Rigidity and symmetry breaking in a Serrin problem with a radial variable exponent

Abstract

We study Serrin's overdetermined torsion problem for the $p(|x|)$-Laplacian in bounded domains containing the origin. An explicit radial solution has flux $-x/n$ and gradient modulus $f(r)=(r/n)^{1/(p(r)-1)}$. Comparison with this solution yields two-sided boundary-gradient bounds and a rigidity criterion: if $f(r)\le f(R_1)$ between the smallest and largest boundary radii $R_1,R_2$, then the domain is the ball $B_{R_1}(0)$. An associated flux identity also gives a quantitative estimate for star-shaped domains. In the opposite direction, given $n\ge2$, $k\ge2$, $R\in(0,n)$, and $\varepsilon>0$, we construct a smooth nondecreasing radial exponent, with $0\le p-2<\varepsilon$ and derivative supported in an arbitrarily thin annulus about $R$, for which a branch of non-spherical, strictly convex overdetermined domains bifurcates from $B_R(0)$ in harmonic degree $k$. The construction rests on a variational Dirichlet-to-Neumann reduction and a transversal eigenvalue crossing.