Monotonicity Formula Research Articles

Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius

Consider the class of optimal partition problems with long range interactions $$\begin{aligned} \inf \left\{ \sum _{i=1}^k \lambda _1(\omega _i):\ (\omega _1,\ldots , \omega _k) \in \mathcal {P}_r(\Omega ) \right\} , \end{aligned}$$ where $$\lambda _1(\cdot )$$ denotes the first Dirichlet eigenvalue, and $$\mathcal {P}_r(\Omega )$$ is the set of open k-partitions of $$\Omega $$ whose elements are at distance at least r: $${{\,\mathrm{dist}\,}}(\omega _i,\omega _j)\ge r$$ for every $$i\ne j$$ . In this paper we prove optimal uniform bounds (as $$r\rightarrow 0^+$$ ) in $$\mathrm {Lip}$$ –norm for the associated $$L^2$$ -normalized eigenfunctions, connecting in particular the nonlocal case $$r>0$$ with the local one $$r \rightarrow 0^+$$ . The proof uses new pointwise estimates for eigenfunctions, a one-phase Alt–Caffarelli–Friedman and the Caffarelli-Jerison-Kenig monotonicity formulas, combined with elliptic and energy estimates. Our result extends to other contexts, such as singularly perturbed harmonic maps with distance constraints.

The nodal set of solutions to some nonlocal sublinear problems

We study the nodal set of stationary solutions to equations of the form (-Delta )^s u = lambda _+ (u_+)^{q-1} - lambda _- (u_-)^{q-1}quad text {in }B_1, where lambda _+,lambda _->0, q in [1,2), and u_+ and u_- are respectively the positive and negative part of u. This collection of nonlinearities includes the unstable two-phase membrane problem q=1 as well as sublinear equations for 1<q<2. We initially prove the validity of the strong unique continuation property and the finiteness of the vanishing order, in order to implement a blow-up analysis of the nodal set. As in the local case s=1, we prove that the admissible vanishing orders can not exceed the critical value k_q= 2s/(2- q). Moreover, we study the regularity of the nodal set and we prove a stratification result. Ultimately, for those parameters such that k_q< 1, we prove a remarkable difference with the local case: solutions can only vanish with order k_q and the problem admits one dimensional solutions. Our approach is based on the validity of either a family of Almgren-type or a 2-parameter family of Weiss-type monotonicity formulas, according to the vanishing order of the solution.

A local description of 2-dimensional almost minimal sets bounded by a curve

We study the local regularity of sliding almost minimal sets of dimension 2 in $R^n$ , bounded by a smooth curve $L$. These are a good way to model soap films bounded by a curve, and their definition is similar to Almgren's. We aim for a local description, in particular near L and modulo $C^{1+$\epsilon$}$ diffeomorphisms, of such sets $E$, but in the present paper we only obtain a full description when $E$ is close enough to a half plane, a plane or a union of two half planes bounded by the same line, or a transverse minimal cone of type $Y$ or $T$. The main tools are adapted near monotonicity formulae for the density, including for balls that are not centered on L, and the same sort of construction of competitors as for the generalization of J. Taylor's regularity result far from the boundary.

Relative expander entropy in the presence of a two-sided obstacle and applications

We study a notion of relative entropy motivated by self-expanders of mean curvature flow. In particular, we obtain the existence of this quantity for arbitrary hypersurfaces trapped between two self-expanders that are asymptotic to the same cone and bound a domain. This allows us to begin to develop the variational theory for the relative entropy functional for the associated obstacle problem. We also obtain a version of the forward monotonicity formula for mean curvature flow proposed by Ilmanen.

Measure upper bounds for nodal sets of eigenfunctions of the bi‐harmonic operator

In this paper, we investigate the measure estimate of nodal set of eigenfunction u $u$ of the bi-harmonic operator, that is, ▵ 2 u = λ 2 u $\hbox{{$\triangle$}} ^2u=\lambda ^2u$ in Ω $\Omega$ with some homogeneous linear boundary conditions. We assume that Ω ⊆ R n $\Omega \subseteq \mathbb {R}^n$ ( n ⩾ 2 $n\geqslant 2$ ) is a C ( n + 1 ) / 2 $C^{(n+1)/2}$ -bounded domain, ∂ Ω $\partial \Omega$ is piecewise analytic, and analytic except a set Γ ⊆ ∂ Ω $\Gamma \subseteq \partial \Omega$ which is a finite union of some compact ( n − 2 ) $(n-2)$ -dimensional submanifolds of ∂ Ω $\partial \Omega$ . The main result of this paper is that the measure upper bound of nodal set of the eigenfunction u $u$ is controlled by λ $\sqrt {\lambda }$ . We first define a frequency function and a doubling index related to the eigenfunction. With the help of establishing the monotonicity formula, doubling conditions and various a priori estimates, we obtain that the ( n − 1 ) $(n-1)$ -dimensional Hausdorff measure of nodal set of the eigenfunction in a ball is controlled by the frequency function and λ $\sqrt {\lambda }$ . In order to further control the frequency function by λ $\sqrt {\lambda }$ , we establish the relationship between the frequency function and the doubling index, and then separate the domain Ω $\Omega$ into two parts: the domain away from Γ $\Gamma$ and the domain near Γ $\Gamma$ , and develop a new iteration argument to deal with the two cases, respectively.

