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This paper establishes isomorphisms for Laplace, biharmonic and Stokes operators in weighted Sobolev spaces. The $$W^{m,p}_{\alpha }({{\mathbb {R}}}^n)$$ W α m , p ( R n ) -spaces are similar to standard Sobolev spaces $$W^{m,p}_{}({\mathbb {R}}^n)$$ W m , p ( R n ) , but they are endowed with weights $$(1+|x|^2)^{\alpha /2}$$ ( 1 + | x | 2 ) α / 2 prescribing functions’ growth or decay at infinity. Although well established in $${{\mathbb {R}}}^n$$ R n [3], these weighted results do not apply in 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 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 $$(1,\infty )$$ ( 1 , ∞ ) , and any real $$\alpha $$ α out of a countable set of critical values for the Stokes, the biharmonic and the Laplace operators.
ANNALI DELL'UNIVERSITA' DI FERRARA – Springer Journals
Published: Sep 16, 2015
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