Book

F. Pacard and T.
Rivière.

**Linear and nonlinear aspects of vortices : the Ginzburg Landau model.**Progress in Nonlinear Differential Equations, 39, Birkäuser. 342 pp. (2000). pdfPapers

[76] M. Kowalczyk, Y. Liu, F.
Pacard and J. Wei. End-to-end construction for the Allen-Cahn equation in the plane. Calc. Variat. and P.D.E. pdf

Abstract
:
In this paper, we construct a wealth of bounded, entire solutions of
the Allen-Cahn equation in the plane. The asymptotic behavior at
infinity of these solutions is determined by 2L half affine lines, in
the sense that, along each of these half affine line, the solution is
close to a suitable translated and rotated copy of a one dimensional
heteroclinic solution. The solutions we construct belong to a smooth
2L-dimensional family of bounded, entire solutions of the Allen-Cahn
equation and, in some sense, they provide a description of a collar
neighborhood of part of the compactification of the moduli space of
2L-ended solutions for the Allen-Cahn equation. Our construction is
very much inspired from a former construction of minimal surfaces by M.
Traizet.

[75] M. Kowalczyk, Y. Liu and F. Pacard. The classification of four ended solutions to the Allen-Cahn equation in the plane. Analysis and PDE (2013). pdf

Abstract
:
In this paper we prove the uniqueness of solutions of the Allen-Cahn
equation which are defined in the plane and have 4 almost parallel
ends. We also prove prove that four ended solutions of the Allen-Cahn
equation in the plane form, up to rigid motions, a one parameter family
parameterized by the angle between the ends.

[74] F. Pacard and H. Rosenberg. Attaching handles to Delaunay nodoïds. Pacific J. Math. 266 (2013), no. 1, 129-183. pdf

Abstract
:
For all m ∈ N − {0}, we prove the existence of a family of genus m,
constant mean curvature surfaces which are complete, immersed in R^3
and have two Delaunay ends asymptotic to nodoïdal ends. Moreover, these
surfaces are invariant under the full dihedral group of isometries
which leave a horizontal regular polygon with m + 1 sides.

[73] F. Pacard, F. Pacella and B. Sciunzi. Solutions of semilinear elliptic equations in tubes. J. Geom. Anal. 24 (2014), no. 1, 445-471. pdf

[73] F. Pacard, F. Pacella and B. Sciunzi. Solutions of semilinear elliptic equations in tubes. J. Geom. Anal. 24 (2014), no. 1, 445-471. pdf

Abstract : Given a smooth compact k-dimensional manifold Λ embedded in R^m, with m ≥ 2 and 1 ≤ k ≤ m − 1, and given e > 0, we deﬁne B_e(Λ) to be the geodesic tubular neighborhood of radius e about Λ. In this paper, we construct positive solutions of the semilinear elliptic equation ∆u + u^p = 0 in B_e(Λ) and u = 0 on ∂B_e(Λ), when the parameter ϵ is chosen small enough. In this equation, the exponent p satisﬁes either p > 1 when n := m − k ≤ 2 or p ∈ (1, (n+2)/(n−2)) when n > 2.

[72] F. Pacard and J. Wei. Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones. Journal of Functional Analysis 264 (2013), pp. 1131-1167. pdf

Abstract : We are interested in entire solutions of the Allen-Cahn equation in Euclidean space, whose level sets are asymptotic to minimal cones. In particular, in dimension 8, we prove the existence of stable solutions of the Allen-Cahn equation whose zero sets are not hyperplanes.

[71] M. Kowalczyk, Y. Liu and F. Pacard. The space of 4-ended solutions to the Allen-Cahn equation in the plane. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 5, 761-781. pdf

Abstract : We are interested in entire solutions of the Allen-Cahn equation ∆u − F'(u) = 0 which have some special structure at inﬁnity. In this equation, the function F is an even, bistable function. The solutions we are interested in have their zero set asymptotic to 4 half oriented aﬃne lines at inﬁnity and, along each of these half aﬃne lines, the solutions are asymptotic to the one dimensional heteroclinic solution : such solutions are called 4-ended solutions. The main result of our paper states that, for any θ ∈ (0, π/2), there exists a 4-ended solution of the Allen-Cahn equation whose zero set is at inﬁnity asymptotic to the half oriented aﬃne lines making the angles θ, π − θ, π + θ and 2π − θ with the x-axis. This paper is part of a program whose aim is to classify all 2k-ended solutions of the Allen-Cahn equation in dimension 2, for k ≥ 2.

