Related papers: Some Estimates for a Generalized Abreu's Equation

We study a generalized Abreu 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.

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.

We study a generalized Abreu Equation in $n$-dimensional polytopes and prove some differential inequalities for homogeneous toric bundles.

This is a survey on extended affine Lie algebras and related types of Lie algebras, which generalize affine Lie algebras.

Some solutions of the Heavenly equations and their generalizations are considered

We present some old and new results on dispersive estimates for Schroedinger equations.

The generalized CR equation $u_{\bar{z}}=au+b\bar{u}+f$ is studied when the coefficients $a$ and $b$ have a finite number of singular points inside the domain. Solutions are constructed via the study of an associated integral operator and…

We study generalized complex Monge-Amp\`ere type equations on closed Hermitian manifolds. We derive {\em a priori} estimates and then prove the existence of admissible solutions. Moreover, the gradient estimate is improved.

We prove some weighted $L_p$ estimates for generalized harmonic extensions in the half-space.

We propose generalized Fermat's conjecture in the framework of arithmetic dynamics, and give evidences. The multi-indexed version is added.

In this paper we prove the global second derivative estimates for the second boundary value problem of the prescribed affine mean curvature equation where the affine mean curvature is only assumed to be in $L^{p}$. Our result extends…

Considering some parameters and by means of an inequality of Hadamard, we derive general half-discrete Hilbert-type inequalities. Then we highlight some special cases.

We give some estimate of type sup*inf for scalar curvature type equations.

We study the solvability of singular Abreu equations which arise in the approximation of convex functionals subject to a convexity constraint. Previous works established the solvability of their second boundary value problems either in two…

We Study versions of Cauchy formula in more general algebras than the complex case.

We derive an expression for the generalized Bernoulli numbers in terms of the Bernoulli numbers involving the (exponential) complete Bell polynomials.

We consider the generalized Hurwitz equation $a_1x_1^2+ \cdots +a_nx_n^2 = dx_1 \cdots x_n-k$ and the Baragar-Umeda equation $ax^2+by^2+cz^2=dxyz+e$ for solvability in integers.

Various methods to find Calabi-Yau differential equations are discussed.

We study the solvability of the second boundary value problem of a class of highly singular, fully nonlinear fourth order equations of Abreu type in higher dimensions under either a smallness condition or radial symmetry.