A critical Moser type inequality with loss of compactness due to infinitesimal shocks

Autor: João Marcos do Ó, Bernhard Ruf, Pedro Ubilla
Rok vydání: 2022
Předmět:
Zdroj: Calculus of Variations and Partial Differential Equations. 62
ISSN: 1432-0835
0944-2669
DOI: 10.1007/s00526-022-02367-5
Popis: We consider a one-dimensional integral inequality of Moser type: set $$\begin{aligned} \quad J_c(v) = \int _{0}^1 {\textrm{e}}^{c(s) v^2(s)} ds \quad \quad \hbox { and consider } \quad \quad \sup _{ \{\int _0^1 |v'|^2 = 1, v(0) = 0\}} J_c(v) \end{aligned}$$ J c ( v ) = ∫ 0 1 e c ( s ) v 2 ( s ) d s and consider sup { ∫ 0 1 | v ′ | 2 = 1 , v ( 0 ) = 0 } J c ( v ) We show that the supremum remains finite up to the optimal coefficient $$c_1(s) = \frac{1}{s}(\log \frac{\textrm{e}}{s} + \log \log \frac{\textrm{e}}{s})\,$$ c 1 ( s ) = 1 s ( log e s + log log e s ) . Indeed, for $$c_\gamma = \frac{1}{s}(\log \frac{\textrm{e}}{s} + \gamma \log \log \frac{\textrm{e}}{s})$$ c γ = 1 s ( log e s + γ log log e s ) , with $$\gamma > 1$$ γ > 1 , the supremum is infinite. For $$c_1$$ c 1 the inequality is critical with loss of compactness: the functional $$J_{c_1}$$ J c 1 fails to be weakly continuous along the infinitesimal Moser sequence $$w_n(t):= t{\sqrt{n}}\ (0 \le t \le \frac{1}{n})\; \ w_n(t) = \frac{1}{\sqrt{n}} \ (\frac{1}{n} \le t \le 1)$$ w n ( t ) : = t n ( 0 ≤ t ≤ 1 n ) w n ( t ) = 1 n ( 1 n ≤ t ≤ 1 ) . Since $$w'(t) = \sqrt{n}\ (0 \le t \le \frac{1}{n})$$ w ′ ( t ) = n ( 0 ≤ t ≤ 1 n ) , one may say that $$w_n$$ w n develops an infinitesimal shock at the origin.
Databáze: OpenAIRE
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