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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:

  1. $u|_H = 0$.
  2. $\Vert u\Vert_{L^2(D_1)} < \varepsilon\Vert u\Vert_{L^2(D)}, \Vert u\Vert_{L^2(D_2)} > \Vert u\Vert_{L^2(D)} - \varepsilon$?

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.

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    $\begingroup$ The solution to the Dirichlet boundary problem is fixed once you fix the boundary data, are you asking whether one can find boundary data so your conditions are satisfied? $\endgroup$ Oct 2, 2015 at 14:39
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    $\begingroup$ BTW, your condition 2 should probably be instead $$\|u\|_{L^2(D_1) } < \epsilon \|u\|_{L^2(D)} $$ As you stated it the answer is trivially yes by taking $u \equiv 0$... $\endgroup$ Oct 2, 2015 at 14:42
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    $\begingroup$ As was pointed out already, this doesn't make sense as stated because only $u=0$ is harmonic and satisfies Dirichlet boundary conditions. $\endgroup$ Oct 2, 2015 at 16:35
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    $\begingroup$ If u = 0 on a hyperplane then u is odd across this hyperplane by Schwarz reflection and unique continuation, so the mass of u is evenly distributed. $\endgroup$ Oct 2, 2015 at 17:54
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    $\begingroup$ @ConnorMooney: ha, now do I feel embarrassed that I didn't write your comment... $\endgroup$ Oct 2, 2015 at 19:14

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