A profile decomposition for the limiting Sobolev embedding. (English) Zbl 1481.46026
Ahusborde, É. (ed.) et al., Fifteenth international conference Zaragoza-Pau on mathematics and its applications. Proceedings of the conference, Jaca, Spain, September 10–12, 2018. Zaragoza: Prensas de la Universidad de Zaragoza. Monogr. Mat. García Galdeano 42, 65-78 (2019).
Summary: For many known non-compact embeddings of two Banach spaces \(E\hookrightarrow F\), every bounded sequence in \(E\) has a subsequence that takes the form of a profile decomposition – a sum of clearly structured terms with asymptotically disjoint supports plus a remainder that vanishes in the norm of \(F\). In this note we construct a profile decomposition for arbitrary sequences in the Sobolev space \(H^{1,2} (M)\) of a compact Riemannian manifold, relative to the embedding of \(H^{1,2}(M)\) into \(L^{2^*}(M)\), generalizing the well-known profile decomposition of
M. Struwe [Math. Z. 187, 511–517 (1984; Zbl 0535.35025); Proposition 2.1] to the case of arbitrary bounded sequences.
For the entire collection see [Zbl 1448.65007].
For the entire collection see [Zbl 1448.65007].
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46B50 | Compactness in Banach (or normed) spaces |
| 58J99 | Partial differential equations on manifolds; differential operators |
| 35B44 | Blow-up in context of PDEs |
| 35A25 | Other special methods applied to PDEs |
