Calculus of Variations and Partial Differential Equations

, Volume 33, Issue 3, pp 329–341

An improved uniqueness result for the harmonic map flow in two dimensions

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Article

DOI: 10.1007/s00526-008-0164-7

Cite this article as:
Rupflin, M. Calc. Var. (2008) 33: 329. doi:10.1007/s00526-008-0164-7

Abstract

Generalizing a result of Freire regarding the uniqueness of the harmonic map flow from surfaces to an arbitrary closed target manifold N, we show uniqueness of weak solutions u H1 under the assumption that any upwards jumps of the energy function are smaller than a geometrical constant \(\epsilon^\star=\epsilon^\star(N)\), thus establishing a conjecture of Topping, under the sole additional condition that the variation of the energy is locally finite.

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© Springer-Verlag 2008