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Beschreibung:
This cumulative thesis discusses various aspects of constrained gradient flows of higher order. The main focus is the analysis of constrained Willmore-type flows of curves and surfaces and especially their asymptotic behavior. These flows yield quasilinear nonlocal geometric evolution equations of fourth order, making them challenging from the perspective of partial differential equations. Chapter 1 provides a brief review on the relevant geometric and analytic concepts needed for the following chapters 2 to 7, each of which contains one research article, Articles A to F. Chapter 2. Article A: F. Rupp. On the Lojasiewicz-Simon gradient inequality on submanifolds. J. Funct. Anal. 279 (8):108708, 2020. 33 pages. We prove a suitable version of the Lojasiewicz–Simon gradient inequality, a fundamental functional analytic tool to study general gradient flows with constraints, which will be essential for the subsequent asymptotic analysis of concrete geometric evolution problems. Chapter 3. Article B: F. Rupp and A. Spener. Existence and convergence of the length-preserving elastic flow of clamped curves, 2020. Preprint. 49 pages. We discuss the length-preserving elastic flow of open curves with clamped boundary conditions and prove existence, parabolic smoothing and convergence for initial data lying merely in the energy space. Chapter 4. Article C: F. Rupp. The volume-preserving Willmore flow, 2020. Preprint. 46 pages. We consider a constrained version of the Willmore flow of immersed closed surfaces which preserves the enclosed volume. For spherical initial data with nonzero volume and Willmore energy below 8π, we show global existence and convergence. Chapter 5. Article D: F. Rupp. The Willmore flow with prescribed isoperimetric ratio, 2021. Preprint. 39 pages. We study the Willmore flow with a constraint on the isoperimetric ratio. This flow describes a dynamical approach to the Canham–Helfrich model for lipid bilayers with zero spontaneous curvature. Under suitable assumptions on the topology and the initial ...