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My notes motivate Lax-Milgram theorem by using the membrane problem.
We take $E[u] = \int_{\Omega} \Big(\frac \sigma 2|\nabla u|^2+\frac \alpha 2u^2-f(x)u\Big)dx$ the functional associated to the problem and we make variations on it. So we define $g(s) = E[u+s\phi]$ with $\phi \in H_0^1(\Omega)$...
A two-dimensional elastic membrane under tension can support transverse vibrations. The properties of an idealized drumhead can be modeled by the vibrations of a circular membrane of uniform thickness, attached to a rigid frame. Due to the phenomenon of resonance, at certain vibration frequencies, its resonant frequencies, the membrane can store vibrational energy, the surface moving in a characteristic pattern of standing waves. This is called a normal mode. A membrane has an infinite number of these normal modes, starting with a lowest frequency one called the fundamental mode.
There exis...
To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of overtones, via the use of mathematical theory. "Can One Hear the Shape of a Drum?" was the title of an article by Mark Kac in the American Mathematical Monthly in 1966, but the phrasing of the title is due to Lipman Bers. These questions can be traced back all the way to Hermann Weyl.
For the 1966 paper that made the question famous, Kac was given the Lester R. Ford Award in 1967 and the Chauvenet Prize in 1968.
The frequencies at which a drumhead can vibrate depends...
