A symmetry result for the torsion equation in an unbounded domain in 3D
An overdetermined problem is obtained by associating to a partial differential equation so many conditions that no solution exists, in general. The situation may be compared, roughly speaking, to a linear algebraic system with more (independent) equations than unknowns. Nevertheless, it may well happen that an overdetermined problem does possess a solution in some particular domain \(\Omega\). For instance, a celebrated result by \textit{J. Serrin} [Arch. Ration. Mech. Anal. 43, 304--318 (1971; Zbl 0222.31007)] and \textit{H. F. Weinberger} [Arch. Ration. Mech. Anal. 43, 319--320 (1971; Zbl 0222.31008)] asserts that the following overdetermined problem: \[ \begin{cases} \Delta u = -1 &\text{in \(\Omega\),} \cr u = 0 &\text{on \(\partial \Omega\),} \cr |\nabla u| = c &\text{on \(\partial \Omega\),} \end{cases} \] where \(\Omega\) is a bounded, smooth domain in \(\mathbb R^N\), \(N \geq 2\), and \(c\) is a positive constant, is solvable if and only if \(\Omega\) is a ball. The case when \(\Omega\) is an \textit{exterior domain}, i.e., a domain whose complement is compact, was investigated by \textit{A. Aftalion} and \textit{J. Busca} [C. R. Acad. Sci., Paris, Sér. I 324, 633--638 (1997; Zbl 0876.35032)] and by \textit{W. Reichel} [Arch. Ration. Mech. Anal. 137, 381--394 (1997; Zbl 0891.35006)]. The reviewed paper is part of a branch of current research on similar problems in unbounded domains: see, for instance, [\textit{F. Hélein} et al., Pacific J. Math. 250, 319--334 (2011; Zbl 1211.35207); \textit{A. Ros} et al., Commun. Pure Appl. Math. 70, 1223--1252 (2017; Zbl 1373.35207); J. Differ. Equations 255, 951--977 (2013; Zbl 1284.35297); \textit{F. Schlenk} and \textit{P. Sicbaldi}, Adv. Math. 229 (2012; Zbl 1233.35147); \textit{M. Traizet}, Geom. Funct. Anal. 24, 690--720 (2014; Zbl 1295.35344)]. To be more precise, let \(u\) be a continuous function over \(\mathbb R^3\) satisfying \(0 \leq u \leq a\) in \(\mathbb R^3\) and such that the set \(\{\, 0 < u < a \,\}\) is connected (here \(a\) is a positive constant). Assume that for each \(R > 0\) the level sets \(\{u = 0\}\) and \(\{u = a\}\) contain a ball of radius \(R\) and that either the superlevel set \(\{u > 0\}\) or the sublevel set \(\{u < a\}\) is included in some half-space \(H\). If \(u\) satisfies \[ \begin{cases} \Delta u = -1 &\text{in }\{\, 0 < u < a \,\}, \cr |\nabla u| = c &\text{on }\partial \, \{\, u = 0 \,\}, \cr |\nabla u| = 0 &\text{on }\partial \, \{\, u = a \,\}, \end{cases} \] then \(c = \sqrt{2a \, }\) and, up to a convenient change of the coordinate frame, we may write \[ \{\, 0 < u < a \,\} = \mathbb R^2 \times (0, \, \sqrt{2a \,}) \] as well as \[ u(x,y,z) = {} - \frac1{\, 2 \,} \, z^2 + \sqrt{2a \,} \, z \quad \text{for }z \in (0, \, \sqrt{2a \,}). \] The proof is based on the comparison of \(u\) with suitable functions, which are in their turn derived from the distance function to the catenoid: this makes the argument \(3\)-dimensional. In particular, the author takes advantage of the fact that when the equatorial radius \(\varepsilon\) of the catenoid is let vanish, then the surface approaches its equatorial plane and not its axis as one may erroneously expect. In order to allow comparison with the solution \(u\), the comparison functions mentioned before are conveniently displaced (\textit{slided}) up to a limiting position. By contrast, the \textit{sliding method} described in \textit{H. Berestycki} and \textit{L. Nirenberg} [Bol. Soc. Bras. Mat. 22, 1--37 (1991; Zbl 0784.35025)] relies on the comparison between the unknown function \(u\) and a translation of the same function \(u\). A two-dimensional result analogous to the present one was proved by \textit{L. Wang} and \textit{L. Wang} [J. Math. Anal. Appl. 407, 550--556 (2013; Zbl 1310.35024)].
scientific article; zbMATH DE number 6756247
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