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While trying to complete the calculations in the question : Value of $u(0)$ of the Dirichlet problem for the Poisson equation I came across a point where I need clarification.

$$F(r) := \int_{B(0:r)}{f(x) \over |x|^{n-2}} dx $$

I want to know when is : $$\int_{R^n\backslash\{0\}} {\left|f(x)\right| \over |x|^{n-2}} dx < \infty$$

That is, what conditions should we impose on $f$ so that ${f \over |x|^{n-2}}$ is continuous and summable ?

I want it to be summable because it will then ensure that

${\partial \over \partial r } F(r) = \int_{\partial B(0:r)}{f(x) \over |x|^{n-2}} dx $ , which is required in my calculations.

As such the only conditions on $f$ are that it is continuous and summable. I'm doubtful about summability because $1 \over {|x|^{n-2}}$ shoots up as we near zero.

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Srinivas K
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1 Answers1

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Outside the origin, $\frac{f(x)}{\lvert x\rvert^{n-2}}$ is continuous as the quotient of two continuous functions. For continuity in the origin, a necessary condition is that $f$ has a zero of "order" at least $n-2$ in $0$, but that is not sufficient. A zero of order strictly greater than $n-2$ would be sufficient, but not necessary.

However, for the result

$$\frac{\partial}{\partial r} \int_{B(0;r)} \frac{f(x)}{\lvert x\rvert^{n-2}}\,dx = \int_{\partial B(0;r)} \frac{f(x)}{\lvert x\rvert^{n-2}}\, dS(x),\; r > 0,\tag{1}$$

it is not necessary that the integrand be continuous in $0$. That $f$ is continuous and integrable is sufficient. The continuity of $f$ means that $f$ is bounded on $\overline{B(0;1)}$, so

$$\int_{\mathbb{R}^n}\left\lvert \frac{f(x)}{\lvert x\rvert^{n-2}}\right\rvert\,dx \leqslant \max \{\lvert f(x)\rvert : \lvert x\rvert \leqslant 1\}\int_{B(0;1)}\frac{dx}{\lvert x\rvert^{n-2}} + \int_{\mathbb{R}^n\setminus B(0;1)} \lvert f(x)\rvert\,dx < \infty$$

since $\lvert x\rvert^{2-n}$ is locally integrable, and $f$ integrable.

Thus $(1)$ holds already under the assumption that $f$ is continuous and integrable.

Daniel Fischer
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  • Thank you.Are there " necessary and sufficient conditions " for (1) to hold where the continuity condition on $f$ can be relaxed ? – Srinivas K Jun 28 '14 at 10:51
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    You can relax the continuity requirement (and also the integrability requirement for $n \geqslant 3$ [at which point I note that we must assume that $n > 1$, since $f(x)\lvert x\rvert$ need not be integrable if $n = 1$], since $\lvert x\rvert^{2-n}$ tends to $0$ as $\lvert x\rvert\to\infty$ then). But I don't know what "necessary and sufficient" conditions could be, if they even exist (apart from the tautological "$(1)$ holds"). – Daniel Fischer Jun 28 '14 at 10:56
  • How to prove that $\int_{B(0;1)}\frac{dx}{\lvert x\rvert^{n-2}}$ is finite ? can we conclude this from local integrability ? – Srinivas K Jun 28 '14 at 11:02
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    Yes, local integrability implies integrability over every compact set (and hence over every bounded measurable set). Here, it's probably easiest to think in polar/spherical coordinates. – Daniel Fischer Jun 28 '14 at 11:24