**[70] F. Pacard. The role of minimal surfaces in the study of the Allen-Cahn equation. Contemp. Math., 570, Amer. Math. Soc., Providence, RI, (2012). pdf**

Abstract
:
In these lectures, we review some recent results on the existence of
solutions of the Allen-Cahn equation in a Riemannian manifold (M; g).
In the case where the ambient manifold M is compact, we provide a
complete proof of the existence of such solutions whose zero set is close to a
given minimal hypersurface. These lectures were given at the Summer
School Santalo 2010.

**[69] M. del Pino, M. Musso and F. Pacard. Solutions of the Allen-Cahn equation which are invariant under screw motion. Manuscripta Mathematica, 138, (2012), 273-286. pdf**

Abstract
: We
prove the existence of entire solutions of the Allen-Cahn equation
whose level sets is a given helicoïd in Euclidean 3 space..

[68] M. del Pino, M. Musso, F. Pacard and A. Pistoia. Torus action on S^n and sign changing solutions for conformally invariant equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 12, No. 1, 209-237 (2013). pdf

[68] M. del Pino, M. Musso, F. Pacard and A. Pistoia. Torus action on S^n and sign changing solutions for conformally invariant equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 12, No. 1, 209-237 (2013). pdf

Abstract
:
We construct sequences of sign changing solutions for some conformally
invariant semilinear elliptic equation which is defined in S^n, with n
≥
4. The solutions we obtain have large energy and concentrate along some
special submanifolds of S^n. For example, in dimension n ≥
4 we obtain sequences of solutions whose energy concentrates along one
great circle or finitely many great circles which are linked (a Hopf
link). In dimension n ≥ 5, we obtain sequences of solutions whose
energy concentrates along a two dimensional torus (a Clifford torus).

[67] M. del Pino, M. Kowalczyk and F.
Pacard. Moduli space theory for the Allen-Cahn
equation in the plane. Trans. Amer. Math. Soc. 365 (2013), no. 2, 721-766. pdf

Abstract
: In
this paper, we study the moduli space of entire solutions to the
Allen-Cahn equation which are defined in the plane and have 2k ends
(this means that the nodal set of the solution is, away from a compact,
asymptotic to 2k half lines).
We prove that, close to any nondegenerate element, this moduli space
has
dimension 2k. We also provide a "refined asymptotic" result for such
solutions.

[66] M. Musso, F. Pacard and J. Wei. Finite energy, sign changing solutions with dihedral symmetry for the stationary non linear Schrödinger equation. J. Eur. Math. Soc. (JEMS) 14 (2012), no. 6, 1923-1953. pdf

Abstract
:
We address the problem of the existence of ﬁnite energy solitary waves
for nonlinear Klein-Gordon or Schrodinger equations. Under
natural conditions on the nonlinearity, we prove the existence of
inﬁnitely many nonradial solutions in any dimension N ≥ 2. Our
result complements earlier works of Bartsch and Willem and Lorca-Ubilla
where solutions invariant under the action of O(2) × O(N −
2) are constructed. In contrast with these later works, the solutions
we construct are
invariant under the action of D_k × O(N − 2) where D_k ⊂
O(2) denotes the subgroup generated by the rotation of angle 2π/k,
for some integer k ≥ 7, but they are not invariant under the
action of O(2) × O(N − 2).

[65]
R. Mazzeo and F. Pacard. Constant
curvature foliations in asymptotically hyperbolic spaces. Rev.
Mat. Iberoamericana Volume 27, Number 1 (2011), 303-333. pdf

Abstract : 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 the
boundary of M and Weingarten foliations in some neighbourhood of
infinity in M. We focus mostly on foliations where each leaf has
constant mean curvature. There is a subtle interplay between the
precise terms in the expansion for the metric g and various properties
of the foliation.

[64] M. del Pino, M. Musso, F. Pacard and A. Pistoia. Large energy entire solutions for the Yamabe equation. Journal of Differential Equations. 251 (2011), 2568-2597. pdf

Abstract : We construct infinitely many sign-changing solutions for the Yamabe equation in S^n.

[63] C. Arezzo, F. Pacard and M. Singer. Extremal Kähler metrics on blow-ups. Duke Math. Journal. Vol. 157 no. 1. (2011), 1-51. pdf

Abstract
: We present some new results about the existence of extremal
Kähler metrics on blow-ups of Kähler manifolds which already carry an
extremal Kähler metric. As a
consequence, we obtain extremal metrics on the blow up
of the m-dimensional complex projective space CP^m at n=1, ... , m+1
linearly independent points.

[62] L.
Hauswirth, F. Helein and F. Pacard. A note on some overdetermined problem. Pacific
J. Maths. Vol. 250, No. 2, (2011), 319-334. pdf

Abstract
:
In this short note, we address the classiﬁcation of all flat surfaces M
with smooth boundary on which there exist positive harmonic
functions having 0 Dirichlet data and constant (nonzero) Neumann data.
In particular, we show that this problem bear strong similarities with
the study os minimal surfaces in Euclidean 3-space. We also provide a
Weierstrass type representation formula for these surfaces.

