Extremals for a Moser-Jodeit exponential inequality.
Let \(W^ n(D)\) be the Sobolev space of functions \(u\) supported in the closure of a bounded domain \(D\) in \(\mathbb R^ n\) whose gradient belongs to \(L^ n(D)\). The classical result of \textit{N. S. Trudinger} [J. Math. Mech. 17, 473--484 (1967; Zbl 0163.36402)] states that for \(u\) in the unit ball of \(W^ n(D)\), there are positive constants \(\alpha\) and \(A\), depending only on the dimension \(n\), such that \[ \int_D\exp(\alpha u^{\frac n{n-1}})\,dx\leq A| D| . \] \textit{J. Moser} [Indiana Univ. Math. J. 20, 1077--1092 (1971; Zbl 0213.13001)], using reduction to the one-dimensional case by symmetrization, established the largest possible value of \(\alpha\) and showed that the Trudinger theorem has extremals. The integer \(n\) now can be replaced by a real number \(q\geq 2\). For \(1<q<2\), the same result was established by \textit{M. Jodeit} [Stud. Math. 44, 545--554 (1972; Zbl 0244.26010)]. Thus, if \(\omega\) is a \(C^1\) function on \([0,\infty)\) such that \(\omega(0)=0\) and \(\int| \omega'| ^ q\leq 1\), then the quantity \[ A(q)=\sup_{\omega}\int_0^\infty\exp(\omega^ p(t)-t)\,dt \] is finite. \textit{L. Carleson} and \textit{S.-Y. A. Chang} [Bull. Sci. Math., II. Sér. 110, 113--127 (1986; Zbl 0619.58013)] proved that this theorem has extremals for integers bigger than or equal to 2. This shows that Moser's improved version of Trudinger's result has extremals, at least when \(D\) is a ball. This result was later extended to the case of arbitrary smooth bounded domains by \textit{M. Flucher} [Comment. Math. Helv. 67, 471--497 (1992; Zbl 0763.58008)]. The main result of the paper under review is that the above mentioned Jodeit-Moser theorem has extremals for every real number \(q>1\). Several applications of this result are discussed.
scientific article; zbMATH DE number 1868176