pacard (main)

Publications

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).

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Papers

[57] M. del Pino, M. Kowalczyk, F. Pacard and J. Wei. The Toda system and multiple-end solutions of autonomous planar elliptic problems. Advances in Maths.

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).

[56] F. Pacard. Geometric aspects of the Allen-Cahn equation. Matematica Contemporanea.

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] R. Mazzeo and F. Pacard. Constant curvature foliations in asymptotically hyperbolic spaces. Revista Matematica Iberoamericana.

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.

[54] 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.

Abstract : We construct new solutions of the Allen-Cahn equation in R2. Given k ≥ 1 we ﬁnd a family of solutions whose zero level sets are asymptotic to 2k straight half lines.

[53] M. del Pino, M. Musso and F. Pacard. Bubbling along boundary geodesics near the second critical exponent. J. of the European Math Society.

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 \Delta 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=\frac{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.

[52] 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.

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.

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

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.

[50] 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).

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.

[49] 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.

Abstract : We present a gluing construction which adds, via a localized deformation,exactly "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.

[48] 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.

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.

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

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.

[46] 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.

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

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 necessary 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.

[44] 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.

Abstract : The sphere S^n contains a simple family of constant mean curvature hypersurfaces of the form \Lambda^{p,q}_a \equiv S^p(a) \times S^q (\sqrt{1-a^2}) for p+q+1 = n and a \in (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.

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.

[42] 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.

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.

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

[40] F. Mahmoudi, R. Mazzeo and F. Pacard. Constant mean curvature hypersurfaces condensing along a submanifold. Geom. Funct. Anal. 16, no 4, (2006) 924-958.

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 \subset 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.

[39] C. Arezzo and F. Pacard. Blowing up and desingularizing Kähler manifolds of constant scalar curvature. Acta Mathematica 196, no 2, (2006) 179-228.

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.

[38] R. Mazzeo and F. Pacard. Maskit combinations of Poincaré-Einstein metrics. Advances in Mathematics 204 no 2, (2006) 379-412.

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 \times S^1 bounds such a space (with conic singularities); putatively, any 3-manifold admitting a metric of positive scalar curvature is of this form.

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

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

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.

[34] 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.

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.

[33] 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).

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.

[32] Y. Ge, R. Jing and F. Pacard. Bubble towers for supercritical semilinear elliptic equations. Journal of Functional Analysis, 2, Vol 221 (2005) 251-302.

Abstract : We construct positive solutions of a semilinear elliptic problem with Dirichet boundary conditions, in a bounded smooth domain of R^N, N \geq 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.

[31] F. Pacard and F. Pimentel. Attaching handles to Bryant surfaces. Journal of the Institute of Mathematics of Jussieu, Vol 3, 3, (2004) 421-459.

[30] 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.

[29] R. Mazzeo and F. Pacard. Poincaré-Einstein metrics and the Schouten tensor. Pacific J. Maths. Vol 212, 1, (2003) 169-185.

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

[27] R. Mazzeo and F. Pacard. Bifurcating nodoids. American Mathematical Society (AMS). Contemp. Math. 314, (2002) 169-186.

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

[25] 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.

[24] R. Mazzeo and F. Pacard. Constant mean curvature surfaces with Delaunay ends. Comm. Analysis and Geometry. 9, 1, (2001), 169-237.

[23] S. Fakhi and F. Pacard. Existence of complete minimal hypersurfaces with finite total curvature. Manuscripta Mathematica. 103, (2000), 465-512.

[22] R. Mazzeo and F. Pacard. Constant scalar curvature metrics with isolated singularities. Duke Math. J, 99, (1999), 3, 353-418.

[21] 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.

[20] 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.

[19] 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.

[18] 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.

[17] 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.

[16] F. Pacard. The Yamabe problem on subdomains of even dimensional spheres. Topological Methods in Nonlinear Anal. 6, (1995), 137-150.

[15] F. Pacard and A. Unterreiter. A variational analysis of the thermal equilibrium state of quantum fluids. Comm. P.D.E. 20 (1995) 885-900.

[14] 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.

[13] 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.

[12] 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.

[11] 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.

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

[09] 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.

[08] F. Pacard. Partial regularity for weak solutions of a nonlinear elliptic equation. Manuscripta Math., 79, (1993) 161-172.

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

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

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

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

[03] F. Pacard. A note on the regularity of weak solutions of -\Delta u=u^p in R^n, n \geq 3. Houston J. Math., 18, 4, (1992) 621-632.

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

[01] F. Pacard. Convergence of surfaces of prescribed mean curvature. Nonlinear Analysis, Vol. 13, (11), (1989) 1269-1281.