Non-homogeneous boundary value problems and applications
Non-homogeneous boundary value problems and applications
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DOI：
10.1007/978-3-642-65393-3
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发表时间：
1972
期刊：
影响因子：
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通讯作者：
J. Lions;E. Magenes
中科院分区：
文献类型：
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作者：
J. Lions;E. Magenes
1.我们首先以一种非常正式的方式描述我们的基本目标。设m是R的一个开子集，边界为am。在m和am上分别引入线性微分算子P和Qj ′ 0 ~ i ∈ ′ V.所谓”非齐次边值问题”是指如下类型的问题：设f和gj ′ 0 ~ i ∈ ′ V分别在函数空间sF和G中给定，F是“m”上的空间，G/s是“am”上的空间;我们在满足（1）Pu= f in m，（2）Qju= gj on am，0~ i（v <<]）的函数空间u/t”on m”中寻找u。Qj可以在am的部分上相同地为零，使得边界条件的数量可以取决于被认为是2的am的部分。我们采取的”工作假设”，对于fEF和gjEG，j的问题（1），（2）承认一个唯一的解决方案u EU/t，它依赖于3连续的数据。但对于全线性问题，空间su/t和{F; G}（自然地联系在一起）有大量的选择。一般地说，我们的目的是确定与问题（1），（2）以”自然”的方式相联系的便于应用的空间族ft和{F; G}，以及这些空间族中u/t和{F; G} j的所有可能的选择.
1. We describe, at first in a very formaI manner, our essential aim. n Let m be an op en subset of R, with boundary am. In m and on am we introduce, respectively, linear differential operators P and Qj'0~ i~'V. By" non-homogeneous boundary value problem" we mean a problem of the following type: let f and gj'0~ i~'v, be given in function space s F and G, F being a space" on m" and the G/s spaces" on am"; j we seek u in a function space u/t" on m" satisfying (1) Pu= f in m,(2) Qju= gj on am, 0~ i~'v «])). Qj may be identically zero on part of am, so that the number of boundary conditions may depend on the part of am considered 2. We take as" working hypothesis" that, for fEF and gjEG, j the problem (1),(2) admits a unique solution u EU/t, which depends 3 continuously on the data. But for alllinear probIems, there is a large number of choiees for the space su/t and {F; G}(naturally linke d together). j Generally speaking, our aim is to determine families of spaces' ft and {F; G}, associated in a" natural" way with problem (1),(2) and con j venient for applications, and also all possible choiees for u/t and {F; G} j in these families.