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Heiko Kröner

Publications and source records attributed to Heiko Kröner.

17 recordsLinked to original sources
arXiv2021

Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.

math.AP↗
arXiv2021

A priori error estimates for finite element approximations of regularized level set flows in higher norms

This paper proves error estimates for $H^2$ conforming finite elements for equations which model the flow of surfaces by different powers of the mean curvature (this includes mean curvature flow). for an adapted scheme originally proposed in [17] for the inverse mean curvature flow. The scheme is based on a known regularization procedure and produces different kinds of errors, a regularization error, a finite element discretization error for the regularized problems and a full error. While in the literature and own previous work different aspects of the aforementioned error types are treated, here, we solely and for the first time focus on the finite element discretization error in the $W^{2,μ}$ norm for the regularized equation analyzing also the dependencies from the regularization parameter.

math.NA↗
arXiv2020

Flowing the leaves of a foliation with normal speed given by the logarithm of general curvature functions

Generalizing results of Chou and Wang \cite{1} we study the flows of the leaves $(M_Θ)_{Θ>0}$ of a foliation of $\mathbb{R}^{n+1}\setminus \{0\}$ consisting of uniformly convex hypersurfaces in the direction of their outer normals with speeds $-\log(F/f)$. For quite general functions $F$ of the principal curvatures of the flow hypersurfaces and $f$ a smooth and positive function on $S^n$ (considered as a function of the normal) we show that there is a distinct leaf $M_{Θ_{*}}$ in this foliation with the property that the flow starting from $M_{Θ_{*}}$ converges to a translating solution of the flow equation. Furthermore, when starting the flow from a leave inside $M_{Θ_{*}}$ it shrinks to a point and when starting the flow from a leave outside $M_{Θ_{*}}$ it expands to infinity. While \cite{1} considered this mechanism with $F$ equal to the Gauss curvature we allow $F$ to be among others the elementary symmetric polynomials $H_k$. We, furthermore, show that such kind of behavior is robust with respect to relaxing certain assumptions at least in the rotationally symmetric and homogeneous degree one curvature function case.

math.DG↗
arXiv2019

Definition and certain convergence properties of a two-scale method for Monge-Ampère type equations

The Monge-Ampère equation arises in the theory of optimal transport. When more complicated cost functions are involved in the optimal transportation problem, which are motivated e.g. from economics, the corresponding equation for the optimal transportation map becomes a Monge-Ampère type equation. Such Monge-Ampère type equations are a topic of current research from the viewpoint of mathematical analysis. From the numerical point of view there is a lot of current research for the Monge-Ampère equation itself and rarely for the more general Monge-Ampère type equation. Introducing the notion of discrete $Q$-convexity as well as specifically designed barrier functions this purely theoretical paper extends the very recently studied two-scale method approximation of the Monge-Ampère itself \cite{NochettoNtogkasZhang2018} to the more general Monge-Ampère type equation as it arises e.g. in \cite{PhillippisFigalli2013} in the context of Sobolev regularity.

math.NA↗
arXiv2019

Li-Yau gradient estimates for curvature flows in positively curved manifolds

We prove differential Harnack inequalities for flows of strictly convex hypersurfaces by powers $p$, $0<p<1$, of the mean curvature in Einstein manifolds with a positive lower bound on the sectional curvature. We assume that this lower bound is sufficiently large compared to the derivatives of the curvature tensor of the ambient space and that the mean curvature of the initial hypersurface is sufficiently large compared to the ambient geometry. We also obtain some new Harnack inequalities for more general curvature flows in the sphere, as well as a monotonicity estimate for the mean curvature flow in non-negatively curved, locally symmetric spaces.

math.DG↗
arXiv2018

Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and $F$ is a positive, strictly monotone and 1-homogeneous curvature function. In particular this class includes the mean curvature $F=H$. We prove that a certain initial pinching condition is preserved and the properly rescaled hypersurfaces converge smoothly to the unit sphere. We show that an example due to Andrews-McCoy-Zheng can be used to construct strictly convex initial hypersurfaces, for which the inverse mean curvature flow to the power $p>1$ loses convexity, justifying the necessity to impose a certain pinching condition on the initial hypersurface.

math.DG↗
arXiv2016

Optimal control of elliptic surface PDEs with pointwise bounds on the state

We consider a linear-quadratic optimization problem with pointwise bounds on the state for which the constraint is given by the Laplace-Beltrami equation (to have uniqueness we add an lower order term) on a two-dimensional surface . By using finite elements we approximate the optimization problem by a family of discrete problems and prove convergence rates for the discrete controls and the discrete states. Furthermore, assuming (roughly spoken) a higher regularity for the control the order of convergence improves. This extends a result known in an Euclidean setting to the surface case.

math.OC↗
arXiv2016

Variational discretization of parabolic control problems on evolving surfaces with pointwise state constraints

We consider a linear-quadratic pde constrained optimal control problem on an evolving surface with pointwise state constraints. We reformulate the optimization problem on a fixed surface and approximate the reformulated problem by a discrete control problem based on a discretization of the state equation by linear finite elements in space and a discontinuous Galerkin scheme in time. We prove error bounds for control and state.

