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). pdf


Papers


[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

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 planeAnn. 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

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. pdf

Abstract : 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. pdf

Abstract : 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. pdf

Abstract : 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.

[41] F. Pacard, Surfaces à courbure moyenne constante. "Images des Mathématiques 2006". pdf

[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. 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. pdf

Abstract : 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). pdf

Abstract : 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. pdf

Abstract : 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

Abstract : We examine the space of conformally compact metrics g on the interior of a compact manifold with boundary which have the property that the k-th elementary symmetric function of the Schouten tensor Ag is constant. When k = 1 this is equivalent to the familiar Yamabe problem, and the corresponding metrics are complete with constant negative scalar curvature. We show for every k that the deformation theory for this problem is unobstructed, so in particular the set of conformal classes containing a solution of any one of these equations is open in the space of all conformal classes. We then observe that the common intersection of these solution spaces coincides with the space of conformally compact Einstein metrics, and hence this space is a finite intersection of closed analyticsubmanifolds.

[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. Bifurcating nodoids. American Mathematical Society (AMS). Contemp. Math. 314, (2002) 169-186. pdf

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.

[30] F. Pacard. Higher dimensional Scherk's hypersurfaces. J. Math. Pures Appl., IX. Sér. 81, No.3, (2002) 241-258. pdf

Abstract : 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. pdf

Abstract : 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.  pdf

Abstract : 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. pdf

Abstract : 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. pdf

Abstract : 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. pdf

Abstract : 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. pdf

Abstract : 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. pdf

Abstract : 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. pdf

Abstract : 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 ≥  3Calc. 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 ≥ 3C. 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