MSc Defense in Mathematics: Albert Jels Eilgaard
MSc Defense in Mathematics
Candidate: Albert Jels Eilgaard
Title: Nonlocal Minimal Surfaces
The Nonlocal Yau Conjecture
Abstract: This thesis concerns the Yau conjecture for nonlocal minimal surfaces, that is, the existence of infinitely many critical points of the fractional perimeter functional in a closed Riemannian manifold. We first discuss essentials of the theory of fractional Sobolev spaces, giving a treatment of the theory for Rn , as well as the case of a general closed manifold. We highlight that in many cases, the results and proofs are essentially the same, when viewed through the lens of Fourier Analysis. We then apply the theory to show the veracity of the nonlocal Yau conjecture. This will be done by applying min-max theory to establish solutions of nonlocal Allen-Cahn equations with ε−parameter going to 0 and bounded Morse index. We will then show that as ε ↘ 0 these converge strongly in suitable Sobolev spaces to the indicator function of a set of finite s−perimeter. Finally, we verify that these sets are critical points of the s−perimeter functional, from which the Nonlocal Yau Conjecture will follow. Furthermore, we see that the Morse index is lower semi-continuous in this limiting procedure.
Vejleder: Niels Martin Møller (GeoTop)
Title: Nonlocal Minimal Surfaces
The Nonlocal Yau Conjecture
Abstract: This thesis concerns the Yau conjecture for nonlocal minimal surfaces, that is, the existence of infinitely many critical points of the fractional perimeter functional in a closed Riemannian manifold. We first discuss essentials of the theory of fractional Sobolev spaces, giving a treatment of the theory for Rn , as well as the case of a general closed manifold. We highlight that in many cases, the results and proofs are essentially the same, when viewed through the lens of Fourier Analysis. We then apply the theory to show the veracity of the nonlocal Yau conjecture. This will be done by applying min-max theory to establish solutions of nonlocal Allen-Cahn equations with ε−parameter going to 0 and bounded Morse index. We will then show that as ε ↘ 0 these converge strongly in suitable Sobolev spaces to the indicator function of a set of finite s−perimeter. Finally, we verify that these sets are critical points of the s−perimeter functional, from which the Nonlocal Yau Conjecture will follow. Furthermore, we see that the Morse index is lower semi-continuous in this limiting procedure.
Vejleder: Niels Martin Møller (GeoTop)