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Frank Pacard

Publications and source records attributed to Frank Pacard.

At least 19 recordsLinked to original sources
arXiv2023

Bouncing Jacobi fields and the Allen-Cahn equation on surfaces

The Allen-Cahn functional is a well studied variational problem which appears in the modeling of phase transition phenomenon. This functional depends on a parameter $\varepsilon >0$ and is intimately related to the area functional as the parameter $\varepsilon$ tends to $0$. In the case where the ambient manifold is a compact surface, we give sufficient assumptions which guarantee the existence of countable families of critical points of the Allen-Cahn functional whose nodal sets converge with multiplicity $2$ to a given embedded geodesic, while their energies and Morse indices stays uniformly bounded, as the parameter $\varepsilon$ tends to $0$. This result is specific to two dimensional surfaces and, for generic metric, it does not occur in higher dimension.

math.AP↗
arXiv2015

Higher codimension isoperimetric problems

We consider a variational problem for submanifolds Q $\subset$ M with nonempty boundary $\partial$Q = K. We propose the definition that the boundary K of any critical point Q have constant mean curvature, which seems to be a new perspective when dim Q \textless{} dim M . We then construct small nearly-spherical solutions of this higher codimension CMC prob-lem; these concentrate near the critical points of a certain curvature function.

math.DG↗
arXiv2015

Free boundary minimal surfaces in the unit 3-ball

In a recent paper A. Fraser and R. Schoen have proved the existence of free boundary minimal surfaces $Σ\_n$ in $B^3$ which have genus $0$ and $n$ boundary components, for all $ n \geq 3$. For large $n$, we give an independent construction of $Σ\_n$ and prove the existence of free boundary minimal surfaces $\tilde Σ\_n$ in $B^3$ which have genus $1$ and $n$ boundary components. As $n$ tends to infinity, the sequence $Σ\_n$ converges to a double copy of the unit horizontal (open) disk, uniformly on compacts of $B^3$ while the sequence $\tilde Σ\_n$ converges to a double copy of the unit horizontal (open) punctured disk, uniformly on compacts of $B^3-\{0\}$.

math.DG↗
arXiv2013

Serrin's Overdetermined Problem and Constant Mean Curvature Surfaces

For all $N \geq 9$, we find smooth entire epigraphs in $\R^N$, namely smooth domains of the form $Ω: = \{x\in \R^N\ / \ x_N > F (x_1,\ldots, x_{N-1})\}$, which are not half-spaces and in which a problem of the form $Δu + f(u) = 0 $ in $Ω$ has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on $\partial Ω$. This answers negatively for large dimensions a question by Berestycki, Caffarelli and Nirenberg \cite{bcn2}. In 1971, Serrin \cite{serrin} proved that a bounded domain where such an overdetermined problem is solvable must be a ball, in analogy to a famous result by Alexandrov that states that an embedded compact surface with constant mean curvature (CMC) in Euclidean space must be a sphere. In lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given CMC surface where Serrin's overdetermined problem is solvable.

math.AP↗
arXiv2012

Solutions of semilinear elliptic equations in tubes

Given a smooth compact k-dimensional manifold Λembedded in $\mathbb {R}^m$, with m\geq 2 and 1\leq k\leq m-1, and given ε>0, we define B_ε(Λ) to be the geodesic tubular neighborhood of radius εabout Λ. In this paper, we construct positive solutions of the semilinear elliptic equation Δu + u^p = 0 in B_ε(Λ) with u = 0 on \partial B_ε(Λ), when the parameter εis chosen small enough. In this equation, the exponent p satisfies either p > 1 when n:=m-k \leq 2 or p\in (1, \frac{n+2}{n-2}) when n>2. In particular p can be critical or supercritical in dimension m\geq 3. As εtends to zero, the solutions we construct have Morse index tending to infinity. Moreover, using a Pohozaev type argument, we prove that our result is sharp in the sense that there are no positive solutions for p>\frac{n+2}{n-2}, n\geq 3, if εis sufficiently small.

math.FA↗
arXiv2011

The space of 4-ended solutions to the Allen-Cahn equation on the plane

An entire solution of the Allen-Cahn equation $Δu=F'(u)$, 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 $u$ looks like the one dimensional, heteroclinic solution. In this paper we initiate a program to classify the four-end solutions of the Allen-Cahn equation in $\R^2$. We show that there exists a one parameter family of solutions containing the saddle solution, for which the angle between the nodal lines is $\fracπ{2}$, as well as solutions for which the angle between the asymptotic half lines is any $θ\in (0, \fracπ{2})$. This justifies the definition of the angle map for a four-end solution $u$, which is the angle $θ=θ(u)\in (0, \fracπ{2})$ between the asymptote to the nodal line in the first quadrant and the x axis. Then we show that on any connected component in the moduli space of four-end solutions the angle map is surjective onto $(0,\fracπ{2})$.

