Riccardo Tione
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
Riccardo Tione has a total of 18 co-authors. Here are the top co-authors by number of shared publications:
- Andrea Marchese
Universität Zürich
4 shared publications - André Guerra
ETH Zurich, Switzerland
4 shared publications - Annalisa Massaccesi
Dipartimento di Tecnica e Gestione dei Sistemi Industriali Università degli Studi di Padova Italy
4 shared publications - Guido De Philippis
New York University, USA
4 shared publications - Antonio De Rosa
Department of Decision Sciences and BIDSA, Bocconi University, Milano, Italy
3 shared publications - Jonas Hirsch
Mathematisches Institut, Universität Leipzig, Leipzig, Germany
3 shared publications
Latest Publications
- Open Access
- 14 August 2026
- Published by Springer Nature in Inventiones Mathematicae
- Vol. 246 (3) , 925-950
On the Lawson-Osserman conjecture
We prove that if $u : B_{1} \subset \mathbb{R}^{2} \rightarrow \mathbb{R}^{n}$ is a Lipschitz critical point of the area functional with respect to outer variations, then $u$ is smooth. This solves a conjecture of Lawson and Osserman from 1977 in the planar case.- 14 August 2026
- Published by Springer Nature in Inventiones Mathematicae
- Vol. 246 (3) , 925-950
Open AccessOn the Lawson-Osserman conjecture
We prove that if $u : B_{1} \subset \mathbb{R}^{2} \rightarrow \mathbb{R}^{n}$ is a Lipschitz critical point of the area functional with respect to outer variations, then $u$ is smooth. This solves a conjecture of Lawson and Osserman from 1977 in the planar case. - Open Access
Constancy of the index for gradient mappings
We show that if the Hessian of a $C^{1,1}$ function has uniformly positive determinant almost everywhere then its index is locally constant, as conjectured by Šverák in 1992. We deduce this result as a consequence of a more general theorem valid for quasiregular gradient mappings.Open AccessConstancy of the index for gradient mappings
We show that if the Hessian of a $C^{1,1}$ function has uniformly positive determinant almost everywhere then its index is locally constant, as conjectured by Šverák in 1992. We deduce this result as a consequence of a more general theorem valid for quasiregular gradient mappings. - 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.
- 4 November 2024
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 248 (6) , 1-17
Regularity and compactness for critical points of degenerate polyconvex energies
- 4 November 2024
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 248 (6) , 1-17
Regularity and compactness for critical points of degenerate polyconvex energies
- 28 October 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Annales de l'Institut Henri Poincaré C, Analyse non linéaire
- Vol. 43 (1) , 127-154
Unique continuation for differential inclusions
We consider the following question arising in the theory of differential inclusions: Given an elliptic set \Gamma and a Sobolev map u whose gradient lies in the quasiconformal envelope of \Gamma and touches \Gamma on a set of positive measure, must u be affine? We answer this question positively for a suitable notion of ellipticity, which for instance encompasses the case where \Gamma \subset \mathbb{R}^{2\times 2} is an elliptic, smooth, closed curve. More precisely, we prove that the distance of \mathrm{D}u to \Gamma satisfies the strong unique continuation property. As a by-product, we obtain new results for non-linear Beltrami equations and recover known results for the reduced Beltrami equation and the Monge–Ampère equation: concerning the latter, we obtain a new proof of the W^{2,1+\varepsilon} -regularity for two-dimensional solutions.
- 28 October 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Annales de l'Institut Henri Poincaré C, Analyse non linéaire
- Vol. 43 (1) , 127-154
Unique continuation for differential inclusions
We consider the following question arising in the theory of differential inclusions: Given an elliptic set \Gamma and a Sobolev map u whose gradient lies in the quasiconformal envelope of \Gamma and touches \Gamma on a set of positive measure, must u be affine? We answer this question positively for a suitable notion of ellipticity, which for instance encompasses the case where \Gamma \subset \mathbb{R}^{2\times 2} is an elliptic, smooth, closed curve. More precisely, we prove that the distance of \mathrm{D}u to \Gamma satisfies the strong unique continuation property. As a by-product, we obtain new results for non-linear Beltrami equations and recover known results for the reduced Beltrami equation and the Monge–Ampère equation: concerning the latter, we obtain a new proof of the W^{2,1+\varepsilon} -regularity for two-dimensional solutions.
- 23 April 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Journal of the European Mathematical Society
- Vol. 27 (12) , 4845-4890
Non-classical solutions of the $p$-Laplace equation
In this paper we settle Iwaniec and Sbordone’s 1994 conjecture concerning very weak solutions to the p -Laplace equation. Namely, on the one hand we show that distributional solutions of the p -Laplace equation in W^{1,r} for p \neq 2 and r>\max\,\{ 1,p-1\} are classical weak solutions if their weak derivatives belong to certain cones. On the other hand, we construct via convex integration non-energetic distributional solutions if this cone condition is not met, thus disproving Iwaniec and Sbordone’s conjecture in general.
