Abstract In this paper, we review recent results on stability and instability in logarithmic Sobolev inequalities, with a particular emphasis on strong norms. We consider several versions of these inequalities on the Euclidean space, for the Lebesgue and the Gaussian measures, and discuss their differences in terms of moments and stability. We give new and direct proofs, as well as examples and discuss the stability of a logarithmic uncertainty principle. Although we do not cover all aspects of the topic, we hope to contribute to establishing the state of the art.

We consider an open, bounded, simply connected (Lipschitz) domain in mathbb {R}^d, which contains a closed polyhedral surface or polygonal contour, referred to as the interface. From this interface, forces are exerted in the normal direction. The forces are continuously distributed over the interface, resulting in an integral expression. This features an important characteristic of the immersed interface method. Since the integral cannot be resolved exactly, one relies on numerical quadrature rules to approximate the integral. Therefore, we consider two different linear elasticity problems with forces over a curve or surface (interface) that is located within the (open) domain of computation: (1) The force is defined by an integral over the interface; (2) The force is defined by a quadrature approximation of the integral over the interface. We prove that the textbf{L}^2-norm of the difference between the solutions from the two elasticity problems is of the same order as the error of quadrature. The results are demonstrated for both bounded and unbounded domains. The proof that we establish relies on the use of: (i) fundamental solutions for linear elasticity, exhibiting singular behaviors (in particular around points of action) and not being in textbf{H}^1, and (ii) on the use of singularity removal principle and the Extended Trace Theorem. Convergence is demonstrated in the textbf{L}^2-norm on curves and manifolds. We show some numerical experiments on the basis of fundamental solutions with a Midpoint quadrature rule in an unbounded and a bounded domain. The numerical experiments confirm our theoretical results. We note that the difference between the interface integral and the quadrature rule over the interface holds for the exact solution in the bulk and not for any discretization carried out in the bulk. Hence, in the numerical finite element-based simulations, the numerical results contain an additional error due to the finite element approach.

Abstract It is well-known that functions in De Giorgi classes are continuous up to the boundary, if this satisfies a geometric density condition, and the Dirichlet boundary datum is itself continuous. We give a new proof of this classical result relying on ideas that were originally proposed by Gariepy and Ziemer to prove the sufficient part of the Wiener criterion for the continuity of solutions to quasilinear elliptic equations of p-Laplace type.

We study the puzzle graphs of hexagonal sliding puzzles of various shapes, and with various numbers of holes. The puzzle graph is a combinatorial model which captures the solvability and the complexity of sequential mechanical puzzles. Questions relating to the puzzle graph have been previously studied and resolved for the 15 Puzzle, which is the most famous—and unsolvable—square sliding puzzle of all time. It is known that for square puzzles such as the 15 Puzzle, solvability depends on a parity property that splits the puzzle graph into two components. In the case of hexagonal sliding puzzles, we get more interesting parity properties that depend on the shape of the boards and on the missing tiles or holes on the board. We show that for large-enough hexagonal, triangular, or parallelogram-shaped boards with hexagonal tiles, all puzzles with three or more holes are solvable. For puzzles with two or more holes, we give a solvability criterion involving both a parity property, and the placement of tiles in tight corners of the board. The puzzle graph is a discrete model for the configuration space of hard tiles (hexagons or squares) moving on different tessellation-based domains. Understanding the combinatorics of the puzzle graph could lead to understanding some aspects of the topology of these configuration spaces.

Abstract In this work we develop and investigate mathematical and computational models that describe drug delivery from a contact lens during wear. Our models are designed to predict the dynamics of drug release from the contact lens and subsequent transport into the adjacent pre-lens tear film and post-lens tear film as well as into the ocular tissue (e.g. cornea), into the eyelid, and out of these regions. These processes are modeled by one dimensional diffusion out of the lens coupled to compartment-type models for drug concentrations in the various accompanying regions. In addition to numerical solutions that are compared with experimental data on drug release in an in vitro eye model, we also identify a large diffusion limit model for which analytical solutions can be written down for all quantities of interest, such as cumulative release of the drug from the contact lens. We use our models to make assessments about possible mechanisms and drug transport pathways through the pre-lens and post-lens tear films and provide interpretation of experimental observations. We discuss successes and limitations of our models as well as their potential to guide further research to help understand the dynamics of ophthalmic drug delivery via drug-eluting contact lenses.

We consider integrals in the sense of Choquet with respect to the δ-dimensional Hausdorff content for continuously differentiable functions defined on open, connected sets in the Euclidean n-space, n≥2, 0<δ≤n. In particular, for these functions we prove Sobolev inequalities in the limiting case p=δ/n and in the case p>δ\\delta $$\\end{document}]]>, here p is the integrability exponent of the absolute value of the gradient of any given function. The results complement previously known Poincaré–Sobolev and Morrey inequalities.