# Variational Formulation of Boundary Value Problems With Unknown on the boundary

Suppose that we have a linear operator equation on $\Omega$ with Lipschitz boundary $\partial \Omega$,

\begin{eqnarray} Lu &=& \frac{\partial u}{\partial t}, u(x,0) &=& u_0 \; \; \mbox{in}\; \Omega \\ \end{eqnarray}

where $L$ is a Parabolic operator. Furthermore, we consider a situation whereby on a portion of the boundary, say $\Gamma \subset \partial \Omega$ we have Robin conditions with an unknown function $\tilde{u} :\partial \Omega \times [0,T] \rightarrow \mathbb{R}$ posed in the following way

$$\tilde{u}_t =\frac{\partial u}{\partial \nu} = cu + \tilde{u}, \; \mbox{on}\; \partial \Omega.$$

Question: Whats the best variational formulation for this problem?

I have investigated a weak - strong formulation, whereby the main operator equation is transformed to its weak form using the second equality of the boundary condition and coupling that with a strong part( classical ) form of the first equality of the boundary condition. Moreover on $\partial \Omega \cap \Gamma^c$ we have Dirichlet boundary data. It is worth noting that we assume suitable regularity for both unknown function.

I have consulted literature on Free boundary problems including the celebrated Stefan Problem, however I am still in the woods as far as figuring out a concrete formulation.

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