相关论文: On the Allen-Cahn equation in the Grushin plane: a…

We present arguments for the existence of both globally regular and black hole solutions of the Einstein equations with a conformally coupled scalar field, in the presence of a negative cosmological constant, for space-time dimensions…

In higher dimensional gauge theory, we need energies with higher power terms of field strength in order to realize point-wise monopoles. We consider new models with higher power terms of field strength and extraordinary kinetic term of…

We prove nonlinear stability for a large class of solutions to the Einstein equations with a positive cosmological constant and compact spatial topology in arbitrary dimensions, where the spatial metric is Einstein with either positive or…

We consider static cosmological solutions along with their stability properties in the framework of a recently proposed theory of massive gravity. We show that the modifcation introduced in the cosmological equations leads to several new…

An entire solution of the Allen-Cahn equation $\Delta u=F'(u)$, where $F$ is an even, bistable function, is called a $2k$-end solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$…

Cylindrically-symmetric solutions in Conformal Gravity are investigated and several new solutions are presented and discussed. Among them, a family of vacuum solutions, generalizations of the Melvin solution and cosmic strings of the…

We consider nonnegative solutions to $-\Delta u=f(u)$ in unbounded euclidean domains, where $f$ is merely locally Lipschitz continuous and satisfies $f(0)<0$. In the half-plane, and without any other assumption on $u$, we prove that $u$ is…

We prove that positive solutions of the superlinear Lane-Emden system in a two-dimensional smooth bounded domain are bounded independently of the exponents in the system, provided the exponents are comparable. As a consequence, the energy…

A long-standing conjecture of De Giorgi asserts that every monotone solution of the Allen--Cahn equation in \(\mathbb{R}^{n+1}\) is one-dimensional if \(n \leq 7\). A stronger version of the conjecture, also widely studied and often called…

The semi-linear, elliptic PDE $AC_{\varepsilon}(u):=-\varepsilon^2\Delta u+W'(u)=0$ is called the Allen-Cahn equation. In this article we will prove the existence of finite energy solution to the Allen-Cahn equation on certain complete,…

In this article one will discuss the system of coupled nonlinear Klein-Gordon equations with different velocities and different masses. The nonlinearity considered is a general quadratic nonlinearity without any restriction. The method is a…

In analogy with the standard derivation of the Schwarzschild solution, we find all static, cylindrically symmetric solutions of the Einstein field equations for vacuum. These include not only the well known cone solution, which is locally…

We study solutions of the inner-variational equation associated with the Dirichlet energy in the plane, given homeomorphic Sobolev boundary data. We prove that such a solution is monotone if and only if its Jacobian determinant does not…

We discuss spherically symmetric static solutions of the Einstein-Klein-Gordon equations for a real scalar field with a mass and a quartic self-interaction term. As for the massless case the solutions have a naked singularity at the origin.…

We analyze existence and properties of solutions of two-dimensional general relativistic initial data sets with a negative cosmological constant, both on spacelike and characteristic surfaces. A new family of such vacuum, spacelike data…

We show that the method used in the Schwarzschild black hole for finding the elementary solution of the electrostatic equation in closed form cannot extend in higher dimensions. By contrast, we prove the existence of static, spherically…

The Cauchy problem of the Cahn-Hilliard equations is studied in three-dimensional space. Firstly, we construct its approximate fourth-order parabolic equation, obtaining the existence of solutions by the Aubin-Lions's compactness lemma.…

The Schwarzschild solution is a complete solution of Einstein's field equations for a static spherically symmetric field. The Einstein's field equations solutions appear in the literature, but in different ways corresponding to different…

A model one-dimensional self consistent steady state collisionless self-gravitating system in which all the particles have the same energy is presented. This has the remarkable property that the position and velocity of the particles…

We construct infinite-dimensional families of non-singular static space times, solutions of the vacuum Einstein-Maxwell equations with a negative cosmological constant. The families include an infinite-dimensional family of solutions with…