[61] F.
Pacard. Constant scalar curvature and
extremal metrics on blow ups. Proceedings of the International
Congress of Mathematicians. Hyderabad, India, (2010). pdf

Abstract
:
In this paper, we report some joint works with C. Arezzo and M. Singer
concerning the construction of extremal Kähler metrics on blow ups at
finitely many points of Kähler manifolds which already carry an
extremal metric. In particular, we give sufficient conditions on the
number and locations of the blow up points for the blow up to carry an
extremal Kähler metric

[60] M. del Pino, M. Kowalczyk, F.
Pacard and J. Wei. The Toda system
and multiple-end solutions of autonomous planar elliptic
problems. Adv. Math. 224, No. 4, (2010), 1462-1516. pdf

Abstract
: 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. This result can be understood as the counterpart, in
this setting, of
various connected sum results in which have been obtained for some
geometric problems (constant scalar curvature problem or constant mean
curvuture surfaces).

[59] M.
del Pino, M. Kowalczyk, F. Pacard and J. Wei. Multiple-end solutions to the Allen-Cahn
equation in R^2. J. Functional Analysis 258 (2010), 458-503. pdf

Abstract :
We construct new solutions of the Allen-Cahn
equation in R∈2. Given k ≥ 1 we ﬁnd a family of solutions whose
zero level sets are asymptotic to 2k straight half lines.

[58] M.
del Pino, M. Musso and F. Pacard. Bubbling
along boundary geodesics near the second critical exponent. J.
Eur. Math. Soc. 12, (2010), 1553-1605. pdf

Abstract
: The role of the second
critical exponent p=\frac{n+1}{n-3},
the Sobolev critical exponent in one dimension less, is investigated
for the classical Lane-Emden-Fowler problem ∆u + u^p
=0, u > 0 under zero Dirichlet boundary conditions, in a
domain in R^n with bounded, smooth boundary. Given \Gamma, a
geodesic of the boundary with negative inner normal curvature we find
that for p=(n+1)/(n-3)-\epsilon, a solution u_\epsilon
such that |\nabla u_\epsilon|^2 converges weakly to a Dirac
measure on \Gamma as \epsilon tends to 0^+ exists, provided
that \Gamma is non-degenerate in the sense of second variations of
length and \epsilon remains away from certain explicit
discrete set of values for which a resonance phenomenon takes
place.

[57] C.
Arezzo and
F. Pacard,

**Blowing up Kähler manifolds with constant scalar curvature II.**Annals of Math. (2), 170, n° 2, (2009), 685-738. pdfAbstract
:
This paper is a continuation of a previous paper on the same subject.
Given a complex
manifold endowed with a Kähler metric with constant scalar
curvature, we prove the existence of Kähler metrics with constant
scalar curvature on the blow up at finitely many points of this
manifold. This paper covers cases that were not covered by the results
of the previous paper. The result now applies to manifolds that
carry nontrivial holomorphic vector fields with zeros, in
which case a sufficient condition on the blow up points is given to
ensure the existence of a Kähler metric on the blow up. Some
applications of our result to the blow up of CP^n at
finitely many points are given.

[56] F. Pacard. Geometric aspects of the Allen-Cahn equation. Matematica Contemporânea, Vol 37, (2009), 91-122. pdf

Abstract : These are lectures I gave during
the Winter School on Nonlinear Analysis (UFRJ-August 2009). In these
notes I describe recent advances on the existence of entire solutions
of some semilinear elliptic equations.

[55] F. Pacard and P. Sicbaldi. Extremal domains for the first eigenvalue of the Laplace-Beltrami operator. Annales de l'Institut Fourier. 59, n° 2, (2009), 515-542. pdf

Abstract
: We prove the existence of extremal domains with small
prescribed
volume for the ﬁrst eigenvalue of Laplace-Beltrami operator in
some Riemannian manifold. These domains are close to geodesic spheres
of small radius centered at a nondegenerate critical point of the
scalar curvature.

[54] F.
Pacard and X. Xu. Constant mean
curvature spheres in Riemannian manifolds. Manuscripta Mathematica. 128, 3
(2009), 275-295. pdf

Abstract
: We prove a multiplicity result for constant mean curvature
embedded
spheres in any Riemannian manifold, provided the mean curvature is
large enough.This result extends a former result by R. Ye when the
scalar curvature function of the manifold has non degenerate critical
points.

[53] P.
Chruschiel, F. Pacard and D. Pollack. Singular
Yamabe metrics and initial data with exactly
Kottler-Schwarzschild-de Sitter ends II. Generic metrics.
Math. Res. Letters. 16, no. 1 (2009) 157-164. pdf

Abstract
: We present a gluing construction which adds, via a localized
deformation, exact "Delaunay" ends to generic metrics with
constant positive scalar curvature. This provides time-symmetric
initial data sets for the vacuum Einstein equations with positive
cosmological constant with exactly Kottler--Schwarzschild--de
Sitter ends.

