@Kitonum That's a very clever solution.  I like it!

@vv Yes, you are right in that the Gateaux differential can be nonlinear in general.

In the problem at hand, however, the functional is Frechet differentiable if we assume q(w) is positive and bounded away from zero.  In that case the Gateaux and Frechet derivatives agree, and the differenial is linear.

@mayzal After making a few adjustments to your worksheet, I got pdsolve() to work and return a module. However, the evaluation of the module fails.  See the attached worksheet.

I have no further ideas on how to proceed.  Perhaps someone else may chime in.

question4-alt.mw

@mayzal One of your PDEs is

diff(X_dimensionless[1](t), t)  = ... P_dimensionless(z, t) ...

The right-hand side depends on z and therefore X_dimensionless[1] depends on z.  Shouldn't that be X_dimensionless[1](z,t)?

Aside: What is the point of the "_dimensionless" suffixes in your variable names?  Remove the horrible clutter by dropping those suffixes.  P(z,t) is as good a name for the dimensionless pressure as P_dimensionless(z,t), and is much easier to read.

@Kitonum The Wikipedia page Bernoulli quadrisection problem gives an example of an isosceles triangle which can be quadrisected in two different ways.

Kitonum, that's an excellent solution to a nontrivial problem.  Congratulations and vote up!

PS: In this post's first paragraph you write that your procedure applies to convex domains, but the examlpes show that it works with non-convex domains as well. Perhaps you meant simply connected?

@C_R I have no Maple Sim so I cannot be of help here, but let us know if you make further progress in what you are doing.

I don't have Maple Sim but I am guessing that what you are seeing is due to Maple Sim's attempt to simulate the dynamics of a massless disk.  Give the disk a little mass (something like 1e-6) and see if that helps.

Here is an interesting exercise: Calculate the vertical component of the force that the floor exerts on the disk. Is that force always possitive?

When that force becomes negative, the disk will lift off the floor!

@acer Filed a bug report.  Included Kitonum's examples as well.

Additionally, submitted a separate SCR suggesting an improvement to implicitplot3d.  Currently, implicitplot is designed to handle variable ranges, as seen in the samples in this thread. Implicitplot3d, however, requires fixed (constant) bounds on the ranges.  That's too limiting and it can be improved.

@Kitonum Good catches!  The graph I plotted in Maple 2021 looked sort of correct, so I did not inversitage it any further.

Thank you all, mmcdara, acer, Kitonum, dharr,  for your varied suggestions.  The substring method seems to be the most direct.  I didn't know about it.  Despite its name, substring extracts characters from both strings and names, as documented in its help page.

My calculation shows, if I am not mistaken, that the ratio of the areas is 

72/5*(sqrt(2)-1)

which is approximately 5.965, not 6 as claimed.

Edit: Oops, I found my error.  Sorry.

@Earl Most any book on calculus of variations includes a discussion of constraints and the use of Lagrange multipliers to enforce them.  The lambda that appears in my formulation is a Lagrange multiplier.  It enforces the constraint z = f(x,y), that is, the requirement that the moving point does not leave the surface.

A quick search on "lagrange multiplier, calculus of variations" on Google, turns up quite a few references.  One of them is a  handbook which I haven't seen but it comes as highly recommended.  You may want to get a copy.

I take this opportunity to make a correction to my original post.  There I took lambda to be a constant.  In general, lambda would be a function of t.  Because of the special structure of the current problem, that error does not affect the solution, but in other situations it may.  So I have attached a corrected worksheet here. I have also included a new animation where contour lines help to better illustrate the particle's motion. 

Worksheet:  hills2.mw

You have:

restart, with(plots):

That's not good.  Here is what the help page on restart says:

It [restart] must be executed in a separate prompt (or line) from all other commands, since all commands in a prompt are passed to the kernel at once; entering other commands in the prompt could cause unexpected results.