# Elliptic equation with Laplace-Beltrami boundary condition

For my research, I've come across the following type of equation (under variational form).

Assume $\Omega\subset\mathbb{R}^d$ is a Lipschitz domain, $\phi \in L^2(\partial \Omega)$ and $\nabla_{\Gamma}$ the tangent gradient operator. Let $V$ be the space

$$V:=\{u \in H^1(\Omega): \nabla_{\Gamma} u \in L^2(\Omega)\}.$$

Find $u\in V$ such that for all $v\in V$, we have $$\int_\Omega \nabla u \cdot \nabla v\,dx + \int_{\partial\Omega} \left(\nabla_{\Gamma} u \cdot \nabla_{\Gamma} v + \phi (v\cdot n)\right)ds = 0.$$

Questions.

• Is $V$, as defined above, a relevant choice of space?
• With this choice of space, do we have existence & uniqueness?