Publications.
  1. Nontrivial solutions to the relative overdetermined torsion problem in a cylinder (with F. Pacella and D. Ruiz), Preprint.

    Abstract. We construct new nontrivial solutions for the relative overdetermined torsion problem in a cylinder D x R, where D is a generic bounded domain of R^{N-1}. Such solutions are defined in domains that bifurcate from a bounded piece of the cylinder.
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  2. A Schiffer-type problem for annuli with applications to stationary planar Euler flows (with A. Enciso, A. J. Fernández and D. Ruiz), Duke Mathematical Journal, to appear.

    Abstract. If on a smooth bounded planar domain there is a nonconstant Neumann Laplace eigenfunction that is locally constant on the boundary, must the domain be a disk or an annulus? We provide a negative answer to this question by constructing a family of nontrivial doubly connected domains with the above property bifurcating from an annulus. Furthermore, our construction implies the existence of continuous, compactly supported stationary weak solutions to the 2D incompressible Euler equations which are not locally radial.
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  3. Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space (with G. Dai and Y. Zhang), Preprint.

    Abstract. We construct nontrivial unbounded domains in the hyperbolic space H^N, with N = 2,3 or 4, bifurcating from the complement of a ball, such that the problem \Delta u - u + u^3 = 0, with overdetermined conditions u=0 and |\nabla u| = 1 at the boundary, admits a positive bounded solution. We also give a condition under which this construction holds for larger dimensions N. This is linked to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems, and, as far as we know, is the first nontrivial example of solution to an overdetermined elliptic problem in the hyperbolic space.
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  4. Modica type estimates and curvature results for overdetermined elliptic problems (with D. Ruiz and J. Wu), Communications in Contemporary Mathematics, to appear.

    Abstract. In this paper, we establish a Modica type estimate for bounded solutions to overdetermined elliptic problems. The presence of the boundary changes the usual form of the Modica estimate for entire solutions. We also discuss the equality case. From such estimates we deduce information about the curvature of the boundary of the domain under certain conditions. The proof uses the maximum principle together with scaling arguments and a careful passage to the limit in the arguments by contradiction.
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  5. Bifurcation domains for the Serrin's overdetermined problem in S^n x R and H^n x R (with G. Dai and F. Morabito), Preprint.

    Abstract. We obtain the existence of a smooth 1-parameter family of noncompact domains in S^n x R and N^n x R bifurcating from the straight cylinder such that the Serrin overdetermined problem has a bounded solution. The domains are rotationally symmetric and periodic with respect to the R-axis of the cylinder. Moreover, we also show that the bifurcation is critical.
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  6. A smooth 1-parameter family of Delaunay-type domains for an overdetermined elliptic problem in S^n x R and H^n x R (with G. Dai and F. Morabito), Potential Analysis, Vol. 60, 2024, 1407–1420.

    Abstract. We prove the existence of a smooth 1-parameter family of Delaunay type domains in S^n x R and H^n x R, bifurcating from the straight cylinder for which the first eigenfunction of the Laplace-Beltrami operator with zero Dirichlet boundary condition also has constant Neumann data at the boundary. This improves a previous result by the second and third author.
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  7. Overdetermined elliptic problems in nontrivial contractible domains of the sphere (with D. Ruiz and J. Wu) Journal de Mathématiques pures et appliquées, Vol. 180, 2023, 151-187.

    Abstract. In this paper, we prove the existence of nontrivial simply connected domains of the sphere that support solutions to an overdetermined elliptic problem These domains are perturbations of topological balls. This shows in particular that Serrin’s theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for simply connected domains.
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  8. Onduloid-type solutions to overdetermined elliptic problems with general nonlinearities (with D. Ruiz and J. Wu), Journal of functional analysis, Vol. 283, n. 12, 2022.

    Abstract. In this paper, we prove the existence of nontrivial unbounded domains bifurcating from the straight cylinder where a general overdetermined elliptic problem with equation ∆u+f(u)=0 has a positive bounded solution. We will prove such result for a very general class of functions f. Roughly speaking, we only ask that the Dirichlet problem in the ball admits a nondegenerate solution. The proof uses a local bifurcation argument.
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  9. A short survey on overdetermined elliptic problems in unbounded domains, in: Cerejeiras, P., Reissig, M., Sabadini, I., Toft, J. (eds) Current Trends in Analysis, its Applications and Computation. Trends in Mathematics(). Birkhäuser, Cham., 2021, 451-461.

    Abstract. In this paper, we prove the existence of nontrivial simply connected domains of the sphere that support solutions to an overdetermined elliptic problem These domains are perturbations of topological balls. This shows in particular that Serrin’s theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for simply connected domains.
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  10. Half-space theorems for the Allen-Cahn equation and related problems (with F. Hamel, Y. Liu, K. Wang and J. Wei), Journal für die reine und angewandte Mathematik, Vol. 770, 2021, 113-133.

    Abstract. In this paper we obtain rigidity results for a non-constant entire solution u of the Allen-Cahn equation in R^n , whose zero level set is contained in a half-space. In dimension n < 4 we prove that the solution must be one-dimensional. In dimension n > 3, we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.
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  11. Existence and regularity for Faber-Krahn minimizers in a Riemannian manifold (with J. Lamboley), Journal de Mathématiques pures et appliquées, Vol. 141, 2020, 137-183.

    Abstract. We study the minimization of the first Dirichlet eigenvalue of the Laplace-Beltrami operator, within the class of open sets of fixed volume in a Riemmanian manifold. We get existence and regularity properties for this spectral shape optimization problem, in a similar fashion as for the isoperimetric problem. We first give an existence result in the context of compact Riemannian manifolds, and we discuss the case of non-compact manifolds by giving a counter-example to existence. We then focus on the regularity theory for this problem, and using the tools coming from the theory of free boundary problems, we show that solutions are smooth up to a possible residual set of co-dimension 5 or higher.
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  12. Solutions to overdetermined elliptic problems in nontrivial exterior domains (with A. Ros and D. Ruiz), Journal of the European Mathematical Society, Vol. 22, n.1, 2020, 253–281.

