Nonuniqueness in Geometric Partial Differential Equations
My project dealt with finding solutions to a particular geometric partial differential equation (PDE) with given Dirichlet boundary values. The geometric nature of this problem meant that in some instances, solutions could be realised as constant mean curvature surfaces with boundary given by the Dirichlet data. To understand what this means, we first need a crash course on surfaces and curvature:
Surfaces are 2-dimensional objects in 3-dimensional space which locally resemble a distorted plane. We restrict our attention to bounded surfaces; surfaces which can be contained within a sufficiently large ball. A closed surface is a surface which completely separates two regions of space. For example, a sphere is a closed surface; there is no way to pass from the interior to the exterior without intersecting the surface itself. If we cut away the bottom half of the sphere, it no longer has this property. Moreover, the resulting surface now has a boundary curve where the two halves used to meet. We call this type of surface a surface with boundary.
We work with a surface mathematically by working with its parametrisation. This is a map from a region in the plane onto the surface. A parametrisation is conformal if it locally preserves angles.
Seldom do we talk about surfaces without also talking about some notion of curvature. The problem in my project was concerned with the mean curvature of a surface. Without getting too technical, we can intuitively think of the mean curvature as a pointwise measure of how much the surface differs from a plane. We say a surface has constant mean curvature (CMC) if the mean curvature takes the same value at every point on the surface, and say it is minimal if this value is zero.
Surfaces can be used to model natural phenomena. For example, if we dip a wire frame into some soap solution, the soap will form a minimal surface with the wire as its boundary. Soap experiments are what inspired the classic Plateau problem: for a given boundary curve, does there exist a zero mean curvature surface with that boundary? This is closely related to the problem in my project.
The physical motivation for my project is the following. Given a boundary curve and a value for the CMC, does there exist a surface with these parameters? Formulating this problem mathematically amounts to solving a PDE. My report looks to answer the questions of existence, uniqueness and regularity of solutions to this PDE. Due to the nature of the formulation, solutions don’t necessarily have the physical realisation that motivated the PDE. If, however, a solution is conformal, it can be interpreted as a parametrisation for a surface satisfying the motivating conditions.
Jacquie Omnet was a recipient of a 2018/19 AMSI Vacation Research Scholarship.