Solution

Bernoulli-type problems are free boundary problems with a transition conditions across the free boundary. The archetype for this class of problems is \begin{equation} \min_{\mathcal K} \int_{B_1} |\nabla v|^2+ Q^2(x)\chi_{\{v>0\}} \qquad {\mathcal K}:=\{v \in W^{1,2}(B_1) :\ v|_{\partial B_1}=g\}, \end{equation} where $g$ may take both negative and positive values, and $Q >0 $, and $C^\alpha$, say. See [1]. Solutions to this problem satisfy $$ \Delta u=0\qquad \text{when $u \neq 0$}, $$ with a transition condition on the free boundary $\{u=0\}$ that can be derived formally in the case of a smooth free boundary and reads $$ (u_\nu^+)^2-(u_\nu^-)^2=Q^2. $$ Here $u_\nu^\pm$ denote the normal derivatives in the inward direction to $\{\pm u>0\}$, so that $u_\nu^\pm$ are both nonnegative.

Because of the jump in the gradient along the free boundary, the optimal possible regularity is Lipschitz.

The proof for Lipschitz regularity for a minimizer uses the strong tool of monotonicity function, see [1]. This tool is not available for general class of functionals, or PDEs. Probably the simplest example is the case of p-Laplacian, or functionals of the type \begin{equation} \min_{\mathcal K} \int_{B_1} |\nabla v|^p+ Q^p(x)\chi_{\{v>0\}} \qquad {\mathcal K}:=\{v \in W^{1,p}(B_1) :\ v|_{\partial B_1}=g\}, \end{equation} with $p>1$.

Therefore a new proof, or approach, to the Lipschitz regularity of solutions to similar problems could be very valuable. There are some recent results in this direction that can be found in the following papers [2], [3].