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 

The problem is to limit the motion of a robot so that if it is commanded to work outside of its workspace, then the robot experiences a graceful degradation of performance depending upon the extent of the workspace violation.  This is demonstrated with Stewart tables, which provide six degrees of freedom.

In this talk, we will show that a geometrical condition on \(2 \times 2\) systems of conservation laws leads to nonuniqueness in the class of 1D continuous functions. This demonstrates that the Liu Entropy Condition alone—which ensures the uniqueness of small BV solutions—is insufficient to guarantee uniqueness in the continuous setting, even within the mono-dimensional frame. We provide examples of systems where this pathology holds, even if they verify stability and uniqueness for small BV solutions. Our proof is based on the convex integration process.

In this talk, we’ll present some recent work on traveling waves in water that carry vortices in their bulk. We show that for any supercritical Froude number (non-dimensionalized wave speed), there exists a continuous one-parameter family of solitary waves with a submerged point vortex in equilibrium. This family bifurcates from an irrotational laminar flow, and, at least for large Froude numbers, it extends up to the development of a surface singularity.

Systems that can be classified as oscillatory media consist of small oscillating elements that interact with each other via some form of coupling. Interest in these systems stems in part from their ability to generate beautiful structures, including target patterns and spiral waves.  When coupling between oscillators occurs over long spatial scales, and for certain parameter values, the set of unstable waven umbers that generate these patterns is no longer constrained to a narrow band.

We prove exact leading-order asymptotic behaviour at the origin for nontrivial solutions of two families of nonlocal equations. The equations investigated include those satisfied by the cusped highest steady waves for both the uni- and bidirectional Whitham equations. The problem is therefore analogous to that of capturing the 120∘ interior angle at the crests of classical Stokes’ waves of greatest height.

The goal of this talk is to discuss spectral theory for quantum Hamiltonians describing  a system of two  identical spinless bosons on the two-dimensional lattice particles.

We assume that the particles are interact via on-site and first and second nearest-neighboring site interactions between the bosons are only nontrivial and that these interactions are of magnitudes \(\gamma\), \(\lambda\), and \(\mu\), respectively.