Related papers: Interior estimates for solutions of Abreu's equati…
We give a uniform estimate for solutions of prescribed scalar curvature type equation in dimension 4.
We study the Dirichlet problem of the Abreu equation. The solutions provide the Kahler metrics of constant scalar curvature on the complex torus.
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 study the Abreu's equation in n-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform K-stability.
We explain a simple construction of solutions to a family of PDE's in two dimensions which includes that defining zero scalar curvature Kahler metrics, with two Killing fields, and the affine maximal equation.
We give Harnack inequalities for solutions of equations of type prescribed scalar curvature in dimensions n $\ge$ 4.
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
In this paper, we investigate interior gradient estimates for solutions to the mean curvature equation $$ \dive \left( \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) = f(\nabla u)$$ under various nonlinear assumptions on the right-hand…
We study the fourth order Schr\"odinger type differential inequality $-\Delta^2 u + \lambda V(x)u \geq a(x)u^q$ with $a,V\in L^1_{loc}(\mathbf{R}^N)$, both nonnegative, and $\lambda>0$. We consider nonnegative solutions without making any…
This paper consists of real and complex affine techniques for studying the Abreu equation on toric surfaces. In particular, an interior estimate for Ricci tensor is given.
The paper develops a continiuty method for solutions of the Abreu equation, which include extremal metrics on toric surfaces. Results are obtained, assuming a hypothesis (the "M-condition") on the solutions.
In this paper, we study the interior gradient estimates for admissible solutions to prescribed curvature equations in hyperbolic space.
The last decades saw growing interest across multiple disciplines in nonlinear phenomena described by partial differential equations (PDE). Integrability of such equations is tightly related with the Painleve property - solutions being free…
We obtain new $L_p$ estimates for subsolutions to fully nonlinear equations. Based on our $L_p$ estimates, we further study several topics such as the third and fourth order derivative estimates for concave fully nonlinear equations,…
We solve Abreu's equation with periodic right hand side, in any dimension. This can be interpreted as prescribing the scalar curvature of a torus invariant metric on an Abelian variety.
We study a generalized Abreu Equation in $n$-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform $K$-stability.
The paper represents the method for construction of the families of particular solutions to some new classes of $(n+1)$ dimensional nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic…
An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in…
A nonlinear inequality is formulated in the paper. An estimate of the rate of growth/decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can…
In this paper there are estimated the derivatives of the solution of an initial boundary value problem for a nonlinear uniformly parabolic equation in the interior with the total variation of the boundary data and the L^{infinity}-norm of…