Minkowski Inequalities via Nonlinear Potential Theory

We prove convergence of solutions to the parabolic Allen–Cahn equation to Brakke's motion by mean curvature in Riemannian manifolds with Ricci curvature bounded from below. Our results hold for a general class of initial conditions and extend previous results from [T. Ilmanen, Convergence of the Allen–Cahn equation to the Brakke's motion by mean curvature, J. Differential Geom. 31 (1993) 417–461] even in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen–Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by nonpositivity of the limiting energy discrepancy and a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, I, uniform estimates, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.; arXiv:1308.0569], to get various density bounds for the limiting measures.

In this paper we study the asymptotic behavior (∈→0) of the Ginzburg-Landau equation: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaDa% aaleaacaWGSbaabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYt% UvgaiuaacqWF1oG8aaGccqGHsislcqqHuoarcaWG1bWaaWbaaSqabe% aacqWF1oG8aaGccqGHRaWkdaWcaaqaaiaaigdaaeaacqWF1oG8daah% aaWcbeqaaGqaaiaa+jdaaaaaaOGaamOzaiaacIcacaWG1bWaaWbaaS% qabeaacqWF1oG8aaGccaGGPaGaeyypa0JaaGimaiaac6caaaa!565F! $$u_l^\varepsilon - \Delta u^\varepsilon + \frac{1}{{\varepsilon ^2 }}f(u^\varepsilon ) = 0.$$ . where the unknownu ∈ is a real-valued function of [0. ∞)× Rd , and the given nonlinear functionf(u) = 2u(u 2−1) is the derivative of a potential W(u) = (u 2−l)2/2 with two minima of equal depth. We prove that there are a subsequence ∈n and two disjoint, open subsetsP, N of (0, ∞) ×R d satisfying % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaCa% aaleqabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaa% cqWF1oG8daWgaaadbaGaamOBaaqabaaaaOGaeyOKH4kcbeGaa4xmam% aaBaaaleaacqWFpepuaeqaaOGaa4xlaiaa+fdadaWgaaWcbaGae8xd% X7eabeaakiaacYcaruqqYLwySbacgaGaa0hiaiaa9bcacaqFGaGaa0% hiaGqaaiaa8fgacaaFZbGaa0hiaGqaciaa75gacqGHsgIRcqGHEisP% caWEUaGaa0hiaaaa!595E! $$u^{\varepsilon _n } \to 1_\mathcal{P} - 1_\mathcal{N} , as n \to \infty . $$ uniformly inP andN (here 1 A is the indicator of the setA). Furthermore, the Hausdorff dimension of the interface Γ = complement of (P∪N) ⊂ (0, ∞)×R d is equal tod and it is a weak solution of the mean curvature flow as defined in [13,92]. If this weak solution is unique, or equivalently if the level-set solution of the mean curvature flow is “thin,” then the convergence is on the whole sequence. We also show thatu ∈n has an expansion of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaCa% aaleqabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaa% cqWF1oG8daWgaaadbaGaamOBaaqabaaaaOGaaiikaiaadshacaGGSa% GaamiEaiaacMcacqGH9aqpcaWGXbWaaeWaaeaadaWcaaqaaiaadsga% caGGOaGaamiDaiaacYcacaWG4bGaaiykaiabgUcaRiaad+eacaGGOa% Gae8xTdi-aaSbaaSqaaGqaciaa+5gaaeqaaOGaaiykaaqaaiab-v7a% YpaaBaaaleaacaGFUbaabeaaaaaakiaawIcacaGLPaaacaGGUaaaaa!5AE5! $$u^{\varepsilon _n } (t,x) = q\left( {\frac{{d(t,x) + O(\varepsilon _n )}}{{\varepsilon _n }}} \right).$$ whereq(r) = tanh(r) is the traveling wave associated to the cubic nonlinearityf, O(∈) → 0 as ∈ → 0, andd(t, x) is the signed distance ofx to thet-section of Γ. We prove these results under fairly general assumptions on the initial data,u 0. In particular we donot assume thatu ∈(0.x) = q(d(0,x)/∈), nor that we assume that the initial energy, e∈(u ∈(0, .)), is uniformly bounded in ∈. Main tools of our analysis are viscosity solutions of parabolic equations, weak viscosity limit of Barles and Perthame, weak solutions of mean curvature flow and their properties obtained in [13] and Ilmanen’s generalization of Huisken’s monotonicity formula.

For surfaces evolving under the inverse mean curvature flow, Geroch observed that the Hawking mass is a Lyapunov function. For weak solutions of the flow, the corresponding monotonicity formula was proved by Huisken and Ilmanen. An analogous formula exists for approximate equations as well, and it provides uniform control of the solutions in certain Sobolev spaces. This helps to construct weak solutions under very weak assumptions on the initial data.

