Elliptic problems on networks with constrictions

Rubinstein, Jacob; Sternberg, Peter; Wolansky, Gershon
August 2006
Calculus of Variations & Partial Differential Equations;Aug2006, Vol. 26 Issue 4, p459
Academic Journal
We investigate the asymptotic behavior of minimizers to sequences of elliptic variational problems posed on thin three-dimensional domains. These domains arise as thin neighborhoods of artibrary graphs that contain severe constrictions near the graph nodes. We characterize an appropriate limit of minimizers as a function of one variable defined on the graph that necessarily minimizes a one-dimensional variational problem. The most salient feature of these limits of minimizers is the emergence of jump discontinuities across the graph nodes. While the approach can handle quite general elliptic problems, we pay particular attention to an application to generalized Josephson junctions within the Ginzburg-Landau theory of superconductivity.


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