an:06532740
Zbl 1331.49014
Fuchs, M.; Tietz, C.
Existence of generalized minimizers and dual solutions for a class of variational problems with linear growth related to image recovery
EN
J. Math. Sci., New York 210, No. 4, 458-475 (2015); translation from Probl. Mat. Anal. 81, 107-120 (2015).
00349811
2015
j
49J45 49N15 26B30 94A08
variational problems; generalized minimizers; dual solutions; functions of bounded variation; image recovery
Summary: We continue the analysis of modifications of the total variation image inpainting method formulated on the space \(BV(\Omega)^M\) and treat the case of vector-valued images where we do not impose any structure condition on the density \(F\) and the dimension of the domain \(\Omega\) is arbitrary. We discuss the existence of generalized solutions of the corresponding variational problem and show the unique solvability of the associated dual variational problem. We establish the uniqueness of the absolutely continuous part \(\nabla^au\) of the gradient of \(BV\)-solutions \(u\) on the domain \(\Omega\) and get the uniqueness of \(BV\)-solutions outside the damaged region \(D\). We also prove new density results for functions of bounded variation and for Sobolev functions.