Calculus of Variations and Geometric Measure Theory

D. Percivale - F. Tomarelli

Smooth and Broken Minimizers of Some Free Discontinuity Problems

created by tomarelli1 on 23 Jan 2018


Published Paper

Inserted: 23 jan 2018
Last Updated: 23 jan 2018

Journal: Springer INdAM Series
Volume: 22
Pages: 431-468
Year: 2017
Doi: 10.1007/978-3-319-64489-9_17

in: Colli, P., Favini, A., Rocca, E., Schimperna, G., Sprekels, J. (Eds.), Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs,


We show that minimizers of free discontinuity problems with energy dependent on jump integrals and Dirichlet boundary conditions are smooth provided a smallness condition is imposed on data. We examine in detail two examples: the elastic-plastic beam and the elastic-plastic plate with free yield lines. In both examples there is a gap between the condition for solvability (safe load condition) and this smallness condition (load regularity condition) which imply regularity and uniqueness of minimizers. Such gap allows the existence of damaged or creased minimizers. Eventually we produce explicit examples of irregular solutions when the load is in the gap.

Keywords: Bounded Hessian functions