# ERC Regularity and Stability in Partial Differential Equations (RSPDE)

## Papers

### 2023

- The Cauchy-Dirichlet Problem for the Fast Diffusion Equation on Bounded Domains (by M. Bonforte, A. Figalli)
- Strong stability of convexity with respect to the perimeter (by A. Figalli, Y. Ru-Ya Zhang)
- Regularity theory for nonlocal obstacle problems with critical and subcritical scaling (by A. Figalli, X. Ros-Oton, J. Serra)
- Regularity Properties of Monotone Measure-preserving Maps (by A. Figalli, Y. Jhaveri)
- Constraint maps with free boundaries: the obstacle case (by A. Figalli, S. Kim, H. Shahgholian)
- Stability of the Faber-Krahn inequality for the Short-time Fourier Transform (by J. Gómez, A. Guerra, J. P. G. Ramos, P. Tilli)
- Sharp Gaussian decay for the one-dimensional harmonic oscillator (by D. Radchenko, J. P. G. Ramos)
- A Faber-Krahn type inequality for log-subharmonic functions in the hyperbolic ball (by D. Kalaj, J. P. G. Ramos)
- A quantitative stability result for the sphere packing problem in dimensions 8 and 24 (by K. J. Böröczky, D. Radchenko, J. P. G. Ramos)
- Quantitative C1-stability of spheres in rank one symmetric spaces of non-compact type (by L. Silini)

### 2022

- Infinite-width limit of deep linear neural networks (by L. Chizat, M. Colombo, X. Fernández-Real, A. Figalli)
- Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence (by J. Dolbeault, M. J. Esteban, A. Figalli, R. L. Frank, M. Loss)
- Uniqueness of solutions to nonlinear Schrödinger equations from their zeros (by Ch. Kehle, J. P. G. Ramos)
- On Gaussian decay rates of harmonic oscillators and equivalences of related Fourier uncertainty principles (by A. Kulikov, L. Oliveira, J. P. G. Ramos)
- The Multilinear Spherical Maximal Function in one dimension (by G. Dosidis, J. P. G. Ramos)
- A Faber-Krahn inequality for wavelet transforms (by J. P. G. Ramos, P. Tilli)
- Local existence of smooth solutions for the semigeostrophic equations on curved domains (by L. Silini)
- Approaching the isoperimetric problem in Hmℂ via the hyperbolic log-convex density conjecture (by L. Silini)
- On the prescribed negative Gauss curvature problem for graphs (by A. Figalli, Ch. Kehle)
- Complete classification of global solutions to the obstacle problem (by S. Eberle, A. Figalli, G. S. Weiss)
- A quantitative stability result for the Prékopa-Leindler inequality for arbitrary measurable functions (by K. J. Böröczky, A. Figalli, J. P. G. Ramos)

### 2021

- Effective dimension of machine learning models (by A. Abbas, D. Sutter, A. Figalli, S. Woerner)
- The singular set in the Stefan problem (by A. Figalli, X. Ros-Oton, J. Serra)
- Free boundary partial regularity in the thin obstacle problem (by F. Franceschini, J. Serra)
- Expansion of the fundamental solution of a second-order elliptic operator with analytic coefficients (by F. Franceschini, F. Glaudo)
- Minkowski inequality for nearly spherical domains (by F. Glaudo)
- Uniform boundedness for finite Morse index solutions to supercritical semilinear elliptic equations (by A. Figalli, Y. Zhang)
- Perturbed Fourier uniqueness and interpolation results in higher dimensions (by J.P.G. Ramos, M. Stoller)
- C∞ partial regularity of the singular set in the obstacle problem (by F. Franceschini, W. Zatoń)