Minkowski Inequalities via Nonlinear Potential Theory

In this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set Omega subset mathbb {R}^n, nge 3. Our proof relies on the discovery of effective monotonicity formulas holding along the level set flow of the p-capacitary potentials associated with Omega , for every p sufficiently close to 1. These formulas also testify the existence of a link between the monotonicity formulas derived by Colding and Minicozzi for the level set flow of Green’s functions and the monotonicity formulas employed by Huisken, Ilmanen and several other authors in studying the geometric implications of the Inverse Mean Curvature Flow. In dimension nge 8, our conclusions are stronger than the ones obtained so far through the latter mentioned technique.

Isoperimetric type inequalities on submanifolds in rotation-symmetric spaces

We study the monotonicity formula involving mean curvature for submanifolds in a rotation-symmetric space $N$. Then we get an isoperimetric type inequality on extrinsic balls of submanifolds in $N$ with a sharp constant.

Doubling inequalities and nodal sets in periodic elliptic homogenization

We prove explicit doubling inequalities and obtain uniform upper bounds (under -dimensional Hausdorff measure) of nodal sets of weak solutions for a family of linear elliptic equations with rapidly oscillating periodic coefficients. The doubling inequalities, explicitly depending on the doubling index, are proved at different scales by a combination of convergence rates, a three-ball inequality from certain “analyticity,” and a monotonicity formula of a frequency function. The upper bounds of nodal sets are shown by using the doubling inequalities, approximations by harmonic functions and an iteration argument.

Monotonicity formulae and F-stress energy

The goal of this work is the application of the f-stress energy of differ-ential forms to study the generalized monotonicity formulae and generalizedvanishing theorems. We obtain some generalized monotonicity formulas forp-formsω∈Ap(ξ), which satisfy the generalizedf-conservation laws, withf∈C∞(M×R) satisfying some conditions

Regularity of shape optimizers for some spectral fractional problems

This paper is dedicated to the spectral optimization problemmin{λ1s(Ω)+⋯+λms(Ω)+ΛLn(Ω):Ω⊂D s-quasi-open} where Λ>0,D⊂Rn is a bounded open set and λis(Ω) is the i-th eigenvalue of the fractional Laplacian on Ω with Dirichlet boundary condition on Rn∖Ω.We first prove that the first m eigenfunctions on an optimal set are locally Hölder continuous in the class C0,s and, as a consequence, that the optimal sets are open sets. Then, via a blow-up analysis based on a Weiss type monotonicity formula, we prove that the topological boundary in D of a minimizer Ω is composed of a relatively open regular part and a closed singular part of Hausdorff dimension at most n−n⁎, for some n⁎≥3. Finally we use a viscosity approach to prove C1,α-regularity of the regular part of the boundary.

On a two-phase free boundary problem ruled by the infinity Laplacian

In this paper we consider a two-phase free boundary problem ruled by the infinity Laplacian. Our main result states that bounded viscosity solutions in B1 are universally Lipschitz continuous in B1/2, which is the optimal regularity for the problem. We make a new use of the Ishii–Lions’ method, which works as a surrogate for the lack of a monotonicity formula and is bound to be applicable in related problems.

A two phase boundary obstacle-type problem for the bi-Laplacian

In this paper we are concerned with a two phase boundary obstacle-type problem for the bi-Laplace operator in the upper unit ball. The problem arises in connection with unilateral phenomena for flat elastic plates. It can also be seen as an extension problem to an obstacle problem for the fractional Laplacian (−Δ)3/2, as first observed in Yang (2013). We establish the well-posedness and the optimal regularity of the solution, and we study the structure of the free boundary. Our proofs are based on monotonicity formulas of Almgren- and Monneau-type.

