Elliptic partial differential equations are important both from application and analysis points of views. In this paper we apply the Closest Point Method to solving elliptic equations on general curved surfaces. Based on the closest point representation of the underlying surface, we formulate an embedding equation for the surface elliptic problem, then discretize it using standard finite differ...

We relate two parameter solutions of the sixth Painlevé equation and finitegap solutions of the Heun equation by considering monodromy on a certain class of Fuchsian differential equations. In the appendix, we present formulae on differentials of elliptic modular functions, and obtain the ellitic form of the sixth Painlevé equation directly.

In what follows by Ω we denote a bounded domain in R n with n ≥ 2, ∂Ω ∈ C 2,α and 0 < α < 1. We let Lu(x) = n i,k=1 a i,k (x) · u xix k (x) + n i=1 a i (x) · u xi (x) + a(x) · u(x) = f (x) to be a linear elliptic differential equation of second order with coefficients a ik , a i , a, f ∈ C 0,α (Ω) and a(x) ≤ 0 for all x ∈ Ω. The paper of J. Schauder " ¨ Uber lineare elliptische Differentialglei...

is responsible for the progress of nonlinear elliptic PDE theory in the last century. Indeed, most early works on nonlinear elliptic problems focused on this particular equation. There are many beautiful existence, uniqueness, regularity theorems for the minimal surface equation, see for example [13]. The graph of a solution to (1.1) is naturally a minimal hypersurface in R. In general, we can ...

Simulations of elliptic particulate suspensions in the planar Poiseuille ow are performed by using the lattice Boltzmann equation. E ects of the multi-particle interaction on the lateral migration and rotational motion of both neutrally and non-neutrally buoyant elliptic particles are investigated. Low and intermediate total particle volume fraction fa = 13%, 25%, and 40% are considered in this...

In his work on Diophantine equations of the form y2=ax4+bx3+cx2+dx+e, Fermat introduced the notion of primitive solutions. In this expository note we intend to interpret this notion more geometrically, and explain what it means in terms of the arithmetic of elliptic curves. The specific equation y2 =x4 + 4x3 + 102 +20x+ 1 was used extensively by Fermat as an example. We illustrate the nowadays ...

We study here the convergence of a finite volume scheme for a coupled system of an hyperbolic and an elliptic equations defined on an open bounded set of IR. On the elliptic equation, a four points finite volume scheme is used then an error estimate on a discrete H norm of order h is proved, where h defines the size of the triangulation. On the hyperbolic equation, one uses an upstream scheme w...

We study the boundedness of the Hausdorff measure of the singular set of any solution for a semi-linear elliptic equation in general dimensional Euclidean space Rn. In our previous paper, we have clarified the structures of the nodal set and singular set of a solution for the semi-linear elliptic equation. In particular, we showed that the singular set is (n − 2)-rectifiable. In this paper, we ...

We compare three nonlinear programming approaches to a well-known elliptic inverse problem in three spatial dimensions. Two of these approaches may be viewed as conventional; the third approach is new and is based on a domain decomposition technique for the solution of the governing elliptic equation. We discuss the beneets that may be obtained from treating the governing diierential equation i...