A second order minimality condition for the Mumford-Shah functional


DOI: 10.1007/s00526-007-0152-3

Cite this article as:
Cagnetti, F., Mora, M.G. & Morini, M. Calc. Var. (2008) 33: 37. doi:10.1007/s00526-007-0152-3


A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on H10(Γ), Γ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of Γ. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.

Mathematics Subject Classification (2000)

49K10 49Q20 

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.SISSATriesteItaly