Department of Mathematics and Statistics
Texas Tech University
We survey recent results regarding curvature-dependent energy functionals, obtained by variational methods. Conditions for criticality and stability are discussed along with applications to fixed and free boundary problems. Results are discussed in general and then specialized to relevant Willmore and p-Willmore surfaces with accompanying simulations.Equilibrium configurations for an energy which is a linear combination of the area and a second term which measures the bending of the boundary have been recently introduced to model thin fluid membranes with elastic boundary components, the Euler-Plateau problem. A natural extension of this problem consists of allowing the boundary curve, regarded as a flexible rod, to twist. This twisting requires an additional term in the energy, leading to the so-called Kirchhoff-Plateau problem. If we replace the rod by a twisted elastic ribbon, then a free boundary equilibrium minimal surface will meet this supporting ribbon at a right angle, so that the twisting energy of the ribbon will be completely determined by the Darboux frame of the surface. Motivated by this, we will discuss equilibrium configurations for the Kirchhoff-Plateau problem where the twisting energy is determined by the Darboux frame.This will be an expository type talk on a field about which the speaker is currently learning. Proving non-uniqueness for PDEs in fluid is a well-known challenging problem, e.g., the Navier-Stokes equations. In fact, until a few decades ago, a consensus was that while classical techniques lead to proofs of uniqueness for sub-critical PDEs, the bag of tricks to prove non-uniqueness for super-critical PDEs was relatively scant. To the best of my knowledge, the term ``convex integration'' was introduced by Gromov, inspired by the work of Nash and Kuiper on isometric embedding theorems. I will survey a series of breakthrough works by De Lellis, Szekelyhidi Jr., and others that have allowed this convex integration reformulated for PDEs to prove non-uniqueness for various PDEs in fluid. In fact, in the spirit of homotopy-principle o Gromov, these new results remarkably proved not only non-uniqueness but the existence of infinitely many solutions (while two is enough to prove non-uniqueness).
The Euler-Plateau problem consists of seeking equilibrium configurations of a functional combining the area of a surface and the bending energy of its boundary. Here, we will discuss an extension of this problem which appears after including the physically relevant total Gaussian curvature term in the energy.The talk is accessible to beginning graduate students. Consider a bilinear form in a vector space. Its index is defined to be the maximal dimension on which it is negative definite. For example, when the space is finite-dimensional, the index of a matrix counts the number of negative eigenvalues. If we reduce to a subspace, how does the index change? In this talk, we'll solve that problem via the theory of Hilbert spaces. The Riesz representation theorem will play a crucial role. The answer is of great importance since a fundamental theme of mathematics is to study optimization problems related to geometric functionals. The generalization of the second derivative test from elementary calculus is called the second variation which is, normally, a bilinear form in a function space. When the index is zero, we are at a local minimum. The presence of constraints is translated to the consideration of a subspace. Therefore, in the next talk in the Applied Math Seminar, I will demonstrate how we can apply the theory here to study capillary surfaces which arise when mixing different fluids. This is joint work with Detang Zhou.We discuss a mixed finite element method for solving time-dependent geometric PDEs on unstructured surface meshes. Using the p-Willmore flow as an example, it is shown that this method effectively produces dissipative schemes which mimic the behavior of their continuous counterparts. Applications are considered, including experimental results which illustrate the effect of the parameter p on flow behavior