[52] C.
Arezzo and F. Pacard. On the
Kähler classes of constant scalar curvature metrics on blow ups. "Aspects analytiques de la
géométrie riemannienne". Série Séminaires et Congrès (SMF).19, (2008),
17-29. pdf

Abstract
: Building on the results of the paper "Blowing up Kähler
manifolds with constant scalar curvature II" we analyse the possible
Kähler classes which carry a constant scalar curvature metric when
small blow ups are considered.

[51] E. Hebey, F. Pacard and D. Pollack, A variational analysis of Einstein--scalar field Lichnerowicz equations on compact Riemannian manifolds. Comm. Mathematicsl Physics Vol 278, 1 (2008), 117-132. pdf

Abstract
:
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.

[50] L.
Hauswirth and F. Pacard, Embedded
minimal surfaces with finite
genus and two limits ends.
Inventiones Mathematicae 169 (3), (2007), 569-620. pdf

Abstract
: Riemann surfaces
constitute a one parameter family of embedded minimal surfaces which
are periodic and have infinitely many horizontal planar ends. The
surfaces in this family are foliated by circles (or straight lines). In
this paper, we prove the existence of a one parameter family of embeded
minimal surfaces which have infinitely many horizontal planar ends and
have genus k, for k = 1, ... , 37.
Riemann surfaces, as their flux is nearly
vertical, can be understood as a sequence of parallel planes connected
by slightly bent catenoidal neks. The surfaces we construct are
obtained by replacing one of these catenoidal necks by a member of the
family of minimal surfaces discovered by C. Costa, D. Hoffman and W.
Meek.

[49] M. del Pino, M. Musso and F. Pacard, Boundary singularities for weak
solutions of semilinear elliptic problems. Journal of Functional Analysis, Vol
253, 1 (2007), 214-272. pdf

Abstract
: In this paper, we are interested in solutions of semilinear
elliptic equations of the form ∆u + u^p =0 which are smooth in
the interior of a domain of R^n and have prescribed boundary
singularities.

[48]
A.
Butscher and F. Pacard, Generalized
doubling constructions for
constant mean curvature hypersurfaces in S^n. Annals of Global
Analysis and Geometry, 32 (2007), 103-123. pdf

Abstract : The
sphere S^n contains
a simple family of constant mean curvature
hypersurfaces of the form S^p (a) × S^q
(\sqrt{1-a^2}) for p+q+1 = n and a ∈ (0,1) called the
generalized Clifford hypersurfaces. This paper demonstrates that new,
topologically non-trivial constant mean curvature 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 perturbation.

[47]
A.
Butscher and F. Pacard, Doubling
constant mean curvature tori
in S^3, Annali de la Scuola Norm. Sup Pisa 5, vol 5 (2006),
611-638. pdf

Abstract
: The Clifford tori in S^3
are a one-parameter family of flat,
two-dimensional, constant mean curvature submanifolds. This paper
demonstrates that new, topologically non-trivial constant mean
curvature surfaces resembling a pair of neighbouring Clifford tori
connected at a sub-lattice of points by small catenoidal bridges
can be constructed by perturbative methods.

[46] S.
Baraket, M. Dammak, T. Ouni and F. Pacard, Singular limits for 4-dimensional
semilinear
elliptic problems with exponential nonlinearity. Annales
de
l'IHP : Analyse non linéaire 24 (6), (2007), 875-896. pdf

Abstract : Using some nonlinear
domain decomposition method, we prove the
existence of singular limits for solution of semilinear
elliptic
problems with exponential nonlinearity in 4 dimensional domains.

[45] S. Kaabachi and F. Pacard, Riemann
minimal surfaces in higher
dimensions. Journal of the Institute of Mathematics of Jussieu, 6 (4)
(2007), 613-637. pdf

Abstract
:
In this paper, we prove the existence of a one parameter family of
minimal hypersurface in R^{n+1}, for n

**≥**2, which generalize the well known "Riemann minimal staircase". The hypersurfaces we obtain are complete, embedded, singly periodic hypersurfaces which have infinitely many parallel hyperplanar ends. By opposition with the 2-dimensional case, they are not foliated by spheres.[44] F. Mahmoudi,
R.
Mazzeo and F. Pacard.

