On Serrin's symmetry result in nonsmooth domains and its applications (Q1950799)

The first part of this interesting paper is a short, well-balanced survey on symmetry of solutions of problems associated to elliptic partial differential equations. Two fundamental achievements in that area are discussed: the result on overdetermined problems proved by \textit{J. Serrin} [Arch. Ration. Mech. Anal. 43, 304--318 (1971; Zbl 0222.31007)] and then by \textit{H. F. Weinberger} [Arch. Ration. Mech. Anal. 43, 319--320 (1971; Zbl 0222.31008)], and the result on the Dirichlet problem in symmetric domains obtained afterwards by \textit{B. Gidas} et al. [Commun. Math. Phys. 68, 209--243 (1979; Zbl 0425.35020)]. Several subsequent extensions are cited, and a reference list with 42 items is provided. Then, the author concentrates on reducing the smoothness assumptions on the domain of the problem, as well as on the solution, that are required by the preceding contributions. In particular, in Theorem~2.1 the following overdetermined fully nonlinear problem is considered: \[ \begin{cases} F(D^2 u, \, \nabla u, \, u) = 0, &x \in D, \\ u = 0,\;|\nabla u| = 0, &x \in \partial D, \\ u > 0, &x \in D, \end{cases} \] where \(F\) is elliptic, smooth enough and rotationally invariant. The interest in the boundary condition \(|\nabla u| = 0\) is motivated by the fact that if, instead, \(|\nabla u| = c \neq 0\) on~\(\partial D\) and if \(u\) is smooth up to the boundary, then the boundary is forced to be smooth. The domain~\(D \subset \mathbb R^N\) is an unknown of the problem. Here, instead of assuming smoothness of~\(\partial D\), reflectional symmetry of~\(D\) with respect to some hyperplane (say, the hyperplane \(\{\, x_1 = 0 \,\}\)) is required. It turns out that if such a problem possesses a sufficiently smooth solution~\(u\), then the domain~\(D\) is a ball, and \(u\) is radially symmetric and radially decreasing. The proof is aimed to show that (after an appropriate choice of the origin) the domain~\(D\) is invariant under rotations that affect two axes only: namely the first one and any of the remaining axes (say, the \(j\)-th), while the other \(N - 2\) axes are kept fixed. This is accomplished by arguing on the function \(w(x) = x_j \, u_{x_1} - x_1 \, u_{x_j}\) and using results in [\textit{H. Berestycki} and \textit{L. Nirenberg}, Bol. Soc. Bras. Mat., Nova Sér. 22, No. 1, 1--37 (1991; Zbl 0784.35025)] and in [\textit{H. Berestycki} et al., Commun. Pure Appl. Math. 47, No. 1, 47--92 (1994; Zbl 0806.35129)]. The results on overdetermined problems are then used to improve a symmetry result for the Dirichlet problem in reflectionally symmetric domains obtained by \textit{P. Poláčic} [Commun. Partial Differ. Equations 36, No. 4--6, 657--669 (2011; Zbl 1241.35083)]. In the two-dimensional case, independent results have been established by \textit{P. Poláčic} [J. Funct. Anal. 262, No. 10, 4458--4474 (2012; Zbl 1243.35074)].