Concentration analysis in Banach spaces (Q2800060)

Concentration compactness principles are a useful tool, for example in the theory of partial differential equations. The authors study general forms of such principles in abstract Banach spaces. The key tool is the notion of \(\Delta\)-convergence: a sequence \((x_k)_{k\in \mathbb{N}}\) in a Banach space \(X\) is said to be \(\Delta\)-convergent to \(x\in X\) if \(\limsup_{k\to \infty}(\| x_k-x\|-\| x_k-y\|)\leq 0\) holds for every \(y\in X\).NEWLINENEWLINE The authors first study the notion of \(\Delta\)-convergence in uniformly convex spaces. For example, they show that in a uniformly convex Banach space \(X\), \(\Delta\)-limits are unique and every bounded sequence has a \(\Delta\)-convergent subsequence. If \(X\) is in addition uniformly smooth, then every \(\Delta\)-convergent sequence is bounded.NEWLINENEWLINEIt is further proved that in a uniformly convex and uniformly smooth Banach space \(X\), \(\Delta\)-convergence is equivalent to weak convergence if and only if \(X\) satisfies the Opial condition (this includes all Hilbert spaces and the spaces \(\ell^p\) for \(1<p<\infty\)).NEWLINENEWLINEThe main result regarding concentration compactness in terms of \(\Delta\)-convergence reads as follows: if \(X\) is uniformly convex and uniformly smooth, \(D_0\) is a group of isometries on \(X\) satisfying a certain property (called dislocation group by the authors), and \(D\subseteq D_0\) with \(\text{id}\in D\), then every bounded sequence \((x_k)_{k\in \mathbb{N}}\) in \(X\) has a \(\Delta\)-profile decomposition with respect to \(D\), i.e., there exist elements \(g_k^{(n)}\in D\) and \(w^{(n)}, r_k\in X\) (for \(n,k\in \mathbb{N}\)) such that the following conditions hold: \(g_k^{(1)}=\text{id}\) for each \(k\), \((h_k^{-1}(r_k))\) is \(\Delta\)-convergent to 0 for every sequence \((h_k)\) in \(D\), \((g_k^{(n)})^{-1}(g_k^{(m)}(x))\to 0\) weakly for every \(x\in X\) and \(n\neq m\), and (after passing to a subsequence \((x_k)\)) one has NEWLINE\[NEWLINEx_k=r_k+\sum_{i=1}^{\infty}g_k^{(i)}(w^{(i)}),NEWLINE\]NEWLINE where the series converges unconditionally and uniformly with respect to \(k\).NEWLINENEWLINEThe authors also prove a version of the Brezis-Lieb lemma with \(\Delta\)-convergence instead of pointwise convergence in the space \(L^p\) for \(p\geq 3\).