    Abstract. In this paper we construct nontrivial exterior domains such that the problem \Delta u - u + u^3 = 0, with overdetermined conditions u=0 and |\nabla u| = 1 at the boundary, admits a positive bounded solution. This result gives a negative answer to the Berestycki- Caffarelli-Nirenberg conjecture on overdetermined elliptic problems in dimension 2, the only dimension in which the conjecture was still open. For higher dimensions, different counterexamples have been found in the literature; however, our example is the first one in the form of an exterior domain.
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  13. A rigidity result for overdetermined elliptic problems in the plane (with A. Ros and D. Ruiz), Communications on Pure and Applied Mathematics, Vol. 70, 2017, 1223-1252.

    Abstract. Let f be a locally Lipschitz function and \Omega a domain whose boundary is unbounded and connected. If there exists a positive bounded solution to equation \Delta u + f(u) = 0, with overdetermined conditions u=0 and |\nabla u| = 1 at the boundary, we prove that \Omega is a half-plane. In particular, we obtain a partial answer to a question raised by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997.
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  14. Delaunay type domains for an overdetermined elliptic problem in S^n x R and H^n x R (with F. Morabito), ESAIM: Control, Optimisation and Calculus of Variations, Vol 22, n. 1, 2016, 1-28.

    Abstract. We prove the existence of a countable family of Delaunay type domains \Omega_j in M^n x R, where M^n is the Riemannian manifold S^n or H^n and n is at least 2, bifurcating from the cylinder B^n x R (where B^n is a geodesic ball of radius 1 in M^n) for which the first eigenfunction of the Laplace-Beltrami operator with zero Dirichlet boundary condition also has constant Neumann data at the boundary. The domains \Omega_j are rotationally symmetric and periodic with respect to the R-axis of the cylinder and as j converges to 0 the domain \Omega_j converges to the cylinder B^n x R.
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  15. New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in a Riemannian manifold with boundary (with J. Lamboley), International Mathematics Research Notices, 2015, n. 18, 8752-8798.

    Abstract. We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifolds with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of the manifold, and their boundary intersects the boundary of the manifold orthogonally.
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  16. Extremal domains for the first eigenvalue in a general compact Riemannian manifold (with E. Delay), Discrete and Continuous Dynamical Systems, Series A, Vol 35, n. 12 (2015), 5799-5825.

    Abstract. We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a nondegenerate critical point of the scalar curvature of the Riemannian manifold was required.
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  17. Extremal domains of big volume for the first eigenvalue of the Laplace-Beltrami operator in a compact manifold, Annales de l'Institut Henri Poincaré (C) Analyse non linéaire, Vol. 31, 2014, 1231-1265.

    Abstract. We prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds, with volume close to the volume of the manifold. If the first (positive) eigenfunction F of the Laplace-Beltrami operator over the manifold is a nonconstant function, these domains are close to the complement of geodesic balls of small radius whose center is close to the point where F attains its maximum. If F is a constant function and the dimension of the manifold is at least 4, these domains are close to the complement of geodesic balls of small radius whose center is close to a nondegenerate critical point of the scalar curvature function.
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  18. Geometry and topology of some overdetermined elliptic problems (with A. Ros), Journal of Differential Equations, Vol. 255, n. 5, 2013, 951-977.

    Abstract. We prove some geometric and topological properties for unbounded domains of the plane that support a positive solution to some elliptic equations, with 0 Dirichlet and constant Neumann boundary condition. Some of such properties are true also in higher dimension. Such properties give a partial answer to a conjecture of Berestycki-Caffarelli-Nirenberg in dimension 2.
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  19. Bifurcating extremal domains for the first eigenvalue of the Laplacian (with F. Schlenk). Advances in Mathematics, Vol. 229, 2012, 602-632.

    Abstract. We prove the existence of a smooth family of noncompact domains of the euclidean space where the first eigenfunction of the Laplacian with 0 Dirichlet boundary condition has also constant Neumann data at the boudary. These domains are rotationally symmetric and periodic with respect to a vertical axe. We determine also the shape of these domains, and precise upper and lower bounds for their period. These domains provide a smooth family of counterexemples to a conjecture of Berestycki-Caffarelli-Nirenberg in dimension bigger or equal then 3.
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  20. New extremal domains for the first eigenvalue of the Laplacian in flat tori, Calculus of Variations and Partial Differential Equations, Vol. 37 n. 3-4, 2010, 329-344.

    Abstract. We prove the existence of nontrivial and noncompact extremal domains for the first eigenvalue of the Laplacian in some flat tori. Such domains can be extended by periodicity to nontrivial and noncompact domains in Euclidean spaces whose first eigenfunction of the Laplacian with 0 Dirichlet boundary condition has also constant Neumann data at the boundary, providing a couterexemple to a conjecture of Berestycki-Caffarelli-Nirenberg in dimension bigger or equal then 3. These domains are close to a straigh cylinder, they are invariant by rotation with respect to the vertical axe, and are not invariant by vertical translations.
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  21. Extremal domains for the first eigenvalue of the Laplace-Beltrami operator (with F. Pacard), Annales de l'Institut Fourier, Vol. 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.
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Ph.D Thesis.
    Domaines extrémaux pour la première valeur propre de l'opérateur de Laplace-Beltrami, Ph.D awarded on 2009 December 8th at the University Paris-Est. Laureate of the Prize of the Conséil General de Val-de-Marne for the best thesis among the universities of Paris.
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