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

[80] R. Mazzeo, F. Pacard
and T. Zolotareva. Higher
codimension isoperimetric problems.

**Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), no 3, 819-851. pdf****Abstract : We consider a variational problem for submanifolds Q ? M with nonempty boundary ?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 < dim M. We then construct small nearly-spherical solutions of this higher codimension constant mean curvature problem; these concentrate near the critical points of a certain curvature function.**

[79] A. Folha, F. Pacard and
T. Zolotareva. Free
boundary minimal surfaces in the unit 3-ball.

**Manuscripta Math. 154 (2017), no 3-4, 359-409. pdf****Abstract : A. Fraser and R. Schoen proved the existence of free boundary minimal surfaces in the unit 3-ball which have genus 0 and n boundary components, for all n ? 3. For large n, we give an independent construction of these minimal surfaces and also prove the existence of free boundary minimal surfaces in the unit 3-ball which have genus 1 and n boundary components. As n tends to infinity, the sequence of free boundary minimal surfaces converges either to a double copy of the unit horizontal (open) disk, uniformly on compacts of the (open) unit 3-ball or to a double copy of the unit horizontal (open) punctured disk, uniformly on compacts of the (open) unit 3-ball a{0}.**

[78] M. del Pino, F. Pacard
and J. Wei. Serrin's
overdetermined problem and constant mean curvature
surfaces.

**Duke Math. J. 164 (2015), no. 14, 2643-2722. pdf****Abstract : For N N?9, we find smooth entire epigraphs in R^N, namely, smooth domains of the form ?:={x?R^N : x_N>F(x_1,...,x_N?1)}, which are not half-spaces and in which a problem of the form ?u+f(u)=0 in ? has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on ??. This answers negatively for large dimensions a question by Berestycki, Caffarelli, and Nirenberg. In 1971, 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 constant mean curvature surface where Serrin’s overdetermined problem is solvable.**

[77] W. Ao, M. Musso, F.
Pacard and J. Wei. Solutions
without any symmetry for semilinear elliptic problems.

**J. Funct. Anal. 270 (2016), no. 3, 884-956. pdf****Abstract : We prove the existence of in?nitely many solitary waves for the non linear Klein-Gordon or Schrödinger equation ?u ? u + u^3 = 0, in R^2 , which have ?nite energy and whose maximal group of symmetry reduces to the identity.**

[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. 52 (2015) no.1-2,
281-302.
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), no.7, 1675-1718. 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 ? embedded in R^m, with m ? 2 and 1 ? k ? m ? 1, and given e > 0, we de?ne B_e(?) to be the geodesic tubular neighborhood of radius e about ?. In this paper, we construct positive solutions of the semilinear elliptic equation ?u + u^p = 0 in B_e(?) and u = 0 on ?B_e(?), when the parameter ? is chosen small enough. In this equation, the exponent p satis?es 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 in?nity. 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 a?ne lines at in?nity and, along each of these half a?ne 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 ? ? (0, ?/2), there exists a 4-ended solution of the Allen-Cahn equation whose zero set is at in?nity asymptotic to the half oriented a?ne lines making the angles ?, ? ? ?, ? + ? and 2? ? ? 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 ?nite
energy solitary waves for nonlinear Klein-Gordon or
Schrodinger equations. Under natural conditions on the
nonlinearity, we prove the existence of in?nitely 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 2?/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 classi?cation
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 ?nd 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 ?rst 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 ?nitely many points which are the critical points of a function whose de?nition 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 gradient theory 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