The Problem
Let $\Omega \subseteq \mathbb{R}^d$ with regular enough boundary $\partial \Omega = \Gamma_1 + \Gamma_2$. Let $K \in \mathbb{R}^{d \times d}$ a symmetric positive-definite matrix.
Consider the following Poisson problem with directional derivative constraint at boundaries:
$$-\Delta u = f \quad \text{in } \Omega$$ $$u = g \quad \text{on } \Gamma_1$$ $$\nabla u \cdot (Kn) = h \quad \text{on } \Gamma_2$$
Is this problem well-posed (e.g. in $H²(\Omega)$) and why ?
What I tried
I tried to investigate:
Uniqueness of solution using the method of energy. Let $w$ the difference of two solutions:
- I tried $\int_\Omega \Vert \nabla w \Vert^2 = \int_{\Gamma_2} w \nabla w \cdot n$, but I do not know how to relate the latter quantity to $\nabla w \cdot (Kn) = 0$ on $\Gamma_2$ to show uniqueness.
- I also tried $\int_\Omega \Vert K^{1/2} w \Vert^2 = -\int_\Omega w \nabla \cdot (K \nabla w)$, but again I cannot conclude.
Existence and uniqueness using the Lax-Milgram theorem, and once again, I am rapidly stuck to concluse because of the last condition