Weighted Sobolev spaces and embedding theorems
By V. Gol’dshtein, A. Ukhlov. 2007
215 citations recorded by the source corpus.
In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A p -condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. The proposed method is based on the concept of 'generalized' quasiconformal homeomorphisms (homeomorphisms with bounded mean distortion). The choice of the homeomorphism type depends on the choice of the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition operators for weighted Sobolev spaces. With the help of these homeomorphism classes the embedding problem for non-smooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic Holder singularities demonstrate the sharpness of our machinery comparatively with known results.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/ba5699f8378e9b624524a7196f2b1777dbd7b92b
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In Applied Mathematics
Citations in the corpus
- Borderline Variational Problems Involving Fractional Laplacians and Critical Singularities (2015, background): “t is remarkable that most of the properties of classical Sobolev spaces, including the embedding theorems have a weighted counterpart as long as the weight is in the Muckenhoupt class A2 see [14] and [23]. Note that H1(Rn+1 + ,y 1−α) – up to a normalization factor– is also isometric to Xα(Rn+1 + ). ”
- On the First Eigenvalues of Free Vibrating Membranes in Conformal Regular Domains (2015, methodology): “The following theorem gives a characterization of composition operators in the classical Sobolev spaces W 1,p (this theorem was proved, but not formulated, in [14,15]): For readers convenience we reproduce here the proof.”