2000 Mathematics Subject Classification: 35A15, 44A15, 26A33The paper is devoted to the study of the Cauchy-type problem for the differential equation [...] involving the Riemann-Liouville partial fractional derivative of order α > 0 [...] and the Laplace operator.

A system of one-dimensional linear parabolic equations coupled by boundary conditions which include additional state variables, is considered. This system describes an electric circuit with distributed parameter lines and lumped capacitors all connected through a resistive multiport. By using the monotony in a space of the form $L^2(0,T;H^1)$, one proves the existence and uniqueness of a variational solution, if reasonable engineering hypotheses are fulfilled.

We study the problem of minimizing ${\int }_{\Omega }F\left(Du\left(x\right)\right)dx$ over the functions $u\in W^{1,1}\left(\Omega \right)$ that assume given boundary values $\phi$ on $\Gamma :=\partial \Omega$. The lagrangian $F$ and the domain $\Omega$ are assumed convex. A new type of hypothesis on the boundary function $\phi$ is introduced: thelower (or upper) bounded slope condition. This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary...

A corrector theory for the strong approximation of gradient fields inside periodic composites made from two materials with different power law behavior is provided. Each material component has a distinctly different exponent appearing in the constitutive law relating gradient to flux. The correctors are used to develop bounds on the local singularity strength for gradient fields inside micro-structured media. The bounds are multi-scale in nature and can be used to measure the amplification of applied...