Parameter-elliptic and parabolic pseudodifferential boundary problems in globalLp sobolev spaces
Parameter-elliptic and parabolic pseudodifferential boundary problems in globalLp sobolev spaces
复制标题
globalLp sobolev 空间中的参数椭圆和抛物线伪微分边界问题
DOI：
10.1007/bf02571889
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发表时间：
1995
影响因子：
0.8
通讯作者：
G. Grubb
中科院分区：
文献类型：
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作者：
G. Grubb
拟微分边值问题理论是在Vishik，L·斯金，Boutet de Monvel，Rempel，Schulze等人的著作中发展起来的，它是包括椭圆边值问题及其解算子的一个大框架。参数相关微积分的引入使抛物型问题的研究成为可能，其应用包括含时的N-S问题(参见[GS L，G-S_2])。本文将参数椭圆型和抛物型微积分推广到Lp-Sobolev空间(1<；p<；oc)，在Lp-Sobolv空间中，参数椭圆型和抛物型微积分只在L2框架下可用。通过处理空间一致的符号类，我们还将结果推广到合适的非紧流形，包括IRn和~+中的外域。在参数依赖的Besov和S、uBessel-位势空间B~和Hp中，建立了微积分中的基本的、有些技术性的步骤和边界算子系统d的映射性质，本文是本文的直接延续。在这里，我们发展了一致Jbrmly参数椭圆组(以及无参数一致椭圆组)的理论。一个有趣的特例是椭圆PS的预解式构造。数据库，有问题了。为了在泛函演算中有用，重要的是要知道，当一个逆存在于某个S，p，#时，它属于微积分。我们对一般的参数椭圆型系统证明了这一点(并包括关于“谱不变性”的一个结果)。然后利用预解考虑建立了这类抛物型初边值问题的可解性结果。
The theory of pseudodifferential boundary problems has been developed, as a large framework including elliptic boundary value problems and their solution operators, in works by Vishik, l~ skin, Boutet de Monvel, Rempel, Schulze, the present author, and others. The introduction of a parameter-dependent calculus (cf.[G1]) made it possible to study also parabolic problems, with applications including the time-dependent Navier-Stokes problem (cf.[GS l, G-S2]). The present work is concerned with the extension of the parameter-elliptic and the parabolic calculus to Lp Sobolev spaces (1< p< oc), where it was earlier available only in an L2 framework. By working with spatially uniform symbol classes, we also extend the results to suitable noncompact manifolds, including exterior domains in IR n and~+. The basic, somewhat technical steps in the calculus, and the mapping properties of the boundary operator systems d, in parameter-dependent Besov and s,, uBessel-potential spaces B~ and Hp, have been established in a joint work with N. Kokholm [GK]; and the present paper is a direct continuation. Here we develop the theory of uniJbrmly parameter-elliptic systems d,(and uniformly elliptic systems without parameter). An interesting special case is the resolvent construction for an elliptic ps. db, problem. For the usefulness in functional calculus it is important to know that an inverse, when it exists for some s, p,#, belongs to the calculus. We show this for general parameterelliptic systems (and include a result on" spectral invariance"). The resolvent considerations are then used to establish solvability results for parabolic initial-boundary value problems of the type