math.AP↗
arXiv2011

The classification of four end solutions of the Allen-Cahn equation on the plane

An entire solution of the Allen-Cahn equation $Δu=f(u)$, where $f$ has exactly three zeros at $\pm 1$ and 0, is balanced and odd, e.g. $f(u)=u(u^2-1)$, is called a $2k$-ended solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$ looks like the one dimensional, heteroclinic solution. In this paper we consider the family of four ended solutions whose ends are almost parallel at $\infty$. We show that this family can be parametrized by the family of solutions of the two component Toda system. As a result we obtain the uniqueness of four ended solutions with almost parallel ends. Combining this result with the classification of connected components in the moduli space of the four ended solutions we can classify all such solutions. Thus we show that four end solutions form, up to rigid motions, a one parameter family. This family contains the saddle solution, for which the angle between the nodal lines is $\fracπ{2}$ as well as solutions for which the angle between the asymptotic half lines of the nodal set is arbitrary small (almost parallel nodal sets).

math.AP↗
arXiv2011

Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones

In this paper, we are interested in bounded, entire, solutions of the Allen-Cahn equation which are defined in Euclidean space and whose zero set is asymptotic to a given minimal cone. In particular, in dimension larger than or equal to 8, we prove the existence of stable solutions of the Allen-Cahn equation whose zero sets are not hyperplanes.

math.AP↗
arXiv2010

Attaching handles to Delaunay nodo\"ıds

For all $m \in \mathbb N - \{0\}$, we prove the existence of a one dimensional family of genus $m$, constant mean curvature (equal to 1) surfaces which are complete, immersed in $\mathbb R^3$ and have two Delaunay ends asymptotic to nodo\"ıdal ends. Moreover, these surfaces are invariant under the group of isometries of $\mathbb R^3$ leaving a horizontal regular polygon with $m+1$ sides fixed.

math.DG↗
arXiv2010

A note on some overdetermined elliptic problem

We define the notion of an exceptional manifold to be a flat Riemannian manifold with boundary which supports a positive harmonic function satisfying simultaneously a zero Dirichlet condition and a constant (nonzero) Neumann condtion at the boundary. We study the two-dimensional case: we present various examples and give a general construction algorithm of such surface by using complex analysis. We deduce a classification of all such surfaces assuming some further natural hypotheses and prove a Bernstein type theorem.

math-ph↗
arXiv2007

Constant curvature foliations on asymptotically hyperbolic spaces

Let $(M,g)$ be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on $\del M$ and Weingarten foliations in some neighbourhood of infinity in $M$. We focus mostly on foliations where each leaf has constant mean curvature, though our results apply equally well to foliations where the leaves have constant $σ_k$-curvature. In particular, we prove the existence of a unique foliation near infinity in any quasi-Fuchsian 3-manifold by surfaces with constant Gauss curvature. There is a subtle interplay between the precise terms in the expansion for $g$ and various properties of the foliation. Unlike other recent works in this area, by Rigger \cite{Ri} and Neves-Tian \cite{NT1}, \cite{NT2}, we work in the context of conformally compact spaces, which are more general than perturbations of the AdS-Schwarzschild space, but we do assume a nondegeneracy condition.

math.DG↗
arXiv2007

The Toda system and multiple-end solutions of autonomous planar elliptic problems

We construct a new class of positive solutions for a classical semilinear elliptic problem in the plane which arise for instance as the standing-wave problem for the standard nonlinear Schrödinger equation or in nonlinear models in Turing's theory biological theory of pattern formation such as the Gray-Scott or Gierer-Meinhardt systems. The solutions we construct have the property that their energy over a ball of radius R grows linearly with R as R tends to infinity. These solutions are strongly related to the solutions of a Toda system.

math.AP↗
arXiv2007

On the Kähler classes of constant scalar curvature metrics on blow ups

In this note we clarify the structure of the moduli space of constant scalar curvature Kaehler metrics as one approaches the boundary of the Kaehler cone on cscK manifolds blown up at finite set of points, in the spirit of the previous work arXiv:math/0504115. Results about which Kaehler classes can be reached and about the position of the blown up points are given.

math.DG↗
arXiv2007

Boundary singularities for weak solutions of semilinear elliptic problems

We construct positive weak solutions of a class of semilinear elliptic equation which vanish in suitable trace sense on the boundary of a given smooth bounded N-dimensional domain, but which are singular at prescribed isolated points of the boundary. Similar constructions are carried out for solutions which are singular on any given embedded submanifold of the boundary.

math.AP↗
arXiv2007

A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds

We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques, in the form of the mountain pass lemma, to the analysis of the Hamiltonian constraint equation, which has been previously studied by other methods.

gr-qc↗
arXiv2006

Generalized Doubling Constructions for Constant Mean Curvature Hypersurfaces in the (n+1)-Sphere

The (n+1)-sphere contains a simple family of constant mean curvature (CMC) hypersurfaces which are products of lower-dimensional spheres called the generalized Clifford hypersurfaces. This paper demonstrates that new, topologically non-trivial CMC hypersurfaces resembling a pair of neighbouring generalized Clifford tori connected to each other by small catenoidal bridges at a sufficiently symmetric configuration of points can be constructed by perturbative PDE methods. That is, one can create an approximate solution by gluing a rescaled catenoid into the neighbourhood of each point; and then one can show that a perturbation of this approximate hypersurface exists which satisfies the CMC condition. The results of this paper generalize those of the authors in math.DG/0511742.

math.DG↗