L. Spolaor

activity

20152026

most citedDirect epiperimetric inequalities for the thin obstacle problem and applications

35 citations · 82 across the 31 of their papers we have counts for

collaborators

42 papers

math.DG2026

On Cylindrical singularities of minimal hypersurfaces

Aria Halavati, Luca Spolaor

In a neighborhood of the origin in $R^{9}$ , we construct a minimal hypersurface with respect to a smooth conformally Euclidean metric, whose singular set consists of the ori…

math.DG2026

Stationary varifolds with singularities II

Camillo De Lellis, Jonas Hirsch, Zachary Lihn +1

For every dimension $m \geq 3$ , we show the existence of an open set $U \subset R^{m + 1}$ , a smooth Riemannian metric $g$ on it, and an integral $m$ -dimensional stationary var…

math.AP2026

Fine structure of the two-phase Bernoulli free boundaries in 2D

Lorenzo Ferreri, Luca Spolaor, Bozhidar Velichkov

We prove that the branching set of a solution to a two-dimensional two-phase Bernoulli problem with constant coefficients is locally finite. We do this via a Weierstrass representa…

math.DG2025

Measure $0$ of the singular set for $2$ -valued stationary hypercurrents

Jonas Hirsch, Luca Spolaor

We prove that the singular set of a multiplicity $2$ integral hypercurrent that is stationary in the sense of varifolds has a singular set of measure zero.

math.DG2025

Stationary and stable varifolds with singularities

Camillo De Lellis, Jonas Hirsch, Luca Spolaor

We construct minimal $m$ -dimensional immersions in $R^{m + 1}$ , equipped with a $C^{1, α}$ metric, $α \in [0, 1)$ , with a sequence of \emph{catenoidal necks} or \emph{floating disks}…

math.AP2025

Carleman estimates for stationary $Q$ -valued maps: a variational approach

Aria Halavati, Luca Spolaor

We prove a Carleman-type estimate for Dirichlet-stationary multivalued functions and apply it to give a different proof of the optimal dimension of the singular set of Dir-minimizi…

math.AP2025

Structure of two-dimensional mod $(q)$ area-minimizing currents near flat singularities: the codimension one case

Anna Skorobogatova, Luca Spolaor, Salvatore Stuvard

We obtain a fine structural result for two-dimensional mod $(q)$ area-minimizing currents of codimension one, close to flat singularities. Precisely, we show that, locally around an…

math.AP2024★ 1 cited

Rectifiability of the singular set and uniqueness of tangent cones for semicalibrated currents

Paul Minter, Davide Parise, Anna Skorobogatova +1

We prove that the singular set of an $m$ -dimensional integral current $T$ in $R^{n + m}$ , semicalibrated by a $C^{2, κ_{0}}$ $m$ -form $ω$ is countably $(m - 2)$ -rectifiable.…

math.AP2024

Unique continuation for nonlinear variational problems

Lorenzo Ferreri, Luca Spolaor, Bozhidar Velichkov

This paper is dedicated to the unique continuation properties of the solutions to nonlinear variational problems. Our analysis covers the case of nonlinear autonomous functionals d…

math.AP2024

On the boundary branching set of the one-phase problem

Lorenzo Ferreri, Luca Spolaor, Bozhidar Velichkov

We consider minimizers of the one-phase Bernoulli free boundary problem in domains with analytic fixed boundary. In any dimension $d$ , we prove that the branching set at the bounda…

math.AP2024

On the fine structure of the solutions to nonlinear thin two-membrane problems in 2D

Lorenzo Ferreri, Luca Spolaor, Bozhidar Velichkov

We prove a structure theorem for the solutions of nonlinear thin two-membrane problems in dimension two. Using the theory of quasi-conformal maps, we show that the difference of th…

math.DG2023

Dimension of the singular set for $2$ -valued stationary Lipschitz graphs

Jonas Hirsch, Luca Spolaor

We prove that the singular set of a $2$ -valued Lipschitz graph that is stationary for the area is of codimension $1$ .

math.AP2023★ 3 cited

Excess decay for minimizing hypercurrents mod $2 Q$

Camillo De Lellis, Jonas Hirsch, Andrea Marchese +2

We consider codimension $1$ area-minimizing $m$ -dimensional currents $T$ mod an even integer $p = 2 Q$ in a $C^{2}$ Riemannian submanifold $Σ$ of the Euclidean space. We prove a suitabl…

math.AP2023

The symmetric (log-)epiperimetric inequality and a decay-growth estimate

Nick Edelen, Luca Spolaor, Bozhidar Velichkov

We introduce a symmetric (log-)epiperimetric inequality, generalizing the standard epiperimetric inequality, and we show that it implies a growth-decay for the associated energy: a…

math.AP2022

Interior regularity for two-dimensional stationary $Q$ -valued maps

Jonas Hirsch, Luca Spolaor

We prove that $2$ -dimensional $Q$ -valued maps that are stationary with respect to outer and inner variations of the Dirichlet energy are Hölder continuous and that the dimension of…

