Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: given $\varepsilon$ small, can we still find a harmonic function $u$ with appropriate Dirichlet boundary conditions on $\partial D$ (meaning, prescription of an appropriate continuous function on $\partial D$) which additionally satisfies:
- $u|_H = 0$.
- $\Vert u\Vert_{L^2(D_1)} < \varepsilon\Vert u\Vert_{L^2(D)}, \Vert u\Vert_{L^2(D_2)} > \Vert u\Vert_{L^2(D)} - \varepsilon$?
Heuristically, can the mass of $u$ be really skewed across the domain, or does it need to be somewhat evenly distributed? Some time back, I heard a talk on "free boundary problems". Is this somehow related to such problems? Thanks!
Comment: Edited after Willie Wong's comment. Connor Mooney's comment answers the question.