Approximation problems for curvature varifolds (Q1296917)

In order to study geometric variational problems involving curvature depending functionals it is necessary to introduce new classes of generalized manifolds; for example, \textit{J. E. Hutchinson} [Indiana Univ. Math. J. 35, 45-71 (1986; Zbl 0561.53008)] defined the space \(W^{2,p}_r\) consisting of all \(r\)-dimensional varifolds with second fundamental form in \(L^p\), and De Giorgi proposed to work in the class \(F_rC^\alpha\) of all Sobolev type manifolds which occur as weak limits (with \(L^p\) bounds on the second fundamental form) of functions which are locally the sum of graphs of characteristic functions of regular functions of \(r\) variables. The present paper adresses a different notion of Sobolev type manifolds, closer to De Giorgi's \(F_rC^\alpha\) classes and denoted by \(F_rW^{2,p}\) (with \(p>r)\) whose members are (locally) sums of characteristic functions of graphs of \(W^{2,p}\) functions of \(r\) variables. The authors then investigate in great detail the relations between the various classes and show that in general these classes do not coincide. In a final section, these results are illustrated through some geometric properties that the various classes may (or may not) share.