The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains

Abstract

Abstract. We introduce certain Sobolev-Besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in Lipschitz domains. In particular, these are used to obtain sharp well-posedness results for the Poisson problem for the Laplacian with mixed boundary conditions on bounded Lipschitz domains which satisfy a suitable geometric condition introduced by R. Brown in (1994). In this context, we obtain results which generalize those by D. Jerison and C. Kenig (1995) as well as E. Fabes, O. Mendez and M. Mitrea (1998). Applications to Hodge theory and the regularity of Green operators are also presented.

Смешанные Задачи В Липшицевой Области Для Сильно Эллиптических Систем 2-Го Порядка

“…в [10]; cм. также [46], [27], [11]). Многие работы посвящены конкретным уравнени-ям или системам, чаще всего это уравнения Лапласа-Гельмгольца или системы теории упругости; кроме названных выше работ, cм.…”

“…Нам оно не понадобится; мы воспользуемся прозрачным подходом в [30] [40] об экстраполяции обратимости операторов, дей-ствующих в интерполяционных шкалах пространcтв. Ср., в частности, [27] (где получены очень общие теоремы об однозначной разрешимости для уравнения Лапласа при дополнительном геометрическом предположении), а также [11]. Отметим, что в этих работах и ряде других авторы выходят за рамки функци-ональных пространств, используемых в нашей статье.…”

Wave diffraction by wedges having arbitrary aperture angle

“…Moreover, among the non-smooth domains, general theories for Lipschitz domains are also available and can be tested in concrete corresponding boundary value problems. Related to this, we would like to refer the works of Costabel [15], Costabel and Stephan [16], Jerrison and Kenig [27,28,29], Kohr, Pintea and Wendland [31], Mitrea and Mitrea [51,52], Mitrea and Taylor [53,54], and Verchota [71].…”

Mixed boundary value problems for the Stokes system

“…While determining the optimal L p -solvability range for the mixed problem (1.1) remains open at the moment, we wish to point out that recent progress in the case when N = ∅ (corresponding to the so-called regularity problem) and D = ∅ (corresponding to the Neumann problem) for the Stokes system in an arbitrary Lipschitz domain Ω ⊂ R n , n ≥ 2, has been made in [21]. For more on the topic of mixed boundary problems in Lipschitz domains the interested reader is referred to [3], [14], [16], [18], [20], and the references cited therein. Let us also mention [15], where value problems for the Navier-Stokes system in polyhedral domains with a variety of mixed boundary conditions are studied and where references to earlier work on this topic can be found.…”