The notion of quasiconvex exposed points is introduced for compact sets of matrices, motivated from the variational approach to material microstructures. We apply the notion to give geometric descriptions of the quasiconvex extreme points for a compact set. A weak version of Straszewicz type density theorem in convex analysis is established for quasiconvex extreme points. Some examples are examined by using known explicit quasiconvex functions.

We establish an approximation theorem for a sequence of linear elastic strains approaching a compact set in ${L}^{1}$ by the sequence of linear strains of mapping bounded in Sobolev space ${W}^{1,p}$. We apply this result to establish equalities for semiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth.

The notion of quasiconvex exposed points is introduced for compact sets of matrices, motivated
from the variational approach to material microstructures.
We apply the
notion to give geometric descriptions of the
quasiconvex extreme points for a compact set. A weak version of Straszewicz type
density theorem in convex analysis is established for quasiconvex extreme points. Some examples
are examined by using known explicit quasiconvex functions.

We establish an approximation theorem for a sequence of
linear elastic strains approaching a compact set in
by the
sequence of linear strains of mapping bounded in Sobolev space
. We apply this result to establish equalities for
semiconvex envelopes for functions defined on linear strains a
construction of quasiconvex functions with linear growth.

In the shape from shading problem of computer vision one
attempts to recover the three-dimensional shape of an object or
landscape from the shading on a single image. Under the
assumptions that the surface is dusty, distant, and illuminated
only from above, the problem reduces to that of solving the
eikonal equation on a domain in ${\mathbb{R}}^{2}$. Despite
various existence and uniqueness theorems for smooth solutions,
we show that this problem is unstable, which is catastrophic for
general numerical algorithms.
...

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