Gershon Wolansky

Apparent contact angles on rough surfaces: the Wenzel equation revisited

Abstract The thermodynamic status of the Wenzel equation is discussed. It is explained and demonstrated by an example that the Wenzel equation may be incorrect. However, it is mathematically proven for sawtooth surfaces that the Wenzel equation becomes the correct one for the apparent contact angle as the drop size becomes infinitely larger than the scale of the roughness. It is shown, that under this condition the drop becomes axisymmetric even if the roughness of the solid surface is not. A complete proof for rough surfaces in general is also feasible, however the mathematical complexity of this proof is beyond the present scope.

On steady distributions of self-attracting clusters under friction and fluctuations

We consider the Vlasov-Fokker-Planek equation with a Newtonian, attracting potential and study its stationary solutions, given by the generalized Lane-Emden equation. In a two-dimensional domain we obtain the existence of a critical mass beyond which the system may admit a gravitational collapse. For a one-dimensional model we prove some results on existence, uniqueness, stability and symmetry-breaking of stationary solutions.

On nonlinear stability of polytropic galaxies

Abstract We consider a certain class of stationary solutions of the Vlasov-Poisson system with a Newtonian potential. These solutions represent some spherically-symmetric, rotating galaxies obeying the generalized Emden-Fowler law. A nonlinear stability under spherically symmetric perturbations is proved. The proof is based on an adaptation of the energy-Casimir method, which implies a uniform estimate on the mass distribution of the perturbed solutions.

Intensity control with a free-form lens.

A family of free-form lenses for intensity control is designed. The lens can shape an incident collimated beam with a given intensity distribution I1 into a new collimated beam with intensity distribution I2. No symmetry is assumed for the two intensity profiles. The key idea is that the lens design problem can be formulated and solved in terms of an optimization process involving a specific action functional. It is further shown that the free-form lens can be manufactured by a surfacing process using a convex tool.

Moser-Trudinger and logarithmic HLS inequalities for systems

We prove several optimal Moser-Trudinger and logarithmic Hardy-Littlewood-Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere

A variational principle in optics.

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Existence and Nonlinear Stability of Stationary States of the Schrödinger–Poisson System

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Combinatorial Ricci Curvature and Laplacians for Image Processing

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Ray mapping and illumination control

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Using well-solvable quadratic assignment problems for VLSI interconnect applications

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