Related papers: On singular Abreu equations in higher dimensions
We investigate the qualitative properties of the weak solutions to the boundary value problems for the hyperbolic fourth-order linear equations with constant coefficients in the plane bounded domain convex with respect to characteristics.…
In this work two-point boundary value problem for one class of second order ordinary differential equations with variable coefficients is solved.
We study a class of second-order boundary-degenerate elliptic equations in two dimensions with minimal regularity assumptions. We prove a maximum principle and a Harnack inequality at the degenerate boundary, and assuming local boundedness,…
H\"older estimates for second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions.
This paper studies an initial boundary value problem for a class of nonlinear Dirac equations with cubic terms and moving boundary. For the initial data with bounded $L^2$ norm and the suitable boundary conditions, the global existence and…
We obtain some rigidity results for overdetermined boundary value problems for singular solutions in bounded domains.
We characterize the behavior of the solutions of linear evolution partial differential equations on the half line in the presence of discontinuous initial conditions or discontinuous boundary conditions, as well as the behavior of the…
We examine the regularity of the extremal solution of the nonlinear eigenvalue problem $\Delta^2 u = \lambda f(u)$ on a general bounded domain $\Omega$ in $ \IR^N$, with the Navier boundary condition $ u=\Delta u =0 $ on $ \pOm$. Here $…
The existence of numerical solutions to a fourth order singular boundary value problem arising in the theory of epitaxial growth is studied. An iterative numerical method is applied on a second order nonlinear singular boundary value…
In the present study we consider an example of a boundary value problem for a simple second order ordinary differential equation, which may exhibit a boundary layer phenomenon. We show that usual central finite differences, which are second…
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.
Boundary value problems for linear stationary dispersive equations of order $2l+1$, $l\in \mathbb{N}$ have been considered on finite intervals $(0,L)$. The existence and uniqueness of regular solutions have been established for general…
An introductory review of algebraic classification of the Weyl tensor and algebraically special solutions in higher dimensions.
In this paper we study some boundary value problems for a fractional analogue of second order elliptic equation with an involution perturbation in a rectangular domain. Theorems on existence and uniqueness of a solution of the considered…
The problem of construction of the boundary conditions for nonlinear equations is considered compatible with their higher symmetries. Boundary conditions for the sine-Gordon, Jiber-Shabat and KdV equations are discussed. New examples are…
In this paper we consider a fully third order nonlinear boundary value problem which is of great interest of many researchers. First we establish the existence, uniqueness of solution. Next, we propose simple iterative methods on both…
In this paper, we establish the second order estimates of solutions to the first initial-boundary value problem for general Hessian type fully nonlinear parabolic equations on Riemannian manifolds. The techniques used in this article can…
We study the periodic boundary value problem associated with the second order nonlinear equation \begin{equation*} u'' + ( \lambda a^{+}(t) - \mu a^{-}(t) ) g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth at zero and sublinear…
We study a boundary value elliptic problem having a lower order nonlinear term with subquadratic growth in the gradient of the solution and possibly singular when the solution vanishes. If the singularity is mild enough (and even in the…
In this work we propose a novel approach to investigate boundary value problems (BVPs) for fully third order differential equations. It is based on the reduction of BVPs to operator equations for the nonlinear terms but not for the…