相关论文: Symmetry of global solutions to a class of fully n…

It is shown that gravity on the line can be described by the two dimensional (2D) Hilbert-Einstein Lagrangian supplemented by a kinetic term for the coframe and a translational {\it boundary} term. The resulting model is equivalent to a…

We establish the existence of strong solutions to a class of nonlinear strongly coupled and uniform elliptic systems consisting of more than two equations. The existence of of nontrivial and non constant solutions (or pattern formations)…

We show that any classical solution of the diffusive Hamilton-Jacobi (DHJ) equation $-\Delta u= |\nabla u|^p$ in a half-space with zero boundary conditions for $1<p\le 2$ is necessarily one-dimensional. This improves the previously known…

We consider weak distributional solutions to the equation $-\Delta_pu=f(u)$ in half-spaces under zero Dirichlet boundary condition. We assume that the nonlinearity is positive and superlinear at zero. For $p>2$ (the case $1<p\leq2$ is…

We prove uniform finite-time existence of solutions to the vacuum Einstein equations in polarized U(1) symmetry which have uniformly positive incoming $H^1$ energy supported on an arbitrarily small set in the 2 + 1 spacetime obtained by…

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…

We derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and…

We explicitly construct the series expansion for a certain class of solutions to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric solutions. We express the Taylor coefficients through some universal combinatorial…

We present some results concerning the solvability of linear elliptic equations in bounded domains with the main coefficients almost in VMO, the drift and the free terms in Morrey classes containing $L_{d}$, and bounded zeroth order…

We prove the non-linear stability of a large class of spherically symmetric equilibrium solutions of both the collisonless Boltzmann equation and of the Euler equations in MOND. This is the first such stability result that is proven with…

A discretized massless wave equation in two dimensions, on an appropriately chosen square lattice, exactly reproduces the solutions of the corresponding continuous equations. We show that the reason for this exact solution property is the…

A new method is given for proving the global existence of the solution to nonlinear Volterra integral equations. A bound on the solution is derived. The results are based on a nonlinear inequality proved by the author earlier.

We obtain the general static solutions of the axially symmetric (2+1)-dimensional Einstein-Maxwell-Dilaton theory by dimensionally reducing it to a 2-dimensional dilaton gravity theory. The solutions consist of the magnetically charged…

We obtain a comparison result for solutions to nonlinear fully anisotropic elliptic problems by means of anisotropic symmetrization. As consequence we deduce a priori estimates for norms of the relevant solutions.

We give formulations of noncommutative two dimensional gravities in terms of noncommutative gauge theories. We survey their classical solutions and show that solutions of the corresponding commutative theories continue to be solutions in…

A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This result is based on an iterative procedure…

We consider the Gelfand problem with rapidly growing nonlinearities in the two-dimensional bounded strictly convex domains. In this paper, we prove the uniformly boundedness of finite Morse index solutions. As a result, we show that there…

We show that the system of vacuum Einstein equations (i.e., Ricci-flat metrics) with two hypersurface-orthogonal, commuting Killing vector fields in $d \ge 5$ dimensions is invariant under the action of a one-parameter Lie group, and the…

$ \newcommand{\schs}{\scriptstyle{\mathsf{S}}_1} $For all $n \ge 1$, we give an explicit construction of $m \times m$ matrices $A_1,\ldots,A_n$ with $m = 2^{\lfloor n/2 \rfloor}$ such that for any $d$ and $d \times d$ matrices…

We consider positive singular solutions of PDEs arising from double phase functionals. Exploiting a rather new version of the moving plane method originally developed by Sciunzi, we prove symmetry and monotonicity properties of such…