Euler equations and trace properties of minimizers of a functional for motion compensated inpainting

<p style='text-indent:20px;'>We compute the Euler equations of a functional useful for simultaneous video inpainting and motion estimation, which was obtained in [<xref ref-type="bibr" rid="b17">17</xref>] as the relaxation of a modified version of the functional proposed in [<xref ref-type="bibr" rid="b16">16</xref>]. The functional is defined on vectorial functions of bounded variations, therefore we also get the Euler equations holding on the singular sets of minimizers, highlighting in particular the conditions on the jump sets. Such conditions are expressed by means of traces of geometrically meaningful vector fields and characterized as pointwise limits of averages on cylinders with axes parallel to the unit normals to the jump sets.</p>

On Inverse Nodal Problem and Multiplicities of Eigenvalues of a Vectorial Sturm-Liouville Problem

An m-dimensional vectorial inverse nodal Sturm-Liouville problem with eigenparameter-dependent boundary conditions is studied. We show that if there exists an infinite sequence ynj,rx,λnj,r2j=1∞ of eigenfunctions which are all vectorial functions of type (CZ), then the potential matrix Qx and A are simultaneously diagonalizable by the same unitary matrix U. Subsequently, some multiplicity results of eigenvalues are obtained.