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

940925振子我们研究了n维实空间中有界域上的奇异半线性椭圆边值问题。这一问题在科学应用中的重要性已得到广泛承认(见W)。这项提议的动机有三个。首先，我们希望通过将方程中的系数函数从Holder连续放宽到Lipschitz可积来推广已有的定理。其次，我们感兴趣的是对方程组实现类似结果的可能性。第三个动力来自我们对单位区间和单位平方的数值结果，这些结果表明多重网格法可以为系数函数的合理选择提供一种有效的求解手段。然而，多重网格完全近似格式的实际收敛证明是非常困难的，而且在多重网格完全近似格式的建立理论方面的文献也非常稀少。我们未来对所讨论问题的求解方法的研究将填补这一空白，并探索新的、可能更有效的技术，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. ***