**Constant mean****curvature hypersurfaces condensing along a submanifold.****Geom. Funct. Anal. 16, no 4, (2006), 924-958. pdf**Abstract
:
We are interested in families of constant mean curvature hypersurfaces,
with mean curvature varying from one member of the family to another,
which `condense' to a submanifold K^k ⊂ M^{m+1} of codimension
greater than 1. Two cases have been studied previously : R. Ye proved
the existence of a local foliation by constant mean curvature
hypersurfaces when K is a point (which is required to be a
nondegenerate critical point of the scalar curvature function); in a
previous paper (see above) R. Mazzeo and I proved existence of a
lamination when K is a nondegenerate geodesic. In this paper we
extend this last result to handle the general case, when K is an
arbitrary nondegenerate minimal submanifold. In particular, this proves
the existence of constant mean curvature hypersurfaces with nontrivial
topology in any Riemannian manifold. This new approach is inspired by
some recent work of A. Malchiodi and M. Montenegro in the contex of
semilinear elliptic partial differential equations.

[43] C.
Arezzo and
F. Pacard.

**Blowing up and desingularizing Kähler manifolds of constant scalar curvature.**Acta Mathematica 196, no 2, (2006), 179-228. pdfAbstract : In
this paper we prove
the existence of Kähler metrics of constant scalar curvature
on blow ups at points and desingularizations of isolated quotient
singularities of compact manifolds and orbifolds which already
carry Kähler
constant scalar curvature metrics. In particular our construction shows
that any blow up (at a finite set of smooth points) of a compact smooth
Kähler manifold (or orbifold) of zero
scalar curvature of discrete type with nonzero first Chern class, has a
Kähler metric of zero constant scalar
curvature, generalizing former construction by Y. Rollin and M. Singer.
And we also prove that any compact complex surface of general type
admits constant scalar curvature Kähler metrics.

[42] R.
Mazzeo and F. Pacard.

**Maskit****combinations of Poincaré-Einstein metrics.**Advances in Mathematics 204 no 2, (2006), 379-412. pdfAbstract :
We establish a boundary connected sum theorem for asymptotically
hyperbolic Einstein metrics, and also show that if the two metrics have
scalar positive conformal infinities, then the same is true for this
boundary join. This construction is also extended to spaces with a
finite number of interior conic singularities, and as a result we show
that any 3-manifold which is a finite connected sum of quotients of S^3
and S^2 × S^1 bounds such a space (with conic
singularities); putatively, any 3-manifold admitting a metric of
positive scalar curvature is of this form.

[40]
F. Pacard, Constant mean curvature
hypersurfaces
in Riemannian manifolds. Riv. Mat. Univ. Parma (7) 4,
(2005),
141-162. pdf

Abstract : This short paper reviews
the results of
the papers "Foliations by
constant mean
curvature tubes" and "Constant
mean curvature
hypersurfaces
condensing along a submanifold". It also describes the strategy
of the proofs.

[39] R. Mazzeo and F. Pacard.

Abstract : In this paper weare interested in families of constant mean curvature hypersurfaces, with mean curvature varying from one member of the family to another, which form (partial) foliations and which ‘condense’ to a submanifold of codimension greater than 1. Our main results concern the existence of such families condensing to a geodesic and, conversely, the geometric nature of the submanifolds to which such families can condense.

**Foliations by constant mean curvature tubes.**Communications in Analysis and Geometry**13**, no 4,**(2005), 633-670. pdf**Abstract : In this paper weare interested in families of constant mean curvature hypersurfaces, with mean curvature varying from one member of the family to another, which form (partial) foliations and which ‘condense’ to a submanifold of codimension greater than 1. Our main results concern the existence of such families condensing to a geodesic and, conversely, the geometric nature of the submanifolds to which such families can condense.

[38]
M. Jleli and F. Pacard.

**An end-to-end construction for compact constant mean curvature surfaces**. Pacific Journal of Maths, 221, no. 1, (2005), 81-108. pdfAbstract :
We give a construction for compact surfaces of constant mean curvature
of genus 3 and higher, based on tools developed for the understanding
of complete noncompact constant mean curvature surfaces. The
construction uses the end-to-end construction developed by J. Ratzkin
to connect (and produce) complete noncompact constant mean curvature
surfaces along their ends as well as the moduli space theory developped
by R. Kusner, R. Mazzeo and D. Pollack.

[37] R. Mazzeo, F. Pacard and D.
Pollack,

**The conformal theory of Alexandrov embedded constant mean curvature surfaces in R^3**. Global Theory of Minimal Surfaces, Clay Mathematics Proceedings, D. Hoffman Edt, AMS (2005). pdfAbstract
: We
prove a general gluing theorem which creates new nondegenerate constant
mean curvature surfaces by attaching half Delaunay surfaces with small
necksize to arbitrary points of any nondegenerate constant mean
curvature surface. The proof uses the method of Cauchy datamatching.
In the second part of this paper, we develop the consequences of this
result and (at least partially) characterize the image of the map which
associates to each complete, Alexandrov-embedded constant mean
curvatursurface
with finite topology its associated conformal structure, which is a
compact Riemann surface with a finite number of punctures. In
particular, we show that this `forgetful' map is surjective when the
genus is zero. This
proves in particular that the constant mean curvature moduli space has a complicated topological
structure. These latter results are closely related to those in
R. Kusner's paper in this
same volume.

