Poisson equation: what shape gives largest area average

[Ric72]

When considering the solution u(x,y) of the poisson equation
u_xx + u_yy = -1 for (x,y) in G
on a 2-dimensional domain G with Dirichlet boundary conditions
u = 0 for (x,y) on boundary of G
I am wondering the following: for what shape of the domain G do I obtain the largest area-average for the solution?

Since this answers the questions of (i) lowest pressure drop in a pipe of given cross-sectional area, (ii) which shape G of a uniformly hot-heating plate has the largest average temperature and (iii) what shape of domain has the largest average exit-time distribution for drunk sailors (random walkers) I am pretty sure someone has figured this out already.

Also, I am pretty sure the answer is a circle. But I lack a proof (or at least an authoritative answer ;-)

Thanks for your help,
Ric

 

[jvicens]

ardo

Dear Ricardo,

Thank you for your interesting question. The solution to the Poisson equation u_xx + u_yy = -1 depends on the shape of the domain G and the boundary conditions. In this case, the boundary conditions are Dirichlet, meaning the value of u is fixed on the boundary of G. This is important to consider when looking for the largest area-average of the solution.

After doing some research, I found that the shape of the domain G that results in the largest area-average for the solution u(x,y) is indeed a circle. This is known as the isoperimetric inequality, which states that among all shapes with the same perimeter, a circle has the largest area.

To understand why this is the case, let's consider the problem in terms of the physical interpretation given in the forum post. The Poisson equation can be used to model the flow of a fluid through a pipe, the temperature distribution on a heating plate, and the movement of particles in a random walk. In all of these cases, the solution u(x,y) represents the quantity we are interested in (pressure, temperature, or exit-time distribution) and the domain G represents the space in which this quantity is defined.

Now, imagine we have two shapes with the same perimeter, one being a circle and the other being a different shape. The boundary conditions dictate that the value of u is fixed on the boundary of G, which means that the solution u(x,y) will be affected by the shape of the boundary. Since a circle has a constant curvature, the solution u(x,y) will be smoother and more evenly distributed compared to a shape with varying curvature. This leads to a larger average value of u for a circle compared to other shapes with the same perimeter.

Therefore, in terms of the physical interpretations given in the forum post, a circle results in the lowest pressure drop in a pipe, the largest average temperature on a heating plate, and the largest average exit-time distribution for random walkers.

I hope this helps to answer your question. If you would like to learn more about the isoperimetric inequality and its applications, I recommend looking into the work of famous mathematicians such as Archimedes and Gauss.