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Kelei Wang

Publications and source records attributed to Kelei Wang.

36 records · Page 2Linked to original sources
arXiv2018

Second order estimates on transition layers

In this paper we establish a uniform $C^{2,θ}$ estimate for level sets of stable solutions to the singularly perturbed Allen-Cahn equation in dimensions $ n\leq 10$ (which is optimal). The proof combines two ingredients: one is the infinite dimensional reduction method which enables us to reduce the $C^{2,θ}$ estimate for these level sets to a corresponding one on solutions of Toda system; the other one uses a small regularity theorem on stable solutions of Toda system to establish various decay estimates on these solutions, which gives a lower bound on distances between different sheets of solutions to Toda system or level sets of solutions to Allen-Cahn equation.

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arXiv2017

The structure of finite Morse index solutions to two free boundary problems in $\mathbb{R}^2$

We give a description of the structure of finite Morse index solutions to two free boundary problems in $\mathbb{R}^2$. These free boundary problems are models of phase transition and they are closely related to minimal hypersurfaces. We show that these finite Morse index solutions have finitely many ends and they converge exponentially to these ends at infinity. As an important tool in the proof, a quadratic decay estimate for the curvature of free boundaries is established.

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arXiv2017

On one phase free boundary problem in $\mathbb{R}^{N}$

We construct a smooth axially symmetric solution to the classical one phase free boundary problem in $\mathbb{R}^{N}$. Its free boundary is of \textquotedblleft catenoid\textquotedblright\ type. This is a higher dimensional analogy of the Hauswirth-Helein-Pacard solution in $\mathbb{R}% ^{2}$ (\cite{Pacard}). The existence of such solution is conjectured in \cite [Remark 2.4]{Pacard}.

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arXiv2017

Finite Morse index implies finite ends

We prove that finite Morse index solutions to the Allen-Cahn equation in $\R^2$ have {\bf finitely many ends} and {\bf linear energy growth}. The main tool is a {\bf curvature decay estimate} on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second order regularity of clustering interfaces for the singularly perturbed Allen-Cahn equation in $\R^n$. Using an indirect blow-up technique, in the spirit of the classical Colding-Minicozzi theory in minimal surfaces, we show that the {\bf obstruction} to the uniform second order regularity of clustering interfaces in $\R^n$ is associated to the existence of nontrivial entire solutions to a (finite or infinite) {\bf Toda system} in $\R^{n-1}$. For finite Morse index solutions in $\R^2$, we show that this obstruction does not exist by using information on stable solutions of the Toda system.

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arXiv2017

On a free boundary problem and minimal surfaces

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension $8$, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that Savin's theorem is optimal.

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arXiv2016

Asymptotics for the fractional Allen-Cahn equation and stationary nonlocal minimal surfaces

This article is mainly devoted to the asymptotic analysis of a fractional version of the (elliptic) Allen-Cahn equation in a bounded domain $Ω\subset\mathbb{R}^n$, with or without a source term in the right hand side of the equation (commonly called chemical potential). Compare to the usual Allen-Cahn equation, the Laplace operator is here replaced by the fractional Laplacian $(-Δ)^s$ with $s\in(0,1/2)$, as defined in Fourier space. In the singular limit $\varepsilon\to 0$, we show that arbitrary solutions with uniformly bounded energy converge both in the energetic and geometric sense to surfaces of prescribed nonlocal mean curvature in $Ω$ whenever the chemical potential remains bounded in suitable Sobolev spaces. With no chemical potential, the notion of surface of prescribed nonlocal mean curvature reduces to the stationary version of the nonlocal minimal surfaces introduced by L.A. Caffarelli, J.M. Roquejoffre, and O. Savin. Under the same Sobolev regularity assumption on the chemical potential, we also prove that surfaces of prescribed nonlocal mean curvature have a Minkowski codimension equal to one, and that the associated sets have a locally finite fractional $2s^\prime$-perimeter in $Ω$ for every $s^\prime\in(0,1/2)$.

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arXiv2016

Global minimizers of the Allen-Cahn equation in dimension $n\geq 8$

We prove the existence of global minimizers of Allen-Cahn equation in dimensions $8$ and above. More precisely, given any strictly area-minimizing Lawson's cones, there are global minimizers whose nodal sets are asymptotic to the cones. As a consequence of Jerison-Monneau's program we establish the existence of many counter-examples to the De Giorgi's conjecture different from the Bombierie-De Giorgi-Giusti minimal graph.

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arXiv2015

A new proof of Savin's theorem on Allen-Cahn equations

In this paper we establish an improvement of tilt-excess decay estimate for the Allen-Cahn equation, and use this to give a new proof of Savin's theorem on the uniform $C^{1,α}$ regularity of flat level sets, which then implies the one dimensional symmetry of minimizers in $\mathbb{R}^n$ for $n\leq 7$. This generalizes Allard's $\varepsilon$-regularity theorem for stationary varifolds to the setting of Allen-Cahn equations.

