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

We study distributional Bernoulli-type conditions in geometrically irregular domains $\mathrm{\Omega }$. Here the zero extension of $u$ to ${\mathbb{R}}^n$ satisfies $$\mathrm{\Delta }u=c{\mathcal{H}}^{n-1}\text{\hspace{0.17em}⁣}{\lfloor }_{{\mathrm{\partial }}^{\ast }\mathrm{\Omega }}-f(u){\mathbf{1}}_{\mathrm{\Omega }}dx$$ 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 $\mathrm{\Omega }\subset {\mathbb{R}}^n$ satisfying this distributional Bernoulli law with $$0<{\mathcal{H}}^{n-1}({\mathrm{\partial }}^{\ast }\mathrm{\Omega })<\mathrm{\infty },{\mathcal{H}}^{n-1}(\mathrm{\partial }\mathrm{\Omega }\setminus {\mathrm{\partial }}^{\ast }\mathrm{\Omega })=0,$$ yet $$\mathrm{e}\mathrm{s}\mathrm{s}\underset{x\in {\mathrm{\partial }}^{\ast }\mathrm{\Omega }}{\sup }\underset{0<r<1}{\sup }\frac{{\mathcal{H}}^{n-1}(B_r(x)\cap {\mathrm{\partial }}^{\ast }\mathrm{\Omega })}{r^{n-1}}=\mathrm{\infty }\mathrm{.}$$ 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 $\mathrm{\Omega }$ 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].