Related papers: Interior Regularity for a generalized Abreu Equati…
We study the Abreu's equation in n-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform K-stability.
In this paper we prove the interior regularity for the solution to the Abreu equation in any dimension assuming the existence of the $C^0$ estimate.
We study a generalized Abreu equation and derive some estimates.
We study a generalized Abreu Equation in $n$-dimensional polytopes and prove some differential inequalities for homogeneous toric bundles.
We study interior regularity of solutions of a generalized stationary Stokes problem in the plane. The main, elliptic part of the problem is given in the form div(A(Du)), where D is the symmetric part of the gradient. The model case is…
In this paper, we study the regularity of solutions to a linear elliptic equation involving a mixed local-nonlocal operator of the form $$Lu - \operatorname{div}\big(a(x)\nabla u(x)\big)= f, \quad \text{in } \Omega \subset \mathbb{R}^n,$$…
We consider a fourth order partial differential equation in n-dimensional space introduced by Abreu in the context of K\"{a}hler metrics on toric orbifolds. Similarity solutions depending only on the radial coordinate in R^n are determined…
We survey some new results regarding a priori regularity estimates for the Boltzmann and Landau equations conditional to the boundedness of the associated macroscopic quantities. We also discuss some open problems in the area. In…
The paper develops various estimates for solutions of a fourth order nonlinear PDE, which corresponds to prescribing the scalar curvature of a toric Kahler metric.
In this paper, we consider a matroid generalization of the stable matching problem. In particular, we consider the setting where preferences may contain ties. For this generalization, we propose a polynomial-time algorithm for the problem…
We study the regularity of solutions of parabolic fully nonlinear nonlocal equations. We proof Holder regularity in space and time and for translation invariant equations and under different assumptions on the kernels Holder regularity for…
Finding equilibria of the finite size Kuramoto model amounts to solving a nonlinear system of equations, which is an important yet challenging problem. We translate this into an algebraic geometry problem and use numerical methods to find…
In this paper, we study the Abreu equation on toric surfaces. In particular, we prove the existence of the positive extremal metric when relative $K$-stability is assumed.
In this paper, we investigate the interior regularity theory for stationary solutions of the supercritical nonlinear elliptic equation $$ -\Delta u=|u|^{p-1}u\quad\text{in }\Omega,\quad p>\frac{n+2}{n-2}, $$ where $…
We propose the generalized quadrature methods for numerical solution of singular integral equation of Abel type. We overcome the singularity using the analytical calculation of the singular integral expression. The problem of solution of…
We study particular solutions of the inner equation associated to the splitting of separatrices on generalized standard maps. An exponentially small complete expression for their difference is obtained. We also provide numerical evidence…
We consider the aggregation equation $u_t= \div(\nabla u-u\nabla \K(u))$ in a bounded domain $\Omega\subset \R^d$ with supplemented the Neumann boundary condition and with a nonnegative, integrable initial datum. Here, $\K=\K(u)$ is an…
We derive a local uniform boundedness result for an elliptic equation having interior singularity.
We construct stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties. In particular, this gives an elliptic generalization of the results of arXiv:1211.1287. We apply them to the computation of the monodromy of…
In this paper, we study the generalized Abreu equation on a Delzant ploytope $\Delta \subset \mathbb{R}^2$ and prove the existence of the constant scalar metrics of homogeneous toric bundles under the assumption of an appropriate stability.