On the Problems with Corners
Søren Fournais, Bernard Helffer · Progress in nonlinear differential equations and their applications · 2009
It has been observed in the physics literature that the transition field $$H_{C_3}$$ (k) is significantly larger when the domain Ω has corners than for samples of the same material but with a smooth cross section. In this chapter, we analyze this phenomenon. It still results in the value of $$H_{C_3}$$ (k) being completely determined by the corresponding linear spectral problem. The experimentally observed change in $$H_{C_3}$$ (k) is due to a decrease of the first eigenvalue for the magnetic Neumann problem when a domain has a corner. Eigenfunctions corresponding to the lowest eigenvalues will be localized near the corners and their leading-order asymptotics for a large field controlled by the model of an infinite sector considered in Section 4.4.