ON p-QUASICONVEX HULLS OF MATRIX SETS

Relaxation of Some Multi { Well Problemskaushik Bhattacharya

Mathematical models of phase transitions in solids lead to the variational problem, minimize R W (Du)dx where W has a multi-well structure: W = 0 on a multi-well set K and W > 0 otherwise. We study this problem in two dimensions in the case of equal determinant, i.e., for K = denotes the (3 2)-matrix formed with the rst two columns of U i. We characterize generalized convex hulls, including the...

Semiconvex Hulls of Quasiconformal Sets

We make some remarks concerning the p-semiconvex hulls of the quasiconformal sets, using a recent significant observation of T. Iwaniec in the paper [7] on the important relation between the regularity of quasiregular mappings in the theory of geometric functions and the notion of Morrey’s quasiconvexity in the calculus of variations. We also point out several partial results on a conjecture in...

On the Stability of Quasiconvex Hulls

F Ur Mathematik in Den Naturwissenschaften Leipzig 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 2 M 22 : F T = F; jF 11 j = a; jF 12 j = b; jF 22 j = cg that was rst considered by Dacorogna&Tanteri Commun. in PDEs 2001]. We obtain explicit formulae for the polyconvex, the quasiconvex, and the rank-one convex hull for ac ? b 2 0 and show in particular that the quasiconvex and the polyconvex hull are diier...

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...