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I have a general question surrounding certain harmonic functions.

I was able to solve the Laplace equation $\Delta f = 0$ in $\mathbb{R^3}$, subject to two spherical (equal radii) boundary conditions, where the function takes on certain scalar values on both surfaces, and it goes to zero in the far field.

My solution is very messy, but there's a certain property that I need but it's difficult to prove. That is, assuming the function depends on spatial coordinates $x,y,z$ as well as distance $d$ between centres of the spheres, will it always be true that the size of the solution $f$ at the midpoint between the two spheres will always decrease monotonically as $d$ gets larger?

To be specific, assume the spheres are aligned on the $x$-axis with the origin being the midpoint and distance $d$ between their centres.

Why is it true that $|f(0,0,0)|$ decreases monotonically as $d$ gets larger?

Physically it makes sense because as $d$ gets larger, the balls should be further away. I have numerical reasoning to believe it's correct too.

I just am not sure how to prove it. Is there an argument involving maximum principle, or properties of harmonic functions anyone knows of? Has this problem already been solved that someone can provide a reference to? I'd appreciate it.

Note that I am also, in general, interested if this question can be extended for any pair of symmetric boundaries. I would propose that it should, and in this case I have considered spheres, though I am not sure how to show this.

Note: I had cross-posted this here on MSE, though since it's a research question I think it's best asked on this site.

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  • $\begingroup$ With two spheres, it's easy to write an explicit Green's function as an infinite series, and the behavior of that ought to tell you what you need to know about the solution. $\endgroup$
    – Buzz
    Commented Apr 29 at 0:46
  • $\begingroup$ I'm not sure I see the explicit Green's function being trivial for two spheres. Would we not need to use a coordinate transformation (eg. bispherical) to obtain this explicitly? I found a paper [here] (asmedigitalcollection.asme.org/appliedmechanics/…), though they consider one sphere inside the other, whereas I'm considering two disjoint spheres. $\endgroup$
    – HtmlProg
    Commented Apr 29 at 1:41
  • $\begingroup$ You can use an infinite sequence of image charges, alternating between the interiors of the two spheres. It's like the limiting problem of two infinite parallel planes, except that the positions and magnitudes of the successive charges are more complicated. However, the asymptotic behavior of the infinite series should be (relatively) straightforward to determine. $\endgroup$
    – Buzz
    Commented Apr 29 at 3:01

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This is obviously not true if you allow general (i.e., varying along a sphere) boundary values. For example, let the boundary values be positive but zero on the hemispheres facing each other. Then $f(0,0,0)$ is positive but tends to zero both as the spheres get close to each other and as $d\to\infty$.

If the boundary value on each sphere is constant, then the result is true. First, you can assume that the constants are the same, say positive, by subtracting a solution with constants differing by sign, which vanishes at the origin. Then, denoting the solution with distance between the spheres $d$ and radii $r$ by $f_{d,r}$, we have for any $a>1,$ $$ f_{d,r}(0)=f_{ad,ar}(0)> f_{ad,r}(0), $$ because the values of $f_{ad,r}$ on the spheres of radius $ar$ are smaller than their constant boundary values on the spheres of radius $r$.

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  • $\begingroup$ Thanks, this is a nice argument! Might be a naïve question: but to show that inequality, you're saying it's since the values of $f_{ad,r}$ on spheres of radius $ar$ are smaller than the boundary values. But none of these spheres are touching the origin, so why could we obtain that inequality, specifically at the origin if we only know that the values on the spheres of radius $r$ are smaller? $\endgroup$
    – HtmlProg
    Commented Apr 29 at 20:19
  • $\begingroup$ @HtmlProg we have $f_{ad,ar}-f_{ad,r}>0$ on these spheres, hence by maximum principle everywhere outside of them. $\endgroup$
    – Kostya_I
    Commented Apr 30 at 7:04
  • $\begingroup$ Yes of course, that was rather trivial. Thanks! $\endgroup$
    – HtmlProg
    Commented Apr 30 at 8:54

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