Antonio De Rosa
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Antonio De Rosa has a total of 60 co-authors. Here are the top co-authors by number of shared publications:
- Andrea Marchese
Universität Zürich
9 shared publications - Guido De Philippis
New York University, USA
9 shared publications - Francesco Ghiraldin
Universität Basel Spiegelgasse 1, CH‐4051 Basel Switzerland
7 shared publications - Maria Colombo
Institute of Mathematics, Swiss Federal Institute of Technology in Lausanne, Lausanne, Switzerland
7 shared publications - Aida Khajavirad
Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, USA
5 shared publications - Riccardo Tione
Dipartimento di Matematica “Giuseppe Peano”, Università degli Studi di Torino, Torino, Italy
3 shared publications
Latest Publications
- Open Access
Anisotropic min-max via phase transitions
We develop a PDE-based approach to the min-max construction of nontrivial integer rectifiable varifolds that are stationary with respect to anisotropic surface energies on closed Riemannian manifolds, in codimension one. Specifically, we study the anisotropic analogue of the Allen-Cahn energy and establish a Modica-type gradient bound for its critical points. Using this in conjunction with certain estimates for stable solutions, we then prove that the energy densities of stable or bounded-Morse-index critical points of its rescalings concentrate along an integer rectifiable varifold that is stationary for the underlying anisotropic integrand. As a consequence, we construct a (possibly singular) anisotropic min-max hypersurface via Allen-Cahn, obtaining an analogue of the result of Hutchinson-Tonegawa in the anisotropic setting.
Open AccessAnisotropic min-max via phase transitions
We develop a PDE-based approach to the min-max construction of nontrivial integer rectifiable varifolds that are stationary with respect to anisotropic surface energies on closed Riemannian manifolds, in codimension one. Specifically, we study the anisotropic analogue of the Allen-Cahn energy and establish a Modica-type gradient bound for its critical points. Using this in conjunction with certain estimates for stable solutions, we then prove that the energy densities of stable or bounded-Morse-index critical points of its rescalings concentrate along an integer rectifiable varifold that is stationary for the underlying anisotropic integrand. As a consequence, we construct a (possibly singular) anisotropic min-max hypersurface via Allen-Cahn, obtaining an analogue of the result of Hutchinson-Tonegawa in the anisotropic setting.
- Open Access
Rigidity of critical points of hydrophobic capillary functionals
We prove the rigidity, among sets of finite perimeter, of volume-preserving critical points of the capillary energy in the half space, in the case where the prescribed interior contact angle is between $90^\circ$ and $120^\circ$. No structural or regularity assumption is required on the finite perimeter sets. Assuming that the ``tangential'' part of the capillary boundary is $\mathcal{H}^n$-null, this rigidity theorem extends to the full hydrophobic regime of interior contact angles between $90^\circ$ and $180^\circ$. Furthermore, we establish the anisotropic counterpart of this theorem under the assumption of lower density bounds.Open AccessRigidity of critical points of hydrophobic capillary functionals
We prove the rigidity, among sets of finite perimeter, of volume-preserving critical points of the capillary energy in the half space, in the case where the prescribed interior contact angle is between $90^\circ$ and $120^\circ$. No structural or regularity assumption is required on the finite perimeter sets. Assuming that the ``tangential'' part of the capillary boundary is $\mathcal{H}^n$-null, this rigidity theorem extends to the full hydrophobic regime of interior contact angles between $90^\circ$ and $180^\circ$. Furthermore, we establish the anisotropic counterpart of this theorem under the assumption of lower density bounds. - 2 June 2025
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 249 (4) , 1-34
Construction of Fillings with Prescribed Gaussian Image and Applications
- 2 June 2025
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 249 (4) , 1-34
Construction of Fillings with Prescribed Gaussian Image and Applications
- 11 February 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 378 (5) , 3393-3444
The double and triple bubble problem for stationary varifolds: The convex case
We characterize the critical points of the double bubble problem in R n \mathbb {R}^n and the triple bubble problem in R 3 \mathbb {R}^3 , in the case the bubbles are convex.
