Uniform Lipschitz regularity of flat segregated interfaces in a singularly perturbed problem

For the singularly perturbed system $$\Delta u_{i,\beta}=\beta u_{i,\beta}\sum_{j\neq i}u_{j,\beta}^2, \quad 1\leq i\leq N,$$ we prove that flat interfaces are uniformly Lipschitz. As a byproduct of the proof we also obtain the optimal lower bound near the flat interfaces, $$\sum_iu_{i,\beta}\geq c\beta^{-1/4}.$$Comment: 11 page

Some remarks on the structure of finite Morse index solutions to the Allen-Cahn equation in R2\mathbb{R}^2

For a solution of the Allen-Cahn equation in $\mathbb{R}^2$, under the natural linear growth energy bound, we show that the blowing down limit is unique. Furthermore, if the solution has finite Morse index, the blowing down limit satisfies the multiplicity one property.Comment: 13 page

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 $$\{\begin{aligned} &\Delta u+ f(u)=0,\ \ \ {in}\ \Omega=\{(x^\prime,x_n): x_n>\varphi (x^\prime)\},\\ &u>0,\ \ \ {in}\ \Omega,\\ &u=0,\ \ \ {on}\ \partial\Omega,\\ &|\nabla u|=const. {on} \partial\Omega. \end{aligned}.$$ 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 $\varphi$ is globally Lipschitz, or $n \leq 8$ and $\frac{\partial u}{\partial x_n} >0$ in $\Omega$. 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}.Comment: comments are welcom

On the uniqueness of solutions of an nonlocal elliptic system

We consider the following elliptic system with fractional laplacian -(-\Delta)^su=uv^2,\ \ -(-\Delta)^sv=vu^2,\ \ u,v>0 \ \mbox{on}\ \R^n, where $s\in(0,1)$ and $(-\Delta)^s$ 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.Comment: 35 page

On solutions with polynomial growth to an autonomous nonlinear elliptic problem

We study the following nonlinear elliptic problem [-\Delta u =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 $\alpha \in {\mathbb R}^n$ and $C>0$ such that [(*) | u- \alpha \cdot x | \leq C.] He also showed the existence of solutions with any prescribed $\alpha \in {\mathbb R}^n$. In this note, we first prove that any solution satisfying (*) with nonzero vector $\alpha$ 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 $|x|^d$.Comment: all comments are welcom

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.Comment: 66 page

Global minimizers of the Allen-Cahn equation in dimension n≥8n\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.Comment: 21 page

On one phase free boundary problem in RN\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}.Comment: 42 page

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 $C^{1,\alpha}$ 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.Comment: 63 pages, author's final version. To appear in J. Eur. Math. So

A generalized one phase Stefan problem as a vanishing viscosity limit

We study the vanishing viscosity limit of a nonlinear diffusion equation describing chemical reaction interface or the spatial segregation interface of competing species, where the diffusion rate for the negative part of the solution converges to zero. As in the standard one phase Stefan problem, we prove that the positive part of the solution converges uniformly to the solution of a generalized one phase Stefan problem. This information is then employed to determine the limiting equation for the negative part, which is an ordinary differential equation.Comment: 18 pages, title changed, accepted for publication in Portugaliae Mathematic