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arXiv2015

On Serrin's overdetermined problem and a conjecture of Berestycki, Caffarelli and Nirenberg

This paper concerns rigidity results to Serrin's overdetermined problem in an epigraph $$ \{\begin{aligned} &Δu+ f(u)=0,\ \ \ {in}\ Ω=\{(x^\prime,x_n): x_n>φ(x^\prime)\},\\ &u>0,\ \ \ {in}\ Ω,\\ &u=0,\ \ \ {on}\ \partialΩ,\\ &|\nabla u|=const. {on} \partialΩ. \end{aligned}. $$ We prove that up to isometry the epigraph must be an half space and that the solution $u$ must be one-dimensional, provided that one of the following assumptions are satisfied: either $n=2$; or $ φ$ is globally Lipschitz, or $n \leq 8$ and $ \frac{\partial u}{\partial x_n} >0$ in $Ω$. In view of the counterexample constructed in \cite{DPW} in dimensions $n\geq 9$ this result is optimal. This partially answers a conjecture of Berestycki, Caffarelli and Nirenberg \cite{BCN}.

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arXiv2014

On the uniqueness of solutions of an nonlocal elliptic system

We consider the following elliptic system with fractional laplacian $$ -(-Δ)^su=uv^2,\ \ -(-Δ)^sv=vu^2,\ \ u,v>0 \ \mbox{on}\ \R^n,$$ where $s\in(0,1)$ and $(-Δ)^s$ is the $s$-Lapalcian. We first prove that all positive solutions must have polynomial bound. Then we use the Almgren monotonicity formula to perform a blown-down analysis to $s$-harmonic functions. Finally we use the method of moving planes to prove the uniqueness of the one dimensional profile, up to translation and scaling.

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arXiv2014

Harmonic approximation and improvement of flatness in a singular perturbation problem

We study the De Giorgi type conjecture, that is, one dimensional symmetry problem for entire solutions of an two components elliptic system in $\mathbb{R}^n$, for all $n\geq 2$. We prove that, if a solution $(u,v)$ has a linear growth at infinity, then it is one dimensional, that is, depending only on one variable. The main ingredient is an improvement of flatness estimate, which is achieved by the harmonic approximation technique adapted in the singularly perturbed situation.

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arXiv2013

On the De Giorgi type conjecture for an elliptic system modeling phase separation

In this paper we study the one dimensional symmetry problem of entire solutions to the problem \[Δu=uv^2,Δv=vu^2,u,v>0 \text{in} \mathbb{R}^n,\] for all $n\geq 2$. We prove that, if a solution $(u,v)$ is a local minimizer and has linear growth at infinity, then it is one dimensional, i.e. depending only on one variable. In the proof we also obtain the global Lipschitz continuity of solutions only under the linear growth assumption.

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arXiv2013

Qualitative Analysis of Rupture Solutions for an MEMS Problem

We prove a sharp Hölder continuity estimates of rupture sets for sequences of solutions of the following nonlinear problem with negative exponent $$ Δu= \frac{1}{u^p}, \ p>1, \ \mbox{in} \ Ω.$$ As a consequence, we prove the existence of rupture solutions with isolated ruptures in a bounded convex domain in $\R^2$.

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arXiv2013

A Monotonicity Formula and a Liouville-type Theorem for a Fourth Order Supercritical Problem

We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ Δ^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where $ p>1$ and $n\ge1$. We give a complete classification of stable and finite Morse index solutions (whether positive or sign changing), in the full exponent range. We also compute an upper bound of the Hausdorff dimension of the singular set of extremal solutions. Our approach is motivated by Fleming's tangent cone analysis technique for minimal surfaces and Federer's dimension reduction principle in partial regularity theory. A key tool is the monotonicity formula for biharmonic equations.

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arXiv2012

On solutions with polynomial growth to an autonomous nonlinear elliptic problem

We study the following nonlinear elliptic problem [-Δu =F^{'} (u) in {\mathbb R}^n] where $F(u)$ is a periodic function. Moser (1986) showed that for any minimal and nonself-intersecting solution, there exist $ α\in {\mathbb R}^n$ and $ C>0$ such that [(*) | u- α\cdot x | \leq C.] He also showed the existence of solutions with any prescribed $α\in {\mathbb R}^n$. In this note, we first prove that any solution satisfying (*) with nonzero vector $α$ must be one dimensional. Then we show that in ${\mathbb R}^2$, for any positive integer $d\geq 1$ there exists a solution with polynomial growth $|x|^d$.

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arXiv2012

On Entire Solutions of an Elliptic System Modeling Phase Separations

We study the qualitative properties of a limiting elliptic system arising in phase separation for Bose-Einstein condensates with multiple states: Δu=u v^2 in R^n, Δv= v u^2 in R^n, u, v>0\quad in R^n. When n=1, we prove uniqueness of the one-dimensional profile. In dimension 2, we prove that stable solutions with linear growth must be one-dimensional. Then we construct entire solutions in $\R^2$ with polynomial growth $|x|^d$ for any positive integer $d \geq 1$. For $d\geq 2$, these solutions are not one-dimensional. The construction is also extended to multi-component elliptic systems.

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