[36] Y. Ge, R. Jing and F.
Pacard.

**Bubble towers for supercritical semilinear elliptic equations**. Journal of Functional Analysis, 2, Vol 221 (2005), 251-302. pdfAbstract
: We construct
positive solutions of a semilinear elliptic problem with Dirichet
boundary conditions, in a bounded smooth domain of R^N, N

**≥**4, when the exponent p is slightly supercritical. The solutions have multiple blow up at ﬁnitely many points which are the critical points of a function whose deﬁnition involves Green’s function. Our result extends the result of Del Pino, Dolbeault and Musso whenthe domain is a ball and the solutions are radially symmetric.[35] F.
Pacard and F. Pimentel.

**Attaching handles to Bryant surfaces**. Journal of the Institute of Mathematics of Jussieu, Vol 3, 3, (2004), 421-459. pdf[34] F.
Pacard and M. Ritoré.

**From constant mean curvature hypersurfaces to the grcdienu |heory of phase transitions**. Journal of Differential Geometry 64 (2003), 359-423. pdf[33] R. Mazzeo and F. Pacard.

**Poincaré-Einstein metrics and the Schouten tensor**. Pacific J. Maths. Vol 212, 1, (2003) 169-185. pdf[32] C.
Arezzo and F. Pacard.

**Minimal embedded n-submanifolds in C^n**. Comm. Pure and Applied Maths, Vol LVI, no 3, (2003) 283-327. pdf[31] R.
Mazzeo and F. Pacard.

Abstract :All complete, axially symmetric surfaces of constant mean curvature in R^3 lie in the one-parameter family of Delaunay surfaces. The elements of this family which are embedded are called unduloids; all other elements, which correspond to negative values of the parameter are immersed and are called nodoids. The unduloids are stable in the sense that the only global constant mean curvature deformations of them are to other elements of this Delaunay family. We prove here that this same property is true for nodoids only when the parameter is sufficiently close to zero (this corresponds to these surfaces having small ‘necksizes’). On the other hand, we show that as the paprameter decreases, infinitely many new families of complete, cylindrically bounded constant mean curvature surfaces bifurcate from this Delaunay family. The surfaces in these branches have only a discrete symmetry group.

**Bifurcating nodoids**. American Mathematical Society (AMS). Contemp. Math. 314, (2002) 169-186. pdfAbstract :All complete, axially symmetric surfaces of constant mean curvature in R^3 lie in the one-parameter family of Delaunay surfaces. The elements of this family which are embedded are called unduloids; all other elements, which correspond to negative values of the parameter are immersed and are called nodoids. The unduloids are stable in the sense that the only global constant mean curvature deformations of them are to other elements of this Delaunay family. We prove here that this same property is true for nodoids only when the parameter is sufficiently close to zero (this corresponds to these surfaces having small ‘necksizes’). On the other hand, we show that as the paprameter decreases, infinitely many new families of complete, cylindrically bounded constant mean curvature surfaces bifurcate from this Delaunay family. The surfaces in these branches have only a discrete symmetry group.

[30] F. Pacard.

**Higher dimensional Scherk's hypersurfaces**. J. Math. Pures Appl., IX. Sér. 81, No.3, (2002) 241-258. pdfAbstract :
In 3-dimensional Euclidean space, Scherk second surfaces are singly
periodic embedded minimal surfaces with four planar ends. In this
paper, we obtain a natural generalization of these minimal surfaces in
any higher dimensional Euclidean space R^{n+1}, for n larger than or
equal to 3. More precisely, we show that there exist (n-1)-periodic
embedded minimal hypersurfaces with four hyperplanar ends. The moduli
space of these hypersurfaces forms a 1-dimensional fibration over the
moduli space of flat tori in R^{n-1}. A partial description of the
boundary of this moduli space is also given.

[29] R.
Mazzeo, F. Pacard and D. Pollack.

**Connected sums of constant mean curvature surfaces in Euclidean 3 space**. J. Reine Angew. Math. 536, (2001), 115-165. pdf[28] R.
Mazzeo and F. Pacard.

**Constant mean curvature surfaces with Delaunay ends.**Comm. Analysis and Geometry. 9, 1, (2001), 169-237. pdf[27] S.
Fakhi and F. Pacard.

**Existence of complete minimal hypersurfaces with finite total curvature.**Manuscripta Mathematica. 103, (2000), 465-512. pdf[26] R. Mazzeo and F. Pacard.

