F Ur Mathematik in Den Naturwissenschaften Leipzig Quasiconvex Hulls in Symmetric Matrices Quasiconvex Hulls in Symmetric Matrices

Quasiconvex Hulls in Symmetric Matrices

We analyze the semiconvex hulls of the subset K in symmetric matrices given by K fF M F F jF j a jF j b jF j cg that was rst considered by Dacorogna Tanteri Commun in PDEs We obtain explicit formulae for the polyconvex the quasiconvex and the rank one convex hull for ac b and show in particular that the quasiconvex and the polyconvex hull are di erent if strict inequality holds For ac b we obta...

Mathematik in den Naturwissenschaften Leipzig Relaxation of three solenoidal wells and characterization of three - phase H - measures

We study the problem of characterizing quasiconvex hulls for three “solenoidal” (divergence free) wells in dimension three when the wells are pairwise incompatible. A full characterization is achieved by combining certain ideas based on Šverák’s example of a “nontrivial” quasiconvex function and on Müller’s wavelet expansions estimates in terms of the Riesz transform. As a by-product, we obtain...

ON p-QUASICONVEX HULLS OF MATRIX SETS

We present some basic properties and equivalent definitions of the p-quasiconvex hull of a given set of matrices. In particular, we completely characterize the p-quasiconvex hull in terms of the W -gradient Young measures studied by [9] and establish an important relationship with the weak convergence in Sobolev spaces. We also give some simple characterization of the p-quasiconvex hulls for ce...

Mathematik in den Naturwissenschaften Leipzig Quasiconvex relaxation of multidimensional control problems with integrands f ( t

We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original prob...

F Ur Mathematik in Den Naturwissenschaften Leipzig Interacting Quantum Fields on a Curved Background