Related papers: A note on the Morse index of 2k-ended phase transi…

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…

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…

In this paper, we prove even symmetry and monotonicity of certain solutions of Allen-Cahn equation in a half plane. We also show that entire solutions with {\it finite Morse index} and {\it four ends} must be evenly symmetric with respect…

In this paper we study axially symmetric solutions of Allen-Cahn equation with finite Morse index. It is shown that there does not exist such a solution in dimensions between $4$ and $10$. In dimension $3$, we prove that these solutions…

In this work we study existence, asymptotic behaviour and stability properties of $O(m) times O(n)$ invariant solutions of the Allen-Cahn equation $Delta u+u(1-u^2)=0$ in $R^m times R^n$ with $m,n ge 2$, $m+n ge 8$. We exhibit four families…

In this article we study the second variation of the energy functional associated to the Allen-Cahn equation on closed manifolds. Extending well known analogies between the gradient theory of phase transitions and the theory of minimal…

The elliptic sine-Gordon equation in the plane has a family of explicit multiple-end solutions (soliton-like solutions). We show that all the finite Morse index solutions belong to this family. We also prove they are non-degenerate in the…

A well-known conjecture of De Giorgi -- motivated by analogy with the Bernstein problem for minimal surfaces -- asserts the rigidity of monotone solutions to the Allen--Cahn equation in $\mathbb{R}^{d+1}$, with $d\leq 7$. We establish close…

The $2^{nd}$ variation formula of the Seiberg-Witten functional is obtained in order to estimate the Morse index of redutible solutions $(A,0)$. It is shown that their Morse index is given by the dimension of the largest negative eigenspace…

We give a Morse-theoretic characterization of simple closed geodesics on Riemannian $2$-spheres. On any Riemannian $2$-sphere endowed with a generic metric, we show there exists a simple closed geodesic with Morse index $1$, $2$ and $3$. In…

In this paper we consider the Allen-Cahn equation $$ -\Delta u = u-u^3 \ \mbox{in} \ {\mathbb R}^3 $$ We prove that for each $k\in\left( \sqrt{2},+\infty\right),$ there exists a solution to the equation which has growth rate $k$, i.e. $$ \|…

An entire solution of the Allen-Cahn equation $\Delta u=F'(u)$, where $F$ is an even, bistable function, is called a $2k$-end solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$…

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…

In this paper, we consider the sharp interface limit of a matrix-valued Allen-Cahn equation, which takes the form: $$\partial_t A=\Delta A-\varepsilon^{-2}( A A^{\mathrm{T}}A-…

We consider the semilinear Lane-Emden problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }B u=0\qquad\qquad\qquad\mbox{ on }\partial B \end{array}\right.\tag{$\mathcal E_p$}…

We consider the equation \[ -\Delta u = |x|^{\alpha} |u|^{p-1}u, \ \ x \in B, \ \ u=0 \quad \text{on} \ \ \partial B, \] where $B \subset {\mathbb R}^2$ is the unit ball centered at the origin, $\alpha \geq0$, $p>1$, and we prove some…

We show that stable solutions $u:\mathbb{R}^4\to (-1,1)$ to the Allen-Cahn equation with bounded energy density (or equivalently, with cubic energy growth) are one-dimensional. This is known to entail important geometric consequences, such…

We prove convergence of solutions to the parabolic Allen-Cahn equation to Brakke's motion by mean curvature in space forms, generalizing previous results from [15] in Euclidean space. We show that a sequence of measures, associated to…

In this paper, we study the relation between the second inner variations of the Allen-Cahn functional and its Gamma-limit, the area functional. Our result implies that the Allen-Cahn functional only approximates well the area functional up…

We consider non-autonomous semilinear elliptic equations of the type \[ -\Delta u = |x|^{\alpha} f(u), \ \ x \in \Omega, \ \ u=0 \quad \text{on} \ \ \partial \Omega, \] where $\Omega \subset {\mathbb R}^2$ is either a ball or an annulus…