On the second eigenfunctions of the Laplacian in R2

A conjecture about the nodal line of a second eigenfunction states that the nodal line of a second eigenfunction divides the domain by intersecting with the boundary of transversely, where is a bounded convex domain ofR ². We prove this conjecture provided has a symmetry. Also, we prove the multiplicity of the second eigenvalue is two at most provided is a bounded convex domain ofR ².Supported in part by NSF DMS 84-09447Home Institution: Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA     

A modified simplex method for multifactor optimization of selectivity in gas chromatography

A computer-assisted advanced simplex method is presented for the simultaneous optimization of multifactor ( stationary phase loading, carrier gas flow rate and column temperature ) for separation of ten compounds in gas chromatography. A three factors factorial design was used. The method was based on a special polynomial established from fifteen preliminary runs, using resolution as the selection criterion, with connection to a general simplex method. Excellent agreement is found between the predicted data and the experimental results, and most of experiments required in the general simplex method can be omitted.     

Uniqueness of conformal metrics with prescribed total curvature in R^2

In this paper, we investigate the solution structure of solutions of where K(x) is a H?lder function in . For a given positive total curvature, we consider the problem of the uniqueness of solutions with this prescribed total curvature. We apply various methods such as the method of moving spheres and the isoperimetric inequality to show the uniqueness for several classes of K. Received December 15, 1998 / Accepted April 23, 1999     

Existence of self-dual non-topological solutions in the Chern–Simons Higgs model

In this paper we investigate the existence of non-topological solutions of the Chern–Simons Higgs model in R². A long standing problem for this equation is: Given N vortex points and β>8π(N+1), does there exist a non-topological solution in R² such that the total magnetic flux is equal to β/2? In this paper, we prove the existence of such a solution if . We apply the bubbling analysis and the Leray–Schauder degree theory to solve this problem.     

Degree counting and Shadow system for Toda system of rank two: One bubbling

We initiate the program for computing the Leray–Schauder topological degree for Toda systems of rank two. This program still contains a lot of challenging problems for analysts. As the first step, we prove that if a sequence of solutions $(u_{1k},u_{2k})$ blows up, then one of $\frac{h_j{e}^{u_{jk}}}{{\int }_M{h}_j{e}^{u_{jk}}d{v}_g}$, $j=1,2$ tends to a sum of Dirac measures. This is so-called the phenomena of weak concentration. Our purposes in this article are (i) to introduce the shadow system due to the bubbling phenomena when one of parameters ${\rho }_i$ crosses 4π and ${\rho }_j?4\pi \mathbb{N}$ where $1\le i\ne j\le 2$; (ii) to show how to calculate the topological degree of Toda systems by computing the topological degree of the general shadow systems; (iii) to calculate the topological degree of the shadow system for one point blow up. We believe that the degree counting formula for the shadow system would be useful in other problems.     

Blow-up for a reaction–diffusion equation with variable coefficient

We study the blow-up behavior for positive solutions of a reaction–diffusion equation with nonnegative variable coefficient. When there is no stationary solution, we show that the solution blows up in finite time. Under certain conditions, we then show that any point with zero source cannot be a blow-up point.