- 11 February 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 378 (5) , 3393-3444
The double and triple bubble problem for stationary varifolds: The convex case
We characterize the critical points of the double bubble problem in R n \mathbb {R}^n and the triple bubble problem in R 3 \mathbb {R}^3 , in the case the bubbles are convex.
- Open Access
- 19 December 2024
- Published by Springer Nature in Mathematical Programming
- Vol. 218 (1-2) , 1-43
Explicit convex hull description of bivariate quadratic sets with indicator variables
We consider the nonconvex set $${{\mathcal {S}}}_n = \{(x,X,z): X = x x^T, \; x (1-z) =0,\; x \ge 0,\; z \in \{0,1\}^n\}$$ , which is closely related to the feasible region of several difficult nonconvex optimization problems such as the best subset selection and constrained portfolio optimization. Utilizing ideas from convex analysis and disjunctive programming, we obtain an explicit description for the closure of the convex hull of $${{\mathcal {S}}}_2$$ in the space of original variables. In order to generate valid inequalities corresponding to supporting hyperplanes of the convex hull of $${{\mathcal {S}}}_2$$ , we present a simple separation algorithm that can be incorporated in branch-and-cut based solvers to enhance the quality of existing relaxations.- 19 December 2024
- Published by Springer Nature in Mathematical Programming
- Vol. 218 (1-2) , 1-43
Open AccessExplicit convex hull description of bivariate quadratic sets with indicator variables
We consider the nonconvex set $${{\mathcal {S}}}_n = \{(x,X,z): X = x x^T, \; x (1-z) =0,\; x \ge 0,\; z \in \{0,1\}^n\}$$ , which is closely related to the feasible region of several difficult nonconvex optimization problems such as the best subset selection and constrained portfolio optimization. Utilizing ideas from convex analysis and disjunctive programming, we obtain an explicit description for the closure of the convex hull of $${{\mathcal {S}}}_2$$ in the space of original variables. In order to generate valid inequalities corresponding to supporting hyperplanes of the convex hull of $${{\mathcal {S}}}_2$$ , we present a simple separation algorithm that can be incorporated in branch-and-cut based solvers to enhance the quality of existing relaxations. - Open Access
Existence and regularity of min-max anisotropic minimal hypersurfaces
In any closed smooth Riemannian manifold of dimension at least three, we use the min-max construction to find anisotropic minimal hyper-surfaces with respect to elliptic integrands, with a singular set of codimension~$2$ vanishing Hausdorff measure. In particular, in a closed $3$-manifold, we obtain a smooth anisotropic minimal surface. The critical step is to obtain a uniform upper bound for density ratios in the anisotropic min-max construction. This confirms a conjecture by Allard [Invent. Math., 1983].
Open AccessExistence and regularity of min-max anisotropic minimal hypersurfaces
In any closed smooth Riemannian manifold of dimension at least three, we use the min-max construction to find anisotropic minimal hyper-surfaces with respect to elliptic integrands, with a singular set of codimension~$2$ vanishing Hausdorff measure. In particular, in a closed $3$-manifold, we obtain a smooth anisotropic minimal surface. The critical step is to obtain a uniform upper bound for density ratios in the anisotropic min-max construction. This confirms a conjecture by Allard [Invent. Math., 1983].