math.OC↗
arXiv2016

Alternative to evolving surface finite element method

ESFEM is a method introduced in order to solve a linear advection-diffusion equation on an evolving two-dimensional surface with finite elements by using a moving grid with nodes sitting on and evolving with the surface. The evolution of the surface is assumed to be given as a smooth one-parameter family of embeddings of a fixed initial surface into $\mathbb{R}^3$ satisfying uniform $C^4$ bounds. We calculate an equivalent transformed equation which is defined on the fixed initial surface and can hence be solved numerically on a fixed grid. We present numerical examples which indicate that both approaches are essentially of the same accuracy.

math.NA↗
arXiv2015

$L^{\infty}$-error estimate for the finite element method on two dimensional surfaces

We approximate the solution of the equation $$ -Δ_S u+u = f $$ on a two-dimensional, embedded, orientable, closed surface $S$ where $-Δ_S$ denotes the Laplace Beltrami operator on $S$ by using continuous, piecewise linear finite elements on a triangulation of $S$ with flat triangles. We show that the $L^{\infty}$-error is of order $O(h^2|\log h|)$ as in the corresponding situation in an Euclidean setting.

math.NA↗
arXiv2015

Numerical approximation of positive power curvature flow via deterministic games

We approximate the level set solution for the motion of an embedded closed curve in the plane with normal speed $\max(0, κ)^{\ga}$ where $κ$ is the curvature of the curve and $\frac{1}{3}<\ga<1$ by the value functions of a family of deterministic two person games. We show convergence of the value functions to the viscosity solution of the level set equation and propose a numerical scheme for the calculation of the value function.

math.NA↗
arXiv2015

Numerical approximation of level set power mean curvature flow

In this paper we investigate the numerical approximation of a variant of the mean curvature flow. We consider the evolution of hypersurfaces with normal speed given by $H^k$, $k \ge 1$, where $H$ denotes the mean curvature. We use a level set formulation of this flow and discretize the regularized level set equation with finite elements. In a previous paper we proved an a priori estimate for the approximation error between the finite element solution and the solution of the original level set equation. We obtained an upper bound for this error which is polynomial in the discretization parameter and the reciprocal regularization parameter. The aim of the present paper is the numerical study of the behavior of the evolution and the numerical verification of certain convergence rates. We restrict the consideration to the case that the level set function depends on two variables, i.e. the moving hypersurfaces are curves. Furthermore, we confirm for specific initial curves and different values of $k$ that the flow improves the isoperimetrical deficit.

math.NA↗
arXiv2014

A note on expansion of convex plane curves via inverse curvature flow

Recently Andrews and Bryan [3] discovered a comparison function which allows them to shorten the classical proof of the well-known fact that the curve shortening flow shrinks embedded closed curves in the plane to a round point. Using this comparison function they estimate the length of any chord from below in terms of the arc length between its endpoints and elapsed time. They apply this estimate to short segments and deduce directly that the maximum curvature decays exponentially to the curvature of a circle with the same length. We consider the expansion of convex curves under inverse (mean) curvature flow and show that the above comparison function also works in this case to obtain a new proof of the fact that the flow exists for all times and becomes round in shape, i.e. converges smoothly to the unit circle after an appropriate rescaling.

math.DG↗
arXiv2013

Finite element approximation of power mean curvature flow

In [21] the evolution of hypersurfaces in $\mathbb{R}^{n+1}$ with normal speed equal to a power $k>1$ of the mean curvature is considered and the levelset solution $u$ of the flow is obtained as the $C^0$-limit of a sequence $u^ε$ of smooth functions solving the regularized levelset equations. We prove a rate for this convergence. Then we triangulate the domain by using a tetraeder mesh and consider continuous finite elements, which are polynomials of degree $\le 2$ on each tetraeder of the triangulation. We show in the case $n=1$ (i.e. the evolving hypersurfaces are curves), that there are solutions $u^ε_h$ of the above regularized equations in the finite element sense, and estimate the approximation error between $u^ε_h$ and $u$. Our method can be extended to the case $n>1$, if one uses higher order finite elements.

math.NA↗
arXiv2012

A note on inverse mean curvatrue flow in cosmological spacetimes

In [8] Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Furthermore, it is shown in [8] that the leaves of the inverse mean curvature flow provide a foliation of the future of the initial hypersurface. We show that this result persists, if we generalize the setting by leaving the mean curvature barrier assumption out. For initial hypersurfaces with sufficiently large mean curvature we can weaken the timelike convergence condition to a physically relevant energy condition.

math.DG↗
arXiv2011

The inverse F-curvature flow in ARW spaces

We consider the so-called inverse $F$-curvature flow (IFCF) $\dot x = -F^{-1}ν$ in ARW spaces, i.e. in Lorentzian manifolds with a special future singularity. Here, $F$ denotes a curvature function of class $(K^*)$, which is homogenous of degree one, e.g. the $n$-th root of the Gaussian curvature, and $ν$ the past directed normal. We prove existence of the IFCF for all times and convergence of the rescaled scalar solution in $C^{\infty}(S_0)$ to a smooth function. Using the rescaled IFCF we maintain a transition from big crunch to big bang into a mirrored spacetime.

math.DG↗