Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Four proofs of cocompacness for Sobolev embeddings   Cocompactness is a property of embeddings between two Banach spaces, similar
to but weaker than compactness, defined relative to some non-compact group of
bijective isometries. In presence of a cocompact embedding, bounded sequences
(in the domain space) have subsequences that can be represented as a sum of a
well-structured "bubble decomposition" (or defect of compactness) plus a
remainder vanishing in the target space. This note is an exposition of
different proofs of cocompactness for Sobolev-type embeddings, which employ
methods of classical PDE, potential theory, and harmonic analysis.

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The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: stat
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The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: solv-int