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Boundary concentration for eigenvalue problems related to the onset of superconductivity. (English) Zbl 0982.35077
The authors deal with the asymptotic behaviour of the eigenvalue $$\mu(h)$$ and corresponding eigenfunction associated with the variational problem $\mu(h)\equiv \inf_{\psi\in H^1(\Omega, \mathbb{C})} {\int_\Omega|(i\nabla+ hA)\psi|^2 dx dy\over \int_\Omega|\psi|^2 dx dy}$ in the regime $$h\gg 1$$. Here, $$A$$ is any vector field with curl equal to $$1$$. The authors show that when the domain $$\Omega$$ is not a disc, the first eigenfunction does not concentrate along the entire boundary. It must be decay to zero with large $$h$$ somewhere along the boundary, while simultaneously decaying at an exponential rate inside the domain.

##### MSC:
 35P15 Estimates of eigenvalues in context of PDEs 35J10 Schrödinger operator, Schrödinger equation 35Q60 PDEs in connection with optics and electromagnetic theory 82D55 Statistical mechanical studies of superconductors
##### Keywords:
eigenvalue; eigenfunction; variational problem; exponential rate
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