Moving-centre monotonicity formulae for minimal submanifolds and related equations

In this chapter we prove Huisken’s monotonicity formula [170](1),(2), which describes how the perimeter of a smooth hypersurface flowing by mean curvature changes when weighted with a suitable heat kernel. Here we limit ourselves to describe only one application of this formula; several other applications and a much wider discussion (in particular related to the study of singularities of mean curvature flow) can be found in [170, 171, 126] and [203].KeywordsMinimal SurfaceCurvature FlowSingular PerturbationDifferential InequalitySigned Distance FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

An important result in the analysis of the mean curvature flow is the following monotonicity formula, which was proved by Huisken in [42]. We recall here the proof of this result and discuss its consequences in the analysis of the singularities.KeywordsHeat KernelMonotonicity FormulaConvexity EstimateConvex HypersurfaceCompact SolutionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A monotonicity formula for mean curvature flow with surgery

For mean curvature flows in Euclidean spaces, Brian White proved a regularity theorem which gives C 2; estimates in regions of spacetime where the Gaussian density is close enough to 1. This is proved by applying Huisken’s monotonicity formula. Here we will consider mean curvature flows in semiEuclidean spaces, where each spatial slice is an m-dimensional graph in R mCn n satisfying a gradient bound stronger than the spacelike condition. By defining a suitable quantity to replace the Gaussian density ratio, we prove monotonicity theorems similar to Huisken’s and use them to prove a regularity theorem similar to White’s.

In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of $\mathbb{R}^{n+1}$ ($n\geqslant2$) is mean convex and star-shaped. Several interesting examples and some hyperbolic evolution equations for geometric quantities of the evolving hypersurfaces have been shown. Besides, under different assumptions for the initial velocity, we can get the expansion and the convergence results of a hyperbolic inverse mean curvature flow in the plane $\mathbb{R}^2$, whose evolving curves move normally.

On the existence in the large of solutions to the one-dimensional, isentropic hydrodynamic equations in a bounded domain.- Initial-boundary value and scattering problems in mathematical physics.- On shape optimization of a turbine blade.- Free boundary problems for the Navier-Stokes equations.- A geometric maximum principle, plateau's problem for surfaces of prescribed mean curvature, and the two dimensional analogue of the catenary.- Finite Elements for the Beltrami operator on arbitrary surfaces.- Comparison principles in capillarity.- Remarks on diagonal elliptic systems.- Quasiconvexity, growth conditions and partial regularity.- The monotonicity formula in geometric measure theory, and an application to a partially free boundary problem.- Isoperimetric problems having continua of solutions.- Harmonic maps - Analytic theory and geometric significance.- Asymptotic behavior of solutions of some quasilinear elliptic systems in exterior domains.- Decomposition theorems and their application to non-linear electro- and magneto-static boundary value problems.- Initial boundary value problems in thermoelasticity.- Applications of variational methods to problems in the geometry of surfaces.- Open problems in the degree theory for disc minimal surfaces spanning a curve in ?3.- On a modified version of the free geodetic boundary-value problem.

Using the stress energy tensor, we establish some monotonicity formulae for vector bundle-valued p-forms satisfying the conservation law, provided that the base Riemannian (resp. K\ahler) manifolds poss some real (resp. complex) p-exhaustion functions. Vanishing theorems follow immediately from the monotonicity formulae under suitable growth conditions on the energy of the p-forms. As an application, we establish a monotonicity formula for the Ricci form of a K\ahler manifold of constant scalar curvature and then get a growth condition to derive the Ricci flatness of the K\ahler manifold. In particular, when the curvature does not change sign, the K\ahler manifold is isometrically biholomorphic to C^m. Another application is to deduce the monotonicity formulae for volumes of minimal submanifolds in some outer spaces with suitable exhaustion functions. In this way, we recapture the classical volume monotonicity formulae of minimal submanifolds in Euclidean spaces. We also apply the vanishing theorems to Bernstein type problem of submanifolds in Euclidean spaces with parallel mean curvature. In particular, we may obtain Bernstein type results for minimal submanifolds, especially for minimal real K\ahler submanifolds under weaker conditions.

We define a local (mean curvature flow) entropy for Radon measures in \mathbb{R}^n or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen–Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and parallel Ricci curvature, this is a consequence of a new monotonicity formula for the parabolic Allen–Cahn equation. As an application, we show that when the entropy of the initial data is small enough (less than twice of the energy of the one-dimensional standing wave), the limit measure of the parabolic Allen–Cahn equation has unit density for all future time.

The entropy of a hypersurface is given by the supremum over all F-functionals with varying centers and scales, and is invariant under rigid motions and dilations. As a consequence of Huisken's monotonicity formula, entropy is non-increasing under mean curvature flow. We show here that a compact mean convex hypersurface with some low entropy is diffeomorphic to a round sphere. We will also prove that a smooth self-shrinker with low entropy is exact a hyperplane.

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn equation that holds useful properties such as the monotonicity formula.

Signed distance from a smooth boundary.- Mean curvature vector and second fundamental form.- First variations of volume integrals and of the perimeter.- Smooth mean curvature flows.- Huisken's monotonicity formula.- Inclusion principle. Local well posedness: the approach of Evans-Spruck.- Grayson's example.- De Giorgi's barriers.- Inner and outer regularizations.- An example of fattening.- Ilmanen's interposition lemma.- The avoidance principle.- Comparison between barriers and a generalized evolution.- Barriers and level set evolution.- Parabolic singular perturbations: formal matched asymptotics, convergence and error estimate.