Symmetry of minimizers with a level surface parallel to the boundary
Let \(\Omega\) be a bounded domain in~\(\mathbb R^N\), and let \(u\) be the minimizer of the functional \(I_\Omega\) defined as follows: \[ I_\Omega(v) = \int_\Omega \Big( f(|Dv(x)|) - v(x) \Big) \, dx, \tag{1.1} \] where \(v\) ranges in the Sobolev space~\(W^{1,\infty}_0\) and \(f\)~satisfies the following assumptions: \noindent (f1) \(f \in C^1([0,+\infty))\) is a convex, nondecreasing function, vanishing at the origin and superlinear at infinity in the sense that \[ \lim_{s \to +\infty} f(s)/s = +\infty; \] (f2) \(f\) is constant over the interval \([0,\sigma]\) for some~\(\sigma \geq 0\), belongs to the Hölder class~\(C^{2,\alpha}((\sigma, \allowbreak +\infty))\) for some \(\alpha \in (0,1)\) and satisfies \(f'(s), f''(s) > 0\) for all \(s \in (\sigma, \allowbreak +\infty)\). \noindent Suppose that for some \(\delta > 0\) the parallel surface ~\(\Gamma_\delta\) (=~the set of all \(x \in \Omega\) whose distance from~\(\mathbb R^N \setminus \Omega\) equals~\(\delta\)) is the boundary of a \(C^1\)-domain \(G \subset\subset \Omega\) satisfying the interior sphere condition. The authors prove that if the minimizer~\(u\) is of class~\(C^1\) in a tubular neighborhood of~\(\Gamma_\delta\), and also constant on~\(\Gamma_\delta\), then \(\Omega\) is a ball (Theorem~1.1). In Section~3 assumption (f2) is relaxed by admitting functions \(s \mapsto f(|s|)\) that are not differentiable at \(s = 0\), and the result is recovered under some restrictions. In the last section of the paper, a possible extension to fully nonlinear elliptic and parabolic equations is outlined. The proof of Theorem 1.1 is based on the moving plane method, which was introduced by \textit{A. D. Aleksandrov} [Transl., Ser. 2, Am. Math. Soc. 21, 341--354, 354--388 (1962; Zbl 0122.39601)] and then by \textit{J. Serrin} [Arch. Ration. Mech. Anal. 43, 304--318 (1971; Zbl 0222.31007)]. See also [\textit{A. D. Aleksandrov}, Ann. Mat. Pura Appl., IV. Ser. 58, 303--315 (1962; Zbl 0107.15603)] and [\textit{H. Berestycki} and \textit{L. Nirenberg}, Bol. Soc. Bras. Mat., Nova Sér. 22, No. 1, 1--37 (1991; Zbl 0784.35025)]. The method, now classical, is adapted to the present problem following [\textit{R. Magnanini} and \textit{S. Sakaguchi}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27, No. 3, 937--952 (2010; Zbl 1194.35209); Math. Methods Appl. Sci. 36, No. 15, 2023--2032 (2013; Zbl 1279.35062)], where radial symmetry for parabolic problems was investigated. The condition on \(u\) of being constant along \(\Gamma_\delta\) should be compared to the Neumann condition \[ \frac{\, \partial u \,}{\, \partial \nu \,} = \text{constant on \(\partial \Omega\),} \] which occurs in Serrin's overdetermined problem. Such a comparison is done in the reviewed paper on p. 2791, bottom. The paper is related to a recent contribution of \textit{G. Crasta} [J. Eur. Math. Soc. (JEMS) 8, No. 1, 139--154 (2006; Zbl 1114.49024)] and to a forthcoming paper by the authors [``Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration'', J. Anal. Math. (to appear)].
scientific article