**Constant scalar curvature metrics with isolated singularities**. Duke Math. J, 99, (1999), 3, 353-418. pdfAbstract
:
We prove the existence of constant positive scalar curvature metrics g
which are complete and conformal to the standard metric on the
complement of Z in S^n, where Z is a disjoint union of submanifolds of
dimensions between 0 and (n−2)/2. The existence of solutions with
isolated singularities occupies the majority of the paper; their
existence was previously established by Schoen, but the proof we give
here is more direct, and provides more information about their
geometry. When Z is discrete we also establish that these solutions
are smooth points in the moduli spaces of all such solutions.

[25] N. Korevaar, R. Mazzeo, F.
Pacard and R. Schoen.

**Refined asymptotics for constant scalar curvature metrics with isolated singularities**. Inventiones Math. 135, 2, (1999) 233-272. pdfAbstract : We
consider the asymptotic behaviour of positive solutions of the
conformal scalar curvature equation, in the neighbourhood of isolated
singularities in the standard Euclidean ball. Although asymptotic
radial symmetry for such solutions was proved some time ago, we present
a much simpler and more geometric derivation of this fact. We also
discuss a refinement, showing that any such solution is asymptotic to
one of the deformed radial singular solutions. Finally we give some
applications of these refined asymptotics, first to computing the
global Pohoˆzaev invariants of solutions on the sphere with isolated
singularities, and then to the regularity of the moduli space of all
such solutions.

[24] F. Pacard, Construction de surfces à courbure moyenne
constante. Sémin. Théor. Spectr. Géom., Vol. 17, Année
1998-1999, 139-157. pdf

[23] S. Baraket and F. Pacard.

**Construction of singular limits for a semilinear elliptic equation in dimension 2**. J. Calc. Variat. and P.D.E., 6, 1, (1998) 1-38. pdf[22] J. Ph. Chancelier, M. Cohen de
Lara and F. Pacard.

**New insights in dynamical moddeling of a secondary settler-II. Dynamical analysis**. Water Research. 31, 8, (1997) 1857-1866. pdfAbstract :
A dynamic model of the settling process in the secondary settler of a
wastewater treatment plant is given by a nonlinear scalar conservation
law for the sludge concentration under the form of a partial
differential equation (PDE). A numerical algorithm is given which also
includes a mathematical model of the aeration tank. Theoretical and
numerical simulations are then compared with real data. The evolution
of the shock corresponding to the rising of a sludge blanket is
described by an ordinary differential equation (ODE). As a consequence,
regulation strategies of the rising of a sludge blanket in case of
important water admission to the plant are proposed. We end briefly
with two possible extensions. A model with two classes of particles in
interaction is introduced to take into account the particle size
change, as well as a model giving the distribution of residence times
to take into account its effect on the velocity.

[21] J. Ph. Chancelier, M. Cohen de Lara,
C. Joannis and F. Pacard.

**New insights in dynamical moddeling of a secondary settler-I. Flux theory and steady states analysis**. Water Research. 31, 8, (1997) 1847-1856. pdfAbstract
:
A dynamic model of the settling process in the secondary settler of a
wastewater treatment plant is given by a nonlinear scalar conservation
law for the sludge concentration under the form of a partial
differential equation (PDE). Theoretical results on stationary
solutions are found to be related with the limiting flux theory,
allowing new insights into this latter theory especially when the
settler is overloaded.

[20] F. Pacard.

**Le problème de Yamabe sur des sous domaines de S^n.**Séminaire sur les Équations aux Dérivées Partielles, 1996-1997, Exp. No. IX, 16 pp., École Polytech., Palaiseau, 1997. pdf[19] R. Mazzeo and F. Pacard.

**A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis**. J. Diff. Geometry. 44, (1996) 331-370. pdfAbstract : The
aim of this paper is to prove the existence of weak solutions to some
semilinear elliptic equation with power nonlinearity u^p which are
positive in a domain of R^n, vanish at the boundary, and have
prescribed isolated singularities. The exponent p is required to lie in
the interval (n/(n−2), (n+2)/(n−2)). We also prove the existence of
solutions which are positive in a domain R^n and which are
singular along arbitrary smooth k-dimensional submanifolds in the
interior of these domains provided p lie in the interval
((n−k)/(n−k−2), (n−k+2)/(n−k−2)). A particular case is when
p=(n+2)/(n−2), in which case solutions correspond to solutions of the
singular Yamabe problem. The method used here is a mixture of different
ingredients used by both authors in their separate constructions of
solutions to the singular Yamabe problem, along with a new set of
scaling techniques.

[18] F. Pacard.

