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Stability And Bifurcation Of Equilibria For The Axisymmetric Averaged Mean Curvature Flow, Jeremy Lecrone
Stability And Bifurcation Of Equilibria For The Axisymmetric Averaged Mean Curvature Flow, Jeremy Lecrone
Department of Math & Statistics Faculty Publications
We study the averaged mean curvature ow, also called the volume preserving mean curvature ow, in the particular setting of axisymmetric surfaces embedded in R3 satisfying periodic boundary conditions. We establish analytic well-posedness of the ow within the space of little-Holder continuous surfaces, given rough initial data. We also establish dynamic properties of equilibria, including stability, instability, and bifurcation behavior of cylinders, where the radius acts as a bifurcation parameter.
On Well-Posedness, Stability, And Bifurcation For The Axisymmetric Surface Diffusion Flow, Jeremy Lecrone, Gieri Simonett
On Well-Posedness, Stability, And Bifurcation For The Axisymmetric Surface Diffusion Flow, Jeremy Lecrone, Gieri Simonett
Department of Math & Statistics Faculty Publications
We study the axisymmetric surface diffusion (ASD) flow, a fourth-order geometric evolution law. In particular, we prove that ASD generates a real analytic semiflow in the space of (2+α)-little-Holder regular surfaces of revolution embedded in R3 and satisfying periodic boundary conditions. Further, we investigate the geometric properties of solutions to ASD. Utilizing a connection to axisymmetric surfaces with constant mean curvature, we characterize the equilibria of ASD. Then, focusing on the family of cylinders, we establish results regarding stability, instability, and bifurcation behavior, with the radius acting as a bifurcation parameter.
Elliptic Operators And Maximal Regularity On Periodic Little-Hölder Spaces, Jeremy Lecrone
Elliptic Operators And Maximal Regularity On Periodic Little-Hölder Spaces, Jeremy Lecrone
Department of Math & Statistics Faculty Publications
We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is satisfied in the setting of periodic little-Hölder spaces, provided the coefficients of the differential operator satisfy minimal regularity assumptions.We address parameter-dependent elliptic equations, deriving invertibility and resolvent bounds which lead to results on generation of analytic semigroups. We also demonstrate that the techniques and results of the paper hold for elliptic differential operators with operator-valued coefficients, in the setting of vector-valued functions.
Continuous Maximal Regularity And Analytic Semigroups, Jeremy Lecrone, Gieri Simonett
Continuous Maximal Regularity And Analytic Semigroups, Jeremy Lecrone, Gieri Simonett
Department of Math & Statistics Faculty Publications
In this paper we establish a result regarding the connection between continuous maximal regularity and generation of analytic semigroups on a pair of densely embedded Banach spaces. More precisely, we show that continuous maximal regularity for a closed operator A : E1 → E0 implies that A generates a strongly continuous analytic semigroup on E0 with domain equal E1.