- Open Access
The ratio-cut polytope and K-means clustering
We introduce the ratio-cut polytope defined as the convex hull of ratio-cut vectors corresponding to all partitions of $n$ points in $\mathbb R^m$ into at most $K$ clusters. This polytope is closely related to the convex hull of the feasible region of a number of clustering problems such as K-means clustering and spectral clustering. We study the facial structure of the ratio-cut polytope and derive several types of facet-defining inequalities. We then consider the problem of K-means clustering and introduce a novel linear programming (LP) relaxation for it. Subsequently, we focus on the case of two clusters and derive sufficient conditions under which the proposed LP relaxation recovers the underlying clusters exactly. Namely, we consider the stochastic ball model, a popular generative model for K-means clustering, and we show that if the separation distance between cluster centers satisfies $\Delta > 1+\sqrt 3$, then the LP relaxation recovers the planted clusters with high probability. This is a major improvement over the only existing recovery guarantee for an LP relaxation of K-means clustering stating that recovery is possible with high probability if and only if $\Delta > 4$. Our numerical experiments indicate that the proposed LP relaxation significantly outperforms a popular semidefinite programming relaxation in recovering the planted clusters.Open AccessThe ratio-cut polytope and K-means clustering
We introduce the ratio-cut polytope defined as the convex hull of ratio-cut vectors corresponding to all partitions of $n$ points in $\mathbb R^m$ into at most $K$ clusters. This polytope is closely related to the convex hull of the feasible region of a number of clustering problems such as K-means clustering and spectral clustering. We study the facial structure of the ratio-cut polytope and derive several types of facet-defining inequalities. We then consider the problem of K-means clustering and introduce a novel linear programming (LP) relaxation for it. Subsequently, we focus on the case of two clusters and derive sufficient conditions under which the proposed LP relaxation recovers the underlying clusters exactly. Namely, we consider the stochastic ball model, a popular generative model for K-means clustering, and we show that if the separation distance between cluster centers satisfies $\Delta > 1+\sqrt 3$, then the LP relaxation recovers the planted clusters with high probability. This is a major improvement over the only existing recovery guarantee for an LP relaxation of K-means clustering stating that recovery is possible with high probability if and only if $\Delta > 4$. Our numerical experiments indicate that the proposed LP relaxation significantly outperforms a popular semidefinite programming relaxation in recovering the planted clusters. - 20 December 2023
- Published by Taylor & Francis in Communications in Partial Differential Equations
- Vol. 49 (1-2) , 15-37
Boundary regularity for anisotropic minimal Lipschitz graphs
- 20 December 2023
- Published by Taylor & Francis in Communications in Partial Differential Equations
- Vol. 49 (1-2) , 15-37
Boundary regularity for anisotropic minimal Lipschitz graphs
- Open Access
- 1 December 2023
- Published by Wiley in Communications on Pure and Applied Mathematics
- Vol. 77 (7) , 3184-3226
The anisotropic min‐max theory: Existence of anisotropic minimal and CMC surfaces
- 1 December 2023
- Published by Wiley in Communications on Pure and Applied Mathematics
- Vol. 77 (7) , 3184-3226
Open AccessThe anisotropic min‐max theory: Existence of anisotropic minimal and CMC surfaces
- Open Access
Explicit convex hull description of bivariate quadratic sets with indicator variables
We consider the nonconvex set $\mathcal S_n = \{(x,X,z): X = x x^T, \; x (1-z) =0,\; x \geq 0,\; z \in \{0,1\}^n\}$, which is closely related to the feasible region of several difficult nonconvex optimization problems such as the best subset selection and constrained portfolio optimization. Utilizing ideas from convex analysis and disjunctive programming, we obtain an explicit description for the closure of the convex hull of $\mathcal S_2$ in the space of original variables. In order to generate valid inequalities corresponding to supporting hyperplanes of the convex hull of $\mathcal S_2$, we present a simple separation algorithm that can be incorporated in branch-and-cut based solvers to enhance the quality of existing relaxations.Open AccessExplicit convex hull description of bivariate quadratic sets with indicator variables
We consider the nonconvex set $\mathcal S_n = \{(x,X,z): X = x x^T, \; x (1-z) =0,\; x \geq 0,\; z \in \{0,1\}^n\}$, which is closely related to the feasible region of several difficult nonconvex optimization problems such as the best subset selection and constrained portfolio optimization. Utilizing ideas from convex analysis and disjunctive programming, we obtain an explicit description for the closure of the convex hull of $\mathcal S_2$ in the space of original variables. In order to generate valid inequalities corresponding to supporting hyperplanes of the convex hull of $\mathcal S_2$, we present a simple separation algorithm that can be incorporated in branch-and-cut based solvers to enhance the quality of existing relaxations.