**The Yamabe problem on subdomains of even dimensional spheres.**Topological Methods in Nonlinear Anal. 6, (1995), 137-150. pdf[17] F. Pacard and A. Unterreiter

**. A variational analysis of the thermal equilibrium state of quantum fluids**. Comm. P.D.E. 20 (1995) 885-900.[16]
J. Ph. Chancelier, M. Cohen de Lara
and
F. Pacard.

**Existence of a solution in an age dependent transport-diffusion P.D.E. : A model of settler**. Math. Models Methods in Appl. Sciences (M3AS). 5, 3 (1995) 267-278. pdfAbstract
:
The modeling of sludge particles settling in the final stage of a waste
water treatment plant may include both transport and diffusion. When
the residence time of sludge particles in the settler is considered,
this leads to a nonlinear age dependent transport-diffusion partial
differential equation (PDE) with nonlocal condition. We investigate the
question of existence of a solution.

[15] J. Ph.
Chancelier, M. Cohen de Lara
and
F. Pacard.

**Equation de Fokker-Planck pour la densité d'un processus aléatoire dans un ouvert régulier.**C. R. Acad. Sci. Paris, t. 321, Série I, (1995) 1251-1256. pdfAbstract
:
To give physical meaning to the boundary conditions of parabolic
partial differential equation, we introduce a diffusion process in some
open set, with different boundary conditions (elastic or non-elastic
reflection, stopping)
and we give the equation satisfied by the density.

[14] J. Ph.
Chancelier, M. Cohen de Lara
and
F. Pacard.

**Analysis of a conservation PDE with discontinuous flux: a model of settler.**SIAM J. Appl. Math. 54 (1994), no. 4, 954–995. pdfAbstract
: A
dynamic model of the settling process in the secondary settler of a
wastewater treatment plant is given by a nonlinear scalar conservation
law for the sludge concentration, where the flux function presents
discontinuities. We analyze this PDE with emphasis both on the
existence of stationary solutions and on the evolution of the shock
corresponding to the rising of a sludge blanket. Theoretical and
numerical simulations are compared with real data. A model with two
classes of particles in interaction is introduced to take into account
the thickening process : it appears to improve the fit with the data.
What is more, regulation strategies of the rising of a sludge blanket
in case of important water admission to the plant are proposed.

[13] F. Pacard.

**Le problème de Yamabe sur des sous domaines de S^6.**C. R. Acad. Sci. Paris, t. 318, Série I, (1994) 639-642.[12] F. Pacard.

**Solutions with high dimensional singular set, to a conformally invariant elliptic equation in R^4 and in R^6**. Comm. Math. Physics, 159, 2, (1994) 423-432. pdf[11] F. Pacard.

**A priori regularity of weak solutions of nonlinear elliptic equations**. Ann. de l'I.H.P, 11, 6, (1994) 693-703. pdf[10] F. Pacard.

**Convergence and partial regularity for weak solutions of some nonlinear elliptic equation: the supercritical case.**Ann. de l'I.H.P., 11, 5, (1994) 537-551. pdf[09] F. Pacard.

**Partial regularity for weak solutions of a nonlinear elliptic equation.**Manuscripta Math., 79, (1993) 161-172. pdf[08] F. Pacard, A regularity criterion for positive weak
solutions of - ∆u = u^p. Commentarii Mathematici Helvetici, 68,
(1993), 73-84. pdf

[07] F. Pacard.

**Existence and convergence of positive weak solutions of -**∆**u = u^{n/(n-2)} in bounded domains of R^n, n ≥ 3**. Calc. Variat. and P.D.E., 1, (1993) 243-265. pdf[06] F. Pacard.

**Radial and non-radial solutions of -**∆**u = \lambda f(u), on an annulus of R^n, n ≥ 3**. J. Diff. Equa., 102, 1, (1993) 103-138. pdf[05] F. Pacard.

**Existence de solutions faibles positives de -**∆**u = u^p dans des ouverts bornés de R^n, n ≥ 3**. C. R. Acad. Sci. Paris, t. 315, Série I, (1992) 793-798. pdf[04] F. Pacard.

**Existence et convergence de solutions faibles positives de -**∆**u = u^{n/(n-2)} dans des ouverts bornés de R^n, n ≥ 3.**C. R. Acad. Sci. Paris t. 314, Série I, (1992) 729-734. pdf[03] F. Pacard.

**A note on the regularity of weak solutions of -**∆**u=u^p in R^n, n****≥****3**. Houston J. Math., 18, 4, (1992) 621-632. pdf[02] F. Pacard.

**Solutions de -**∆**u = \lambda e^u ayant des singularités prescrites**. C. R. Acad. Sci. Paris, t. 311, Série I, (1990) 317-320. pdf[01] F.
Pacard.

**Convergence of surfaces of prescribed mean curvature**. Nonlinear Analysis, Vol. 13, (11), (1989) 1269-1281. pdf