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Metric methods for heteroclinic connections. (English) Zbl 1388.49021

Summary: We consider the problem \(\int_{\mathbb{R}}\frac12|\dot{y}|^2+W(\gamma)\mathrm{d}t\) among curves connecting two given wells of \(W\geq0\), and we reduce it, following a standard method, to a geodesic problem of the form \(\int_0^1 K(\gamma)|\dot{\gamma}|\mathrm{d}t\) with \(K=\sqrt{2W}\). We then prove the existence of curves minimizing this new action just by proving that the distance induced by \(K\) is proper (i.e., its closed balls are compact). The assumptions on \(W\) are minimal, and the method seems robust enough to be applied in the future to some PDE problems.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
49J45 Methods involving semicontinuity and convergence; relaxation
54E35 Metric spaces, metrizability
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
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