math.AP2022

A strong maximum principle for minimizers of the one-phase Bernoulli problem

Nick Edelen, Luca Spolaor, Bozhidar Velichkov

We prove a strong maximum principle for minimizers of the one-phase Alt-Caffarelli functional. We use this to construct a Hardt-Simon-type foliation associated to any 1-homogenous…

math.AP2022★ 4 cited

Fine structure of the singular set of area minimizing hypersurfaces modulo $p$

Camillo De Lellis, Jonas Hirsch, Andrea Marchese +2

Consider an area minimizing current modulo $p$ of dimension $m$ in a smooth Riemannian manifold of dimension $m + 1$ . We prove that its interior singular set is, up to a relatively c…

math.AP2021

(Quasi-)conformal methods in two-dimensional free boundary problems

Guido De Philippis, Luca Spolaor, Bozhidar Velichkov

In this paper we study the local behavior of solutions to some free boundary problems. We relate the theory of quasi-conformal maps to the regularity of the solutions to nonlinear…

math.AP2021

Rectifiability and almost everywhere uniqueness of the blow-up for the vectorial Bernoulli free boundaries

Guido De Philippis, Max Engelstein, Luca Spolaor +1

We prove that for minimizers of the vectorial Alt-Caffarelli functional the two-phase singular set of the free boundary is rectifiable and the blow-up is unique almost everywhere o…

math.AP2021

Area minimizing hypersurfaces modulo $p$ : a geometric free-boundary problem

Camillo De Lellis, Jonas Hirsch, Andrea Marchese +2

We consider area minimizing $m$ -dimensional currents $mod (p)$ in complete $C^{2}$ Riemannian manifolds $Σ$ of dimension $m + 1$ . For odd moduli we prove that, away from a clos…

math.AP2020

Quantitative Stability for Minimizing Yamabe Metrics

Max Engelstein, Robin Neumayer, Luca Spolaor

On any closed Riemannian manifold of dimension $n \geq 3$ , we prove that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close, in a qua…

math.AP2020★ 1 cited

On the logarithmic epiperimetric inequality for the obstacle problem

Luca Spolaor, Bozhidar Velichkov

We give three different proofs of the log-epiperimetric inequality at singular points for the obstacle problem. In the first, direct proof, we write the competitor explicitly; the…

math.DG2020★ 3 cited

Singular behavior and generic regularity of min-max minimal hypersurfaces

Otis Chodosh, Yevgeny Liokumovich, Luca Spolaor

We show that for a generic $8$ -dimensional Riemannian manifold with positive Ricci curvature, there exists a smooth minimal hypersurface. Without the curvature condition, we show t…

math.AP2019

Regularity of the free boundary for the two-phase Bernoulli problem

Guido De Philippis, Luca Spolaor, Bozhidar Velichkov

We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As…

On Cylindrical singularities of minimal hypersurfaces

Stationary varifolds with singularities II

Fine structure of the two-phase Bernoulli free boundaries in 2D

Measure 0 of the singular set for 2-valued stationary hypercurrents

Stationary and stable varifolds with singularities

Carleman estimates for stationary Q-valued maps: a variational approach

Structure of two-dimensional mod(q) area-minimizing currents near flat singularities: the codimension one case

Rectifiability of the singular set and uniqueness of tangent cones for semicalibrated currents

Unique continuation for nonlinear variational problems

On the boundary branching set of the one-phase problem

On the fine structure of the solutions to nonlinear thin two-membrane problems in 2D

Dimension of the singular set for 2-valued stationary Lipschitz graphs

Excess decay for minimizing hypercurrents mod 2Q

The symmetric (log-)epiperimetric inequality and a decay-growth estimate

Interior regularity for two-dimensional stationary Q-valued maps

A strong maximum principle for minimizers of the one-phase Bernoulli problem

Fine structure of the singular set of area minimizing hypersurfaces modulo p

(Quasi-)conformal methods in two-dimensional free boundary problems

Rectifiability and almost everywhere uniqueness of the blow-up for the vectorial Bernoulli free boundaries

Area minimizing hypersurfaces modulo p: a geometric free-boundary problem

Quantitative Stability for Minimizing Yamabe Metrics

On the logarithmic epiperimetric inequality for the obstacle problem

Singular behavior and generic regularity of min-max minimal hypersurfaces

Regularity of the free boundary for the two-phase Bernoulli problem

Measure $0$ of the singular set for $2$ -valued stationary hypercurrents

Carleman estimates for stationary $Q$ -valued maps: a variational approach

Structure of two-dimensional mod $(q)$ area-minimizing currents near flat singularities: the codimension one case

Dimension of the singular set for $2$ -valued stationary Lipschitz graphs

Excess decay for minimizing hypercurrents mod $2 Q$

Interior regularity for two-dimensional stationary $Q$ -valued maps

Fine structure of the singular set of area minimizing hypersurfaces modulo $p$

Area minimizing hypersurfaces modulo $p$ : a geometric free-boundary problem