Friis Transmission Equation and Wavelength dependence The "modern" form of the Friis Transmission Equation states that :
$$\frac{P_r}{P_t} = G_t G_r \left( \frac{\lambda}{4 \pi d} \right)^2$$ where $G_t$, $G_r$ are unitless antenna gains, $\lambda$ is the wavelength "representing the effective aperture area of the receiving antenna", according to Wikipedia, and $d$  is the distance separating the antennas.
If I were to consider perfect isotropic antennas, of gain $1.0$ (or $0$ dBi), this could be rewritten as :
$$P_r=P_t\left( \frac{\lambda}{4 \pi d} \right)^2$$
Under this form, it could be understood that the biggest the wavelength, the biggest the received power at the receiving antenna. I'm pretty sure this is not actually the case, but I'm having trouble understanding why.
My current understanding is that the gains $G_t$ and $G_r$ are functions of $\lambda$, but this collides with the assumption of perfect isotropy of the antennas, meaning a value of $G_t$ and $G_r$ equal to $1$.
 A: The confusion is caused by your fixing directivity. A receive antenna's directivity is proportional to its effective receive area $A_r$, so that if it is immersed in a flux of density $S$ then, assuming perfect matched conditions, it receives $P_r=SA_r$ power, and $S=\frac{P_tG_t}{4\pi d^2}$. The effective receive directivity is then defined as $G_r=\frac{4\pi A_r}{\lambda^2}$, or $A_r = G_r\frac{\lambda^2}{4\pi}$. It can also be shown that the receive and transmit directivities can be equal under quite general conditions but not always. Some pulse radar antennas do behave that way. Note that I use the term directivity and not gain for the latter also depends on the RF plumbing behind the antenna.
Just based on dimensional considerations the effective area must be proportional to $\lambda^2$ the wavelength being the only natural length scale. It is a lot more difficult to understand from where the $4\pi$ comes in the denominator, but for details you should consult Slater whose argument you rarely find spelled out anywhere else.
In practice, for any antenna you are usually given $\lambda$ (the operating frequency) and the available effective area to occupy, and consequently the maximum directivity available to you to close the receive link.

You can also immediately see based on dimensional argument that a formula, such as $\frac{P_r}{P_t}=k G_t G_r \frac {\lambda^2}{d^2}$ should hold for dimensionless number $k$ that depends on the geometry of the antennas. That is so because you know that the received power must be inversely proportional to $d^2$ and the factors of $G_t$ and $G_r$ are just there to express how much energy flux (power) goes into the maximum transmit/receive directions, then the only available dimensional quantity to make the ratio a dimensionless number is $\lambda^2$
