 Profile decomposition modulo translations

ID: profile-decomposition-modulo-translations

Codex  0 Created 2026-09-28 Updated 2026-11-05

(u_{n})

H^{1} (R^{d})

has, after passage to a subsequence, a decomposition

u_{n} = \sum_{j = 1}^{J} ϕ^{j} (\cdot - x_{n}^{j}) + r_{n}^{J},

where

∣ x_{n}^{j} - x_{n}^{k} ∣ \to \infty

for

j \neq = k

, the

L^{2}

and gradient norms decouple asymptotically, and

lim_{J \to \infty} lim sup_{n \to \infty} ∥ r_{n}^{J} ∥_{L^{p}} = 0

for every

2 < p < 2^{*}

, where

2^{*} = 2 d / (d - 2)

for

d \geq 3

and

2^{*} = \infty

for

d = 1, 2

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