MANY-FACETED ATTACK ON THE COMPLEX GINZBURG-LANDAU EQUATION

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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.