We consider stationary solutions of the incompressible Navier-Stokes equations in two dimensions. We give a detailed description of the fluid flow in a half-plane by using a mathematical setup within which the idea of a change of type from an elliptic to a parabolic partial differential equation can be made precise.

We establish an extension of Liouville’s classical representation theorem for solutions of the partial differential equation ∆u = 4 e2u and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps with prescribed critical points and specified boundary behaviour. For instance, we show that for every Blaschke sequence {zj} in the unit disk there is always a Blasc...

A double sub-equation method is presented for constructing complexiton solutions of nonlinear partial differential equations (PDEs). The main idea of the method is to take full advantage of two solvable ordinary differential equations with different independent variables. With the aid of Maple, one can obtain both complexiton solutions, combining elementary functions and the Jacobi elliptic fun...

In this article, we study the existence of nonnegative solutions for the elliptic partial differential equation −[M(‖u‖p1,p)] ∆pu = f(x, u) in Ω, u = 0 on ∂Ω, where Ω ⊂ RN is a bounded smooth domain, f : Ω×R+ → R is a discontinuous nonlinear function.

Higher order elliptic partial differential equations with Dirichlet boundary conditions in general do not satisfy a maximum principle. Polyharmonic operators on balls are an exception. Here it is shown that in R small perturbations of polyharmonic operators and of the domain preserve the maximum principle. Hence the Green function for the clamped plate equation on an ellipse with small eccentri...

We study a seawater intrusion problem in a confined aquifer. This process can be formulated as a coupled system of partial differential equations which includes an elliptic and a degenerate parabolic equation. Existence results of weak solutions, under realistic assumptions, are established through time discretization combined with parabolic regularization.

In this paper, based on the generalized Jacobi elliptic function expansion method,we obtain abundant new complex doubly periodic solutions of the double Sine-Gordon equation (DSGE), which are degenerated to solitary wave solutions and triangle function solutions in the limit cases,showing that this new method is more powerful to seek exact solutions of nonlinear partial differential equations i...

We investigate the numerical solution of a boundary control problem with elliptic partial differential equation by the hp-finite element method. We prove exponential convergence with respect to the number of unknowns for an a-priori chosen discretization. Here, we have to prove that derivatives of arbitrary order of the solution are in suitably chosen weighted Sobolev spaces. Numerical experime...

Motivated by the partial differential equations of mixed type that arise in the reduction of the Einstein equations by a helical Killing vector field, we consider a boundary value problem for the helically-reduced wave equation with an arbitrary source in 2+1 dimensional Minkowski spacetime. The reduced equation is a second-order partial differential equation which is elliptic inside a disk and...

Abstract The Monge-Ampère equation det D 2 u = e is completely nonlinear and elliptic, the convexity estimates for solution of elliptical partial differential very important. We establish a inequality by constructing an auxiliary function give two with 0 boundary value condition.