Publication

Graphical solutions to one-phase free boundary problems

Xavier Fernandez-Real Girona, Hui Yu
2023
Journal Articles

Abstract

We study viscosity solutions to the classical one-phase problem and its thin counterpart. In low dimensions, we show that when the free boundary is the graph of a continuous function, the solution is the half-plane solution. This answers, in the salient dimensions, a one-phase free boundary analogue of Bernstein's problem for minimal surfaces. As an application, we also classify monotone solutions of semilinear equations with a bump-type nonlinearity.

Official source

https://infoscience.epfl.ch/entities/publication/5a17e49f-4eeb-4d70-aecf-4f4bbadfb7ad

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Xavier Fernandez-Real Girona, Hui Yu
2023
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/5a17e49f-4eeb-4d70-aecf-4f4bbadfb7ad

About this result

Related concepts (27)

Viscosity

The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity is defined scientifically as a force multiplied by a time divided by an area. Thus its SI units are newton-seconds per square metre, or pascal-seconds. Viscosity quantifies the internal frictional force between adjacent layers of fluid that are in relative motion.

Minimal surface

In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame.

Boundary layer

In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condition (zero velocity at the wall). The flow velocity then monotonically increases above the surface until it returns to the bulk flow velocity. The thin layer consisting of fluid whose velocity has not yet returned to the bulk flow velocity is called the velocity boundary layer.

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