In the book [2] there are discussions and few results on overdetermined problems with matching Cauchy data. However, there the free boundary is a priori assumed to be $C^1$ and the authors derive higher order regularity.

A similar type of problem is the one of matching Cauchy data from one side of a domain, which can be formulated as follows: ...

Let $(A, \tau)$ be a complex topological algebra(e.g. a Fréchet algebra) . Let $S(A)$ denote the set of all multiplicative functionals $\psi\colon A\to \mathbb{C}$. ...

Consider again an IDLA on $$D:=\{ x_1 > k |x_2|\} \cap \frac{1}{n} \mathbb Z^2, \qquad \hbox{for some } k > 0.$$ Observe that now $ \mathbf 0 \notin D$.

Governing rules:

i) Initially all sites within $D$ are filled with one particle each.

ii) Particles walking from $z= (1,0)$ are allowed to walk out from $D$ and occupy new sites. ...

In the model described before we let $d=2$ and $z=(1,0)$. We shall now consider an IDLA in a wedge shaped domain, such as $$D:=\{ x_1 \geq k |x_2|\} \cap \frac{1}{n} \mathbb Z^2, \qquad \hbox{ for some } k > 0.$$ Governing rules:

i) Initially $D \setminus z$ has no particles. ...

It has been numerically established by Marco Frasca a decade ago [3] that the second order quasi-linear non-homogeneous equations $$ \frac{d^2 w}{d t^2} + N(w) = f(t), ~~ t > 0, $$ subject to homogeneous Cauchy conditions, admits the following solution called short time expansion: $$ w(t) = \int_0^t G(t - \tau) f(\tau) d \tau + \sum_{n = 1}^\infty a_n \int_0^t G(t - \tau) f(\tau) (t - \tau)^n d \tau. $$ Here $a_n$ are determined in terms of the quantities $w^{(k)}(0)$, $G$ is the general solution of the following quasi-linear equation: $$ \frac{d^2 G}{d t^2} + N(G) = s \delta(t). $$ Here $s$ is an error correcting factor introduced from numerical considerations, $\delta$ is the Dirac function. Due to the similarity with the linear case, $G$ is referred to as nonlinear Green's function. ...

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. ...

Consider the Dirichlet problem $$\cases{F(D^2u)=0 &in $\Omega$\cr u=\phi &on $\partial\Omega\;,$\cr}(1)$$ where $\Omega \subset {\bf R}^n$ is a bounded domain with smooth boundary $\partial \Omega$, $\phi$ is a continuous function on $\partial\Omega$ and $F$ is a fully nonlinear uniformly elliptic operator, i.e., we assume that $F$ is a $C^1$ function defined on the space of ${n\times n}$ symmetric matrices satisfying the uniform ellipticity condition: there exists a constant $C=C(F)\ge 1$ (called an ellipticity constant) such that $${1\over C}|\xi|^2\le F_{u_{ij}}\xi_i\xi_j\le C |\xi |^2\;, \forall\xi\in {\bf R}^n\;.$$ Here, $u_{ij}$ denotes the partial derivative $\partial^2 u/\partial x_i\partial x_j$. A function $u$ is called a classical solution of (1) if $u\in C^2(\Omega)$ and $u$ satisfies (1). Actually, any classical solution of (1) is a smooth ($C^{\alpha +3}$) solution, provided that $F$ is a smooth $(C^\alpha )$ function of its arguments. ...