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Let consider the Laplace equation in unbounded domain in $\Omega = \{ x \in \mathbb{R}^n ; |x|>1\}$

$$-\Delta f(x) =0 \ \ \text{in} \ \Omega$$

I want to prove if $f \in C^2 (\overline{\Omega})$ satisfies the above equation and $$\int_{\Omega}|f(x)|^2 dx + \int_{\Omega} |\nabla f(x)|^2 dx \ < \ \infty$$

then $f=0$.

I could prove in case of $n=1$ or $|f(x)|=0 \ \text{on} \ |x|= 1$ or $|\nabla f(x)|=0 \ \text{on} \ |x|= 1$ with the use of integration by parts or the maximum principle.

But in general case, I have no idea.

Any advice would be appreciated.

aot
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  • What exactly do you mean with $f = 0$, i.e., for what $\boldsymbol{x}$ should that hold? Consider for instance $$f(\boldsymbol{x} = \begin{cases} \exp\Big(- 1 /\big(\Vert \boldsymbol{x} \Vert^2 -1 \big)\Big) &\Vert \boldsymbol{x} \Vert < 1 \\ 0 & \Vert \boldsymbol{x} \Vert \geq 0 \end{cases}$$ This is clearly not $f(\boldsymbol{x} ) \equiv 0 $. Based on your assumptions, I am not sure you can control $f$ in $\Vert \boldsymbol{x} \Vert \leq 1$ . – Dan Doe Feb 03 '22 at 07:53
  • @Dan Doe Yes I agree, I think the conclusion should be $u=0$ in $\mathbb R^n \setminus B_1$. – JackT Feb 03 '22 at 08:53
  • @Dan Doe Thank you for pointing that out. – aot Feb 03 '22 at 14:27

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I think your question needs revision. Here is what I think the revised statement should be:

Proposition: Suppose that $u\in C^2(\mathbb R^n)\cap L^2(\mathbb R^n)$ satisfies $\Delta u = 0$ in $\mathbb R^n \setminus B_1$, $u=0$ on $\partial B_1$. Then $u=0$ in $\mathbb R^n \setminus B_1$.

I will give two counter-examples which justify why I think these changes are appropriate. Then I will prove this proposition.


Why can we only conclude $u=0$ in $\mathbb R^n \setminus B_1$? This was answered by Dan Doe in the comments. In fact, any smooth function $v:B_{1/2} \to \mathbb R$ can be smoothly extended so that $v=0$ in $\mathbb R^n \setminus B_1$. Hence, there is absolutely no way to say anything about $u$ in $B_1$.


Why do we need $u=0$ on $\partial B_1$? Let $u(x) = \vert x \vert^{2-n}$ be the fundamental solution. Then $\Delta u=0$ in $\mathbb R^n \setminus B_1$ (it is harmonic everywhere but $0$). Then $u^2 = \vert x \vert^{4-2n}$ which is integrable in $\mathbb R^n \setminus B_1$ provided that $n>4$. Moreover, $\vert \nabla u \vert^2 = (n-2)^2 \vert x \vert^{2-2n}$ which is integrable provided that $n>2$. Hence, for $n>4$ the fundamental solution contradicts your statement.


Proof of Proposition: Let $\eta \in C^1(B_{2r}\setminus B_1)$ be nonnegative such that $\eta =0$ on $\partial B_{2r}$. Since $u$ vanishes on $\partial B_{1}$, integration by parts gives that \begin{align*} 0 &= \int_{B_{2r}\setminus B_1} \eta^2 u \Delta u \, dx \\ &= - \int_{B_{2r} \setminus B_1} \nabla (\eta^2 u) \cdot \nabla u \, dx \\ &=- 2\int_{B_{2r} \setminus B_1} \eta u \nabla \eta \cdot \nabla u \, dx - \int_{B_{2r} \setminus B_1} \eta^2 \vert \nabla u \vert^2 \, dx. \end{align*} Rearranging, applying the Cauchy-Schwarz inequality then applying the Hölder inequality gives \begin{align*} \int_{B_{2r} \setminus B_1} \eta^2 \vert \nabla u \vert^2 \, dx &=- 2\int_{B_{2r} \setminus B_1} \eta u \nabla \eta \cdot \nabla u \, dx \\ &\leqslant 2\int_{B_{2r} \setminus B_1} \eta \vert u \vert \vert \nabla \eta \vert\cdot \vert \nabla u \vert \, dx \\ &\leqslant 2\bigg ( \int_{B_{2r} \setminus B_1} \eta^2 \vert \nabla u\vert^2 \, dx\bigg ) ^{1/2} \bigg (\int_{B_{2r} \setminus B_1} u^2 \vert \nabla \eta \vert^2\, dx \bigg )^{1/2} \end{align*} This implies the following (small) variant on the classical Caccioppoli inequality: \begin{align*} \int_{B_{2r} \setminus B_1} \eta^2 \vert \nabla u \vert^2 \, dx &\leqslant 4\int_{B_{2r} \setminus B_1} u^2 \vert \nabla \eta\vert^2 \, dx . \end{align*}

Fix $\varepsilon >0$ small and choose $\eta$ such that $$\eta (x) = \begin{cases} 1, &\text{if } x\in B_r \setminus B_1, \\ \frac 1 r (2r-\vert x \vert^2 ), &\text{if } x \in B_{2r} \setminus B_r. \end{cases} $$ Then, since $\vert \nabla \eta \vert = \frac1r$ in $B_{2r} \setminus B_r$\begin{align*} \int_{B_{r} \setminus B_1} \vert \nabla u \vert^2 \, dx \leqslant\int_{B_{2r} \setminus B_1} \eta^2 \vert \nabla u \vert^2 \, dx \leqslant 4\int_{B_{2r} \setminus B_1} u^2 \vert \nabla \eta\vert^2 \, dx\leqslant \frac 4 r\| u \|_{L^2(\mathbb R^n)}^2.\end{align*} Sending $r\to \infty$, we obtain $\| \nabla u \|_{L^2(\mathbb R^n \setminus B_1)}=0$, so $u$ is a constant. Then, since $u$ is in $L^2$ it must be zero.

JackT
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  • Thank you for your comment. But I already prove in case of $f=0 \ \text{on} \ |x|=1$ with the use of integration by parts or the maximum principle. I want to prove general case. – aot Feb 03 '22 at 14:42
  • @aot JackT's second counterexample shows that isn't possible. – Ali Feb 03 '22 at 14:47
  • @JackT Thank you for the useful comment. – aot Feb 03 '22 at 15:03
  • I also want to know the case $n = 2, 3$. When $n=1$, the statement holds in general. – aot Feb 03 '22 at 15:12