Inserted: 28 may 2007
Last Updated: 16 oct 2009
Journal: Math. Models Methods Appl. Sci.
DOI : 10.1142S0218202509000411X
We focus the minimization of 1D free discontinuity problem with second order energy dependent on gradient-jump integrals but not on the cardinality of the discontinuity set, in the framework of $L^\infty\,$load. The related energies are not lower semi continuous in $BH$. Nevertheless we show that if a safe load condition is fulfilled then minimizers exist and they belong actually to $SBH,$ say their second derivative has no Cantor part. If in addition a stronger condition on load is fulfilled then minimizer is unique and belongs to $H^2.$ Moreover we can always select one minimizer whose number of plastic hinges does not exceed 2 and is the limit of minimizers of penalized problems. When the load stays in the gap between safe load and regularity condition then minimizers with hinges are allowed; if in addition the load is symmetric and strictly positive then there is uniqueness of minimizer, the hinges of such minimizer are exactly two and they are located at the endpoints.
Keywords: free discontinuity problems, plastic hinges