- 23 April 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Journal of the European Mathematical Society
- Vol. 27 (12) , 4845-4890
Non-classical solutions of the $p$-Laplace equation
In this paper we settle Iwaniec and Sbordone’s 1994 conjecture concerning very weak solutions to the p -Laplace equation. Namely, on the one hand we show that distributional solutions of the p -Laplace equation in W^{1,r} for p \neq 2 and r>\max\,\{ 1,p-1\} are classical weak solutions if their weak derivatives belong to certain cones. On the other hand, we construct via convex integration non-energetic distributional solutions if this cone condition is not met, thus disproving Iwaniec and Sbordone’s conjecture in general.
- Open Access
Regularity and compactness for critical points of degenerate polyconvex energies
We study Lipschitz critical points of the energy $\int_Ωg(\det D u) \, d x$ in two dimensions, where $g$ is a strictly convex function. We prove that the Jacobian of any Lipschitz critical point is constant, and that the Jacobians of sequences of approximately critical points converge strongly. The latter result answers in particular an open problem posed by Kirchheim, Müller and Šverák in 2003.Open AccessRegularity and compactness for critical points of degenerate polyconvex energies
We study Lipschitz critical points of the energy $\int_Ωg(\det D u) \, d x$ in two dimensions, where $g$ is a strictly convex function. We prove that the Jacobian of any Lipschitz critical point is constant, and that the Jacobians of sequences of approximately critical points converge strongly. The latter result answers in particular an open problem posed by Kirchheim, Müller and Šverák in 2003. - Open Access
Unique continuation for differential inclusions
We consider the following question arising in the theory of differential inclusions: given an elliptic set $Γ$ and a Sobolev map $u$ whose gradient lies in the quasiconformal envelope of $Γ$ and touches $Γ$ on a set of positive measure, must $u$ be affine? We answer this question positively for a suitable notion of ellipticity, which for instance encompasses the case where $Γ\subset \mathbb R^{2\times 2}$ is an elliptic, smooth, closed curve. More precisely, we prove that the distance of $D u$ to $Γ$ satisfies the strong unique continuation property. As a by-product, we obtain new results for nonlinear Beltrami equations and recover known results for the reduced Beltrami equation and the Monge--Ampère equation: concerning the latter, we obtain a new proof of the $W^{2,1+\varepsilon}$-regularity for two-dimensional solutions.Open AccessUnique continuation for differential inclusions
We consider the following question arising in the theory of differential inclusions: given an elliptic set $Γ$ and a Sobolev map $u$ whose gradient lies in the quasiconformal envelope of $Γ$ and touches $Γ$ on a set of positive measure, must $u$ be affine? We answer this question positively for a suitable notion of ellipticity, which for instance encompasses the case where $Γ\subset \mathbb R^{2\times 2}$ is an elliptic, smooth, closed curve. More precisely, we prove that the distance of $D u$ to $Γ$ satisfies the strong unique continuation property. As a by-product, we obtain new results for nonlinear Beltrami equations and recover known results for the reduced Beltrami equation and the Monge--Ampère equation: concerning the latter, we obtain a new proof of the $W^{2,1+\varepsilon}$-regularity for two-dimensional solutions. - Open Access
On singular strictly convex solutions to the Monge-Ampère equation
We show the existence of a strictly convex function $u: B_1 \to \mathbb{R}$ with associated Monge-Ampère measure represented by a function $f$ with $0 < f < 1$ a.e. whose Hessian has a singular part. This extends the work [13] and answers an open question of [14,Sec. 6.2(1)].Open AccessOn singular strictly convex solutions to the Monge-Ampère equation
We show the existence of a strictly convex function $u: B_1 \to \mathbb{R}$ with associated Monge-Ampère measure represented by a function $f$ with $0 < f < 1$ a.e. whose Hessian has a singular part. This extends the work [13] and answers an open question of [14,Sec. 6.2(1)]. - 14 September 2023
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 151 (12) , 5283-5297
Improved regularity of second derivatives for subharmonic functions
In this note, we prove that if a subharmonic function Δ u ≥ 0 \Delta u\ge 0 has pure second derivatives ∂ i i u \partial _{ii} u that are signed measures, then their negative part ( ∂ i i u ) − (\partial _{ii} u)_- belongs to L 1 L^1 (in particular, it is not singular). We then show that this improvement of regularity cannot be upgraded to L p L^p for any p > 1 p > 1 . We finally relate this problem to a natural question on the one-sided regularity of solutions to the obstacle problem with rough obstacles.
- 14 September 2023
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 151 (12) , 5283-5297
Improved regularity of second derivatives for subharmonic functions
In this note, we prove that if a subharmonic function Δ u ≥ 0 \Delta u\ge 0 has pure second derivatives ∂ i i u \partial _{ii} u that are signed measures, then their negative part ( ∂ i i u ) − (\partial _{ii} u)_- belongs to L 1 L^1 (in particular, it is not singular). We then show that this improvement of regularity cannot be upgraded to L p L^p for any p > 1 p > 1 . We finally relate this problem to a natural question on the one-sided regularity of solutions to the obstacle problem with rough obstacles.