Mathematical Sciences: A Singular Semilinear Elliptic Boundary Value Problem in Fluid Theory

研究了n维实空间中有界区域上的一类奇异半线性椭圆型边值问题。这个问题在科学应用中的重要性已得到广泛认识（见W）。提出这一建议的动机有三个方面。首先，我们将方程中的系数函数由Holder连续放宽为Lipschitz可积，从而推广已有的定理。其次，我们感兴趣的是在方程组中获得类似结果的可能性。第三个动力来自于我们对单位间隔和单位平方的数值结果，这些结果表明多重网格方法可以为系数函数的合理选择提供有效的解决方法。然而，对于多网格全近似格式（FAS）的实际收敛性证明是非常困难的，而且在FAS方法的建立理论方面的文献也非常稀少。我们对所讨论问题的解决方法的未来研究将填补这一空白，并研究新的、也许更有效的技术，如多层投影（PML）方法来处理这类非线性问题。＊＊＊

940925 Shaker We study a singular semilinear elliptic boundary value problem in a bounded domain in the n dimensional real space. The importance of this problem in scientific applications has been widely recognized (see W ). The motivation for this proposal is three-fold. First we wish to generalize the existing theorems by relaxing the coefficient function in the equation from being Holder continuous to Lipschitz integrable. Second we are interested in possibilities of achieving similar results for systems of equations. A third impetus proceeds from our numerical results on the unit interval and unit square which have shown that multigrid methods can provide an efficient means of solution for reasonable choices of the coefficient function. However, an actual convergence proof for the multigrid Full Approximation Scheme (FAS) is very hard to obtain and the literature is remarkably sparse in the area of founding theory for the FAS method. Our future research into solution methods for the discussed problems will be to fill this gap and to investigate new and perhaps more efficient techniques such as the Multilevel Projection (PML) methods to deal with this type of nonlinear problems. ***