An elastic surface resists not only changes in curvature but also tangential stretches and shears.  In classical plate and shell theories, e.g., due to von Karman, the latter two strain measures are approximated infinitesimally.  We motivate our approach via the phenomenon of wrinkling in highly stretched elastomers.  We postulate a novel, physically reasonable class of stored-energy densities, and we prove various existence theorems based on the direct method of the calculus of variations.

The work is devoted to the studies of the existence of
solutions of an integro-differential equation in the case of the double
scale anomalous diffusion with the sum of the two negative Laplacians
raised to two distinct fractional powers in \(\mathbb{R}^d\), \(d=4,5\). The proof of the
existence of solutions is based on a fixed point technique. Solvability
conditions for the non-Fredholm elliptic operators in unbounded domains
are used.

In this talk we will survey some old and new results on maximum principles for P-functions and their applications to the study of partial differential equations. More precisely, we will show how one can employ the maximum principle in problems of physical or geometrical interest, in order to get the shape of some free boundaries, isoperimetric inequalities, symmetry results, convexity results and Liouville type results.

We compute asymptotics of  eigenvalues approaching the bottom of the continuous spectrum, and associated resonances, for Schrödinger operators in dimension two for which the potential depends on a parameter.  We distinguish persistent eigenvalues, which have associated resonances,  from disappearing ones, which do not.  We illustrate the significance of this distinction by computing corresponding scattering phase asymptotics and numerical Breit--Wigner peaks.  While we concentrate on the case of the circular well for