Serrin's overdetermined theorem and weak Bernoulli laws without Alt--Caffarelli regularity

Abstract

We study distributional Bernoulli-type conditions in geometrically irregular domains $\hat{I}\text{c◯}$. Here the zero extension of $u$ to ${\mathbb{R}}^n$ satisfies $$\hat{I}”u=c{\mathcal{H}}^{n-1}{\lfloor }_{{\partial }^{\ast }\hat{I}\text{c◯}}-f(u)1_{\hat{I}}\text{c◯}dx$$ in the distributional sense. This is a weak version of the one-phase Bernoulli free boundary condition, which avoids the uniform Lipschitz/density assumptions of classical Alt--Caffarelli theory. We prove that for every $n\ge 2$ and every $f\in C^2(\mathbb{R})$ with $f(0)>0$, there exist bounded, non-spherical, finite-perimeter domains $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ satisfying this distributional Bernoulli law with $$0<{\mathcal{H}}^{n-1}({\partial }^{\ast }\hat{I}\text{c◯})<\infty ,{\mathcal{H}}^{n-1}(\partial \hat{I}\text{c◯}\setminus {\partial }^{\ast }\hat{I}\text{c◯})=0,$$ yet $$\mathrm{ess}\underset{x\in {\partial }^{\ast }\hat{I}\text{c◯}}{\sup }\underset{0<r<1}{\sup }\frac{{\mathcal{H}}^{n-1}(B_r(x)\cap {\partial }^{\ast }\hat{I}\text{c◯})}{r^{n-1}}=\infty .$$ This shows the key constraint is not absence of a reduced boundary, but failure of uniform all-scale surface density bounds. For $f\equiv 1$, these results yield counterexamples to the weak Serrin-type overdetermined problems in all dimensions, proving the distributional Bernoulli law alone cannot replace the uniform growth/density conditions core to Alt--Caffarelli theory. On the other hand, we prove a planar rigidity result: Within the Smirnov class, the associated harmonic quadrature identity forces $\hat{I}\text{c◯}$ to be a disk. Thus, for the constant-source Serrin/Bernoulli law, Smirnov regularity is the threshold for weak Bernoulli rigidity in ${\mathbb{R}}^2$, while uniform upper density bounds form the threshold for $n\ge 3$ according to [23].

55 pages, The author would like to express gratitude to Prof. Dmitry Khavinson for drawing his attention to the literature on harmonic quadrature identities in planar domains, possibly with finite connectivity