Regularity of the solutions of elliptic systems in polyhedral domains

نویسندگان

  • Serge Nicaise
  • S. Nicaise
چکیده

The solution of the Dirichlet problem relative to an elliptic system in a polyhedron has a complex singular behaviour near edges and vertices. Here, we show that this solution has a global regularity in appropriate weighted Sobolev spaces. Some useful embeddings of these spaces into classical Sobolev spaces are also established. As applications, we consider the Lamé, Stokes and Navier-Stokes systems. The present results will be applied in a forthcoming work to the constructive treatment of these problems by optimal convergent finite element method. 1 Preliminaries Let Ω ⊂ R be a bounded Lipschitz domain whose boundary Γ is a straight polyhedron. On Ω and Γ, we shall consider the usual Sobolev spaces H(Ω) and H(Γ), s ∈ R, with respective norms and semi-norms denoted by ‖ · ‖s,Ω or | · |s,Ω and ‖ · ‖s,Γ or | · |s,Γ (see [5] for the precise definition). ◦ H s (Ω) is the closure in H(Ω) of D(Ω), the space of C∞ functions with compact support in Ω. We take as interior operators ADN-elliptic systems of multi-degree m = (m1, · · · , mN), homogeneous with constant coefficients as explained below, with Dirichlet boundary conditions. Received by the editors Septembre 1995. Communicated by J. Mawhin. 1991 Mathematics Subject Classification : 35B40, 35B65, 35J67.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Sobolev Spaces and Regularity for Polyhedral Domains

We prove a regularity result on polyhedral domains P ⊂ R using the weighted Sobolev spaces Ka (P). In particular, we show that there is no loss of Ka –regularity for solutions of strongly elliptic systems with smooth coefficients. In the proof, we identify Ka (P) with the Sobolev spaces on P associated to the metric r P gE , where gE is the Euclidean metric and rP(x) is a smoothing of the Eucli...

متن کامل

Sobolev Spaces on Lie Manifolds and Regularity for Polyhedral Domains

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping properties of pseudodifferential operators. A tubular neighborhood theorem for...

متن کامل

Weighted Sobolev spaces and regularity for polyhedral domains

We prove a regularity result for the Poisson problem −∆u = f , u|∂P = g on a polyhedral domain P ⊂ R 3 using the Babuška–Kondratiev spaces Ka (P). These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges [4, 29]. In particular, we show that there is no loss of Ka –regularity for solutions of strongly elliptic systems with smooth coefficients. We also es...

متن کامل

The Solvability of Concave-Convex Quasilinear Elliptic Systems Involving $p$-Laplacian and Critical Sobolev Exponent

In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.

متن کامل

Weighted analytic regularity in polyhedra

We explain a simple strategy to establish analytic regularity for solutions of second order linear elliptic boundary value problems. The abstract framework presented here helps to understand the proof of analytic regularity in polyhedral domains given in the authors’ paper in Math. Models Methods Appl. Sci. 22 (8) (2012). We illustrate this strategy by considering problems set in smooth domains...

متن کامل

A new embedding result for Kondratiev spaces and application to adaptive approximation of elliptic PDEs

In a continuation of recent work on Besov regularity of solutions to elliptic PDEs in Lipschitz domains with polyhedral structure, we prove an embedding between weighted Sobolev spaces (Kondratiev spaces) relevant for the regularity theory for such elliptic problems, and TriebelLizorkin spaces, which are known to be closely related to approximation spaces for nonlinear n-term wavelet approximat...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 1997