MANY-FACETED ATTACK ON THE COMPLEX GINZBURG-LANDAU EQUATION

1)讨论了在有界区域上取Lebesgue空间L-p （p>2）初值的复Ginzburg-Landau方程的初边值问题。作为推论，我们可以重新表述Ginibre-Velo（1997）的结果，其中初始值取自Sobolev空间H-1。此外，如果我们取H-m （m是大于等于2的整数）的初始值，我们可以削弱对非线性项幂指数的限制。同时也考虑了平稳问题。(2)给出了具有奇异一阶系数的薛定谔算子的拟m-活动的一个新的充分条件。这一结果被认为是对已故加藤俊雄教授的结果的改进。(3)得到了有界域上拉普拉斯算子的特征函数e_n的估计。| (e_n) (x) |以特征值的常数倍N/4次方为界。这个N=1的指数出现在一维谐振子的薛定谔算子的特征函数的估计中。

1) Initial-boundary value problems for the complex Ginzburg-Landau equation on a bounded domain is discussed when we take the initial values from the Lebesgue space L-p (p>2). As a corollary we can reformulate the result of Ginibre-Velo (1997) in which the initial values are taken from the Sobolev space H-1. In addition we can weaken the restriction on the exponents of the power of the nonlinear term if we take the initial values from H-m (m is a integer greater than or equal to 2). The stationary problems are also considered. (2) We have presented a new sufficient condition which guarantees the quasi-m-accretivity of Schroedinger operators with singular first-order coefficients. The result is regarded as an improvement of that by late Professor Tosio Kato. (3) We have obtained the estimates of the eigenfunctions e_n of the Laplace operator on a bounded domain. | (e_n) (x) | is bounded by the constant multiple of the eigenvalue to the power of N/4. This exponent with N=1 appears in the estimates of the eigenfunctions of the Schroedinger operator of the one-dimensional harmonic ocsilltor.