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Existence result for a new variational problem in one-dimensional segmentation theory. (English) Zbl 0778.49013

The following one-dimensional segmentation problem for some \(g\in L^ 2\) is considered: \[ F(u,J_ 1,J_ 2):=\int^ b_ a| u''|^ 2 dt+\int^ b_ a| u-g|^ 2 dt+\alpha \# (J_ 1)+\# \beta(J_ 2)\to\min, \] where \(u\) belongs to the intersection of the spaces \(C^ 0(I\backslash J_ 1)\) and \(C^ 2\bigl(I\backslash (J_ 1\cup J_ 2)\bigr)\), \(I:=[a,b]\), \(\#\) is the counting measure on the reals \({\mathfrak R}\), \(J_ 1\) is the cardinality of the finite set \(S_ u\) of all jumps of \(u\) and \(J_ 2\) is the cardinality of the finite set \(S_{u'}\) of all jumps of \(u'\) which doesn’t belong to \(S_ u\). The paper is the first attempt to make a piecewise smooth approximation replacing the first derivative term \(\int^ b_ a | u'|^ 2 dt\) by the cardinality of the jumps of the first derivative \(u'\) different from the jumps of the function \(u\).
Considering the Sobolev-Hilbert space \(H^ 2\) on the subintervals of continuity, the author shows that for positive constants \(\alpha\) and \(\beta\) the lower level sets of the objective are sequentially compact in \(L^ 1\), more precisely:
Any sequence \((u_ n)_{n\in N}\) belonging to such a level set contains a subsequence converging strongly in \(L^ 1\) to a \(u\) of the above structure such that, additionally, \(\dot u_ n\to\dot u\) a.e. and \(\ddot u_ n\to\ddot u\) weakly in \(L^ 2\). In Chapter 3 she shows that the objective is \(L^ 1\) (sequentially) lower semicontinuous on such considered level sets. Using illustrative examples after both the theorems she demonstrates that the assumptions \(\alpha>0\), \(\beta>0\) and for the semicontinuity the assumption \(0<\beta\leq\alpha\leq 2\beta\) is essential. Combining the compactness and the semicontinuity property she proves an existence theorem for the above stated problem. Again she shows that the existence of a solution is not ensured whenever the assumptions of the existence theorem are violated.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
41A99 Approximations and expansions
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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