Monotonicity Formula On Cigar Soliton

In this paper, we prove a monotonicity formula on cigar soliton. Using the monotonicity, we obtain a three-ball type theorem. There might be some people who have proved the three-ball theorem on Cigar Soliton before, but we use a new method. We add a weight function in the norm of u. By doing so, we make the calculation process much easier because when using integral by part, the term that integrates at the surface of ball would disappear as the weight function we add would be zero.

On the entropy of parabolic Allen–Cahn equation

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.

Monotonicity Formula and Classification of Stable Solutions to Polyharmonic Lane–Emden Equations

Abstract In this paper, we consider polyharmonic Lane–Emden equations $$ \begin{equation*} (-\Delta )^m u=|u|^{p-1}u, \ \ \ \mbox{in} \ \ \ {\mathbb R}^n, \end{equation*}$$where $m\geq 3$. We classify the stable or stable outside a compact set solutions when $m=3$ or $4$ for any dimensions and when $m\geq 5$ for large dimensions. In the process, we exhibit the general Joseph–Lundgren exponent (including both local and nonlocal cases) in a concise form and prove related properties. The key ingredient of the proof of the classification is a monotonicity formula for general polyharmonic equations, which may have application in regularity theory for higher-order elliptic equations.

A curve shortening equation with time-dependent mobility related to grain boundary motions

A curve shortening equation related to the evolution of grain boundaries is presented. This equation is derived from the grain boundary energy by applying the maximum dissipation principle. Gradient estimates and large time asymptotic behavior of solutions are considered. In the proof of these results, one key ingredient is a new weighted monotonicity formula that incorporates a time-dependent mobility.

Uniform Rectifiability, Elliptic Measure, Square Functions, and ε-Approximability Via an ACF Monotonicity Formula

Abstract Let $\Omega \subset{{\mathbb{R}}}^{n+1}$, $n\geq 2$, be an open set with Ahlfors regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with real, merely bounded and possibly nonsymmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if $L^*$ is the operator in divergence form associated with the transpose matrix of $A$, then $\partial \Omega $ is uniformly $n$-rectifiable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^*v=0$ in $\Omega $ is $\varepsilon $-approximable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^*v=0$ in $\Omega $ satisfies a suitable square-function Carleson measure estimate. Moreover, we obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called “$S&amp;lt;N$” estimates, and another in terms of a suitable corona decomposition involving $L$-harmonic and $L^*$-harmonic measures. We also prove that if $L$-harmonic measure and $L^*$-harmonic measure satisfy a weak $A_\infty $-type condition, then $\partial \Omega $ is $n$-uniformly rectifiable. In the process we obtain a version of the Alt-Caffarelli-Friedman monotonicity formula for a fairly wide class of elliptic operators which is of independent interest and plays a fundamental role in our arguments.

Minimizing cones for fractional capillarity problems

We consider a fractional version of Gauß capillarity energy. A suitable extension problem is introduced to derive a boundary monotonicity formula for local minimizers of this fractional capillarity energy. As a consequence, blow-up limits of local minimizers are shown to subsequentially converge to minimizing cones. Finally, we show that in the planar case there is only one possible fractional minimizing cone, the one determined by the fractional version of Young’s law.

Almost minimizers for the thin obstacle problem

We consider Anzellotti-type almost minimizers for the thin obstacle (or Signorini) problem with zero thin obstacle and establish their \(C^{1,\beta }\) regularity on the either side of the thin manifold, the optimal growth away from the free boundary, the \(C^{1,\gamma }\) regularity of the regular part of the free boundary, as well as a structural theorem for the singular set. The analysis of the free boundary is based on a successful adaptation of energy methods such as a one-parameter family of Weiss-type monotonicity formulas, Almgren-type frequency formula, and the epiperimetric and logarithmic epiperimetric inequalities for the solutions of the thin obstacle problem.

The harmonic heat flow of almost complex structures

We define and study the harmonic heat flow for almost complex structures which are compatible with a Riemannian structure ( M , g ) (M, g) . This is a tensor-valued version of harmonic map heat flow. We prove that if the initial almost complex structure J J has small energy (depending on the norm | ∇ J | |\nabla J| ), then the flow exists for all time and converges to a Kähler structure. We also prove that there is a finite time singularity if the initial energy is sufficiently small but there is no Kähler structure in the homotopy class. A main technical tool is a version of monotonicity formula, similar as in the theory of the harmonic map heat flow. We also construct an almost complex structure on a flat four tori with small energy such that the harmonic heat flow blows up at finite time with such an initial data.