For my research, I've come across the following type of equation (under variational form) : Find u in V such that for all V in V : $\int_\Omega \nabla u \cdot \nabla v\,dx + \int_{\partial\Omega} \nabla_{\Gamma} u \cdot \nabla_{\Gamma} v + \phi (v\cdot n)\,ds = 0$

With $\Omega$ a Lipschitz domain and $\phi \in L^2(\partial \Omega)^d$.

So far, I have considered V to be $\{u \in H^1(\Omega), \nabla_{\Gamma} u \in L^2(\Omega)^d\}$.

• Is this a relevant choice of space ?
• With this choice of space, do we have existence & uniqueness ?