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Weighted \(L^p\)-theory for Poisson, biharmonic and Stokes problems on periodic unbounded strips of \({{\mathbb {R}}}^n\). (English) Zbl 1361.46032

Summary: This paper establishes isomorphisms for Laplace, biharmonic and Stokes operators in weighted Sobolev spaces. The \(W^{m,p}_{\alpha}({{\mathbb {R}}}^n)\)-spaces are similar to standard Sobolev spaces \(W^{m,p}({\mathbb {R}}^n)\), but they are endowed with weights \((1+|x|^2)^{\alpha /2}\) prescribing functions’ growth or decay at infinity. Although well established in \({{\mathbb {R}}}^n\) [C. Amrouche et al., J. Math. Pures Appl. (9) 73, No. 6, 579–606 (1994; Zbl 0836.35038)], these weighted results do not apply under the specific hypothesis of periodicity. This kind of problem appears when studying singularly perturbed domains (roughness, sieves, porous media, etc): when zooming on a single perturbation pattern, one often ends with a periodic problem set on an infinite strip. We present a unified framework that enables a systematic treatment of such problems in the context of periodic strips. We provide existence and uniqueness of solutions in our weighted Sobolev spaces. This gives a refined description of the solution’s behavior at infinity which is of importance in the multi-scale context. The isomorphisms are valid for any relative integer \(m\), any \(p\) in \((1,\infty)\), and any real \(\alpha \) out of a countable set of critical values for the Stokes, the biharmonic and the Laplace operators.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46N20 Applications of functional analysis to differential and integral equations
35Q30 Navier-Stokes equations
35C15 Integral representations of solutions to PDEs

Citations:

Zbl 0836.35038
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Full Text: DOI

References:

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