Related papers: Solving the brachistochrone and other variational …
Motivated by the evaporation of soap films, which has a significant effect on their lifetime, we performed an experimental study on the evaporation of vertical surfaces with model systems based on hydrogels. From the analogy between heat…
The article provides a pedagogical review aimed at graduate students in materials science, physics, and applied mathematics, focusing on recent developments in the subject. Following a brief summary of concepts from complex analysis, the…
A non-linear theory for the plastic deformation of prismatic bodies is constructed which interpolates between Prandtl's linear soap-film approximation and N\'adai's sand-pile model . Geometrically Prandtl's soap film and N\'adai's wavefront…
We describe the extension of a "viscous froth" model to the dynamics of a wet foam in a Hele-Shaw cell. The two-dimensional model includes the friction experienced by the soap films as they are dragged along the cell walls, while retaining…
After a brief introduction to several variational problems in the study of shapes of thin thickness structures, we deal with variational problems on 2-dimensional surface in 3-dimensional Euclidian space by using exterior differential…
An adaptive proximal method for a special class of variational inequalities and related problems is proposed. For example, the so-called mixed variational inequalities and composite saddle problems are considered. Some estimates of the…
Many objects in nature and industry are wrapped in a thin sheet to enhance their chemical, mechanical, or optical properties. There are similarly a variety of methods for wrapping, from pressing a film onto a hard substrate, to using…
Exact solutions of a classical problem of a plane unsteady potential flow of an ideal incompressible fluid with a free boundary are presented. The fluid occupies a semi-infinite strip bounded by the free surface (from above) and (from the…
An emerging theme across many domains of science and engineering is materials that change shape, often dramatically. Determining their structure involves solving a shape optimization problem where a given energy functional is minimized with…
Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the…
Viscous contact problems describe the time evolution of fluid flows in contact with a surface from which they can detach and reattach. These problems are of particular importance in glaciology, where they arise in the study of grounding…
We develop a variational method for interacting surface systems with higher-form global symmetries. As a natural extension of the conventional second-quantized Hamiltonian of interacting bosons, we explicitly construct a second-quantized…
We present a mathematical and numerical framework for thin-film fluid flows over planar surfaces including dynamic contact angles. In particular, we provide algorithmic details and an implementation of higher-order spatial and temporal…
We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric…
The construction of stochastic solutions for nonlinear partial differential equations is a powerful method to obtain new exact results and to develop efficient numerical algorithms, in particular when domain decomposition techniques are…
Based on the mathematical-physical model of pavement mechanics, a multilayer elastic system with interlayer friction conditions is constructed. Given the complex boundary conditions, the corresponding variational inequalities of the partial…
Understanding the stability of foam films and foams remains a challenge, despite the considerable efforts provided by the scientific community to refine their physical descriptions. This persistent difficulty underscores the interplay of…
We study a class of second order variational inequalities with bilateral constraints. Under certain conditions we show the existence of a unique viscosity solution of these variational inequalities and give a stochastic representation to…
We study the capillary wave dynamics of a single viscoelastic supported film and of a double layer of immiscible viscoelastic supported films. Using both simple scaling arguments and a continuum hydrodynamic theory, we investigate the…
A metallic film of arbitrary thickness is considered. We show that the problem of description of surface plasma oscillations (surface plasmons) with reflection boundary conditions allows analytic solution. Besides, this problem allows…