subspace of smooth functions with compact support. It is shown in this work that for bilinear Calderón–Zygmund operators

We discuss the Hörmander multiplier theorem for $L^p$ boundedness of Fourier multipliers in which the multiplier belongs to a fractional Sobolev space with smoothness $s$. We show that this theorem does not hold in the limiting case $|1/p-1/2|=s/n$.

We consider the anisotropic version of an extension problem studied by Caffarelli and Silvestre. We present the anisotropic fractional Laplacian and prove the Almgren’s frequency formula obtained by Caffarelli and Silvestre for the anisotropic case.

We consider a partially overdetermined problem in a sector-like domain $\Omega$ in a cone $\Sigma$ in $\mathbb{R}^N$, $N\geq 2$, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that $\Omega$ is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces $\Gamma$

We construct Zariski K3 surfaces of Artin invariant 1, 2 and 3 in many characteristics. In particular, we prove that any supersingular Kummer surface is Zariski if $p\not\equiv 1$ mod 12. Our methods combine different approaches such as quotients by the group scheme $\alpha_p$, Kummer surfaces, and automorphisms of hyperelliptic curves.

We study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of $f$-minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number. When the first Betti number is large, this improves index estimates known in literature. In

In this note we give a new proof of the sharp constant $C = e^{-1/2} + \int_0^1 e^{-x^2/2}\,dx$ in the weak (1, 1) inequality for the dyadic square function. The proof makes use of two Bellman functions $\mathbb{L}$ and $\mathbb{M}$ related to the problem, and relies on certain relationships between $\mathbb{L}$ and $\mathbb{M}$, as well as the boundary values of these functions, which we find explicitly

Loewner hulls are determined by their real-valued driving functions. We study the geometric effect on the Loewner hulls when the driving function is composed with a random time change, such as the inverse of an $\alpha$-stable subordinator. In contrast to SLE, we show that for a large class of random time changes, the time-changed Brownian motion process does not generate a simple curve. Further we

We study the inverse problem of determining a magnetic Schrödinger operator in an unbounded closed waveguide from boundary measurements. We consider this problem with a general closed waveguide in the sense that we only require our unbounded domain to be contained into an infinite cylinder. In this context we prove the unique recovery of the magnetic field and the electric potential associated with

A set $\mathcal{V}$ in the tridisk $\mathbb{D}^3$ has the polynomial extension property if for every polynomial $p$ there is a function $\phi$ on $\mathbb{D}^3$ so that $\| \phi \|_{\mathbb{D}^3} = \| p \|_{\mathcal{V}}$ and $\phi |_{\mathcal{V}} = p|_{\mathcal{V}}$. We study sets $\mathcal{V}$ that are relatively polynomially convex and have the polynomial extension property. If $\mathcal{V}$ is one-dimensional

We prove that the spectrum constructed by González-Meneses, Manchón and the second author is stably homotopy equivalent to the Khovanov spectrum of Lipshitz and Sarkar at its extreme quantum grading.

We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for the very degenerate elliptic operator $\mathcal{P}^+_{1}$ mapping a function $u$ to the maximum eigenvalue of its Hessian matrix. The aim is to show that, at least for square type domains having fixed volume, the symmetry of the domain maximizes the principal eigenvalue, contrary to what happens for the Laplacian.

If $f, g, h\colon \mathbb{R}^n\longrightarrow\mathbb{R}_{\geq 0}$ are non-negative measurable functions such that $h(x+y)$ is greater than or equal to the $p$-sum of $f(x)$ and $g(y)$, where $-1/n\leq p\leq\infty$, $p\neq0$, then the Borell–Brascamp–Lieb inequality asserts that the integral of $h$ is not smaller than the $q$-sum of the integrals of $f$ and $g$, for $q=p/(np+1)$. In this paper we obtain