Variational problems on optimal geometry in physics

This work aims to provide insight into shape optimization methods. Three problems, one in electrostatics, second in quantum mechanics and the third in contiuum dynamics, are tackled. We introduce a volume-preserving transformation, which can be interpreted as twisting and bending. Using variational methods, it is shown that if a coaxial capacitor is slightly twisted or bent then its capacitance increases. Next, we apply a special, twisting, case of the transformation to a cylindrical quantum waveguide and prove via spectral theory that the cylindrical waveguide has a lower energy of the ground state than any twisted waveguide. We also offer a comparison of two mathematical models of steady viscous fluid motion in a pipe by different authors. In one model, the cylindrical pipe optimizes dissipated energy, in the other, however, it does not.