A sharp Trudinger-Moser type inequality for unbounded domains in \(\mathbb R^2\)
The classical Trudinger-Moser inequality states that for a bounded domain \(\Omega \subset {\mathbb R}^2\), \[ \sup_{\| u \| _{D} \leq 1} \int_{\Omega}(\exp^{\alpha u^2} -1) \, dx = c(\Omega) < +\infty \quad \text{for } \alpha \leq 4 \pi,\tag{1} \] where \(\| u \| _{D} = ( \int_{\Omega} | \nabla u | ^2 \, dx )^{1/2}\) is the usual Dirichlet norm in \( H^1_0(\Omega)\). Since the supremum in the preceding inequality becomes infinite for domains \(\Omega\) such that \( | \Omega | = \infty\), the Trudinger-Moser inequality does not hold for unbounded domains. Moreover, this inequality is optimal, since, when \(\alpha > 4\pi\), then the corresponding supremum is \(+ \infty\). In this paper, the author shows that if the Dirichlet norm is replaced with the standard Sobolev norm, \(\| u \| _{S} = ( \| u \| _{D}^2 + \| u \| _{L^2}^2 )^{1/2}\) in \( H^1_0(\Omega)\), the inequality holds for any domain with a bound independent of the domain. More precisely, he proves the following Theorem. There exists a constant \(d > 0\) such that, for any domain \(\Omega \subset {\mathbb R}^2\), \[ \sup_{\| u \| _{S} \leq 1} \int_{\Omega}(\exp^{4 \pi u^2} -1)\,dx \, \leq \, d,\tag{2} \] where the growth factor \(4\pi\) is sharp. Next, the author presents a result due to \textit{L. Carleson} and \textit{S.-Y. A. Chang} [Bull. Sci. Math. 110, 113--127 (1986; Zbl 0619.58013)] that describes the highest possible level of non-compactness of normalized (with respect to the Dirichlet norm) concentrating sequences. Then, he defines the Carleson-Chang limit as the maximal limit of Sobolev-normalized concentrating sequences. Thereafter, in a second theorem, the author finds an upper bound of the Carleson-Chang limit for any domain, with sharp estimates for \( \Omega = {\mathbb R}^2\) and for the ball \( \Omega = B_R(0)\). Thus, in a third theorem, the author is able to show, using a restatement of a concentration-compactness alternative of Lions, that the supremum in (2) is achieved for \( \Omega = {\mathbb R}^2\) and for the ball \( \Omega = B_R(0)\).
Sobolev normalized concentrating sequences
scientific article; zbMATH DE number 2141630