We present the necessary conditions for the existence of the Kolwankar-Gangal local fractional derivatives (KG-LFD) and introduce more general but weaker notions of LFDs by using limits of certain integral averages of the difference-quotient. By applying classical results due to Stein and Zygmund (1965) [16] we show that the KG-LFD is almost everywhere zero in any given intervals. We generalize some of our results to higher dimensional cases and use integral approximation formulas obtained to design numerical schemes for detecting fractional dimensional edges in signal processing.

We introduce the notions of lower and upper quadratic compensated convex transforms C l 2,λ (f) and C u 2,λ (f) respectively and the mixed transforms by composition of these transforms for a given function f : R n → R and for possibly large λ > 0. We study general properties of such transforms, including the so-called 'tight' approximation of C l 2,λ (f) to f as λ → +∞ and compare our transforms with the well-known Moreau-Yosida regularization (Moreau envelope) and the Lasry-Lions regularization. We also study analytic and geometric properties for both the quadratic lower transform C l 2,λ (dist 2 (x, K)) of the squared-distance function to a compact set K and the quadratic upper transform C u 2,λ (f) for any convex function f of at most quadratic growth. We show that both C l 2,λ (dist 2 (x, K)) and C u 2,λ (f) are C 1,1 approximations of the original functions for large λ > 0 and C u 2,λ (f) remains convex. Explicitly calculated examples of quadratic transforms are given, including the lower transform of squared distance function to a finite set and upper transform for some non-smooth convex functions in mathematical programming.

We establish the existence of infinitely many weak solutions for the the one-dimensional version of the well-known and widely used Perona-Malik anisotropic diffusion equation model in image processing. We establish the existence result under the homogeneous Neumann condition with smooth nonconstant initial values. Our method is to convert the problem into a partial differential inclusion problem.

We define the set K_{q,e} ⊂ K of quasiconvex extreme points for compact sets K ⊂ M^{N×n} and study its properties. We show that K_{q,e} is the smallest generator of Q(K) -the quasiconvex hull of K , in the sense that Q(K_{q,e}) = Q(K) , and that for every compact subset W ⊂ Q(K) with Q(W) = Q(K) , K_{q,e} ⊂ W . The set of quasiconvex extreme points relies on K only in the sense that Q\left(K\right)_{q,e} \subset K_{q,e} \subset \overline{[Q\left(K\right)_{q,e}]} . We also establish that K_e ⊂ K_{q,e} , where K_e is the set of extreme points of C(K) -the convex hull of K . We give various examples to show that K_{q,e} is not necessarily closed even when Q(K) is not convex; and that for some nonconvex Q(K) , K_{q,e} = K_e . We apply the results to the two well and three well problems studied in martensitic phase transitions.