### 2020

- Sharp quantitative stability for isoperimetric inequalities with homogeneous weights (by E. Cinti, F. Glaudo, A. Pratelli, X. Ros-Oton, J. Serra)
- Lipschitz Regularity in Vectorial Linear Transmission Problems (by A. Figalli, S. Kim, H. Shahgholian)
- The power of quantum neural networks (by A. Abbas, D. Sutter, Ch. Zoufal, A. Lucchi, A. Figalli, S. Woerner)
- A scale-dependent notion of effective dimension (by O. Berezniuk, A. Figalli, R. Ghigliazza, K. Musaelian)
- The Thin Obstacle Problem: A Survey (by X. Fernández-Real)
- Stable cones in the thin one-phase problem (by X. Fernández-Real, X. Ros-Oton)
- Singular solutions for fractional parabolic boundary value problems (by H. Chan, D. Gómez-Castro, J. L. Vázquez)
- Lamellar phase solutions for diblock copolymers with nonlocal diffusions (by H. Chan, M. Jamshid Nejad, J. Wei)
- A note on discrete Heisenberg uniqueness pairs for the parabola (by F. Gonçalves, J.P.G. Ramos)
- Bounds for the Lonely Runner Problem via Linear Programming (by F. Gonçalves, J.P.G. Ramos)
- Blow-up phenomena in nonlocal eigenvalue problems: when theories of L1 and L2 meet (by H. Chan, D. Gómez-Castro, J. L. Vázquez)
- Sharp gradient stability for the Sobolev inequality by A. Figalli, Y. Zhang)
- Uniqueness of entire ground states for the fractional plasma problem (by H. Chan, M. del Mar González, Y. Huang, E. Mainini, B. Volzone)
- Duality of capacities and Sobolev extendability in the plane (by Y. Zhang)

### 2019

- Generic regularity of free boundaries for the obstacle problem (by A. Figalli, X. Ros-Oton, J. Serra)
- Stable solutions to semilinear elliptic equations are smooth up to dimension 9 (by X. Cabré, A. Figalli, X. Ros-Oton, J. Serra)
- On the obstacle problem for the 1D wave equation (by X. Fernández-Real, A. Figalli)
- Free boundary regularity for almost every solution to the Signorini problem (by X. Fernández-Real, X. Ros-Oton)
- ODE-methods in non-local equations (by W. Ao, H. Chan, A. DelaTorre, M. A. Fontelos, M. Del Mar González, J. Wei)
- An analytic construction of singular solutions related to a critical Yamabe problem (by H. Chan, A. DelaTorre)
- Ahlfors reflection theorem for
*p*-morphisms (by P. Koskela, P. Pankka, Y. Zhang) - Strong stability for the Wulff inequality with a crystalline norm (by A. Figalli, Y. Zhang)
- Symmetry results for critical anisotropic p-Laplacian equations in convex cones (by G. Ciraolo, A. Figalli, A. Roncoroni)
- On the sharp stability of critical points of the Sobolev inequality (by A. Figalli, F. Glaudo)
- On the optimal map in the 2-dimensional random matching problem (by L. Ambrosio, F. Glaudo, D. Trevisan)
- Regularity of monotone transport maps between unbounded domains (by D. Cordero-Erausquin, A. Figalli)
- Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains (by M. Bonforte, A. Figalli)

### 2018

- On global solutions to semilinear elliptic equations related to the one-phase free boundary problem (by X. Fernández-Real, X. Ros-Oton)
- Partial Regularity of Solutions to the Second Boundary Value Problem for Generated Jacobian Equations (by Y. Jhaveri)
- Bound state solutions for the supercritical fractional Schrödinger equation (by W. Ao, H. Chan, M. del Mar Gonzalez, J. Wei)
- Regularity of minimal surfaces with lower dimensional obstacles (by X. Fernández-Real, J. Serra)
- On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program (by W. Ao, H. Chan, A. DelaTorre, M. A. Fontelos, M. del Mar Gonzalez, J. Wei)
- On the Singular Set in the Thin Obstacle Problem: Higher Order Blow-ups and the Very Thin Obstacle Problem (by X. Fernández-Real, Y. Jhaveri)
- Finer estimates on the 2-dimensional matching problem (by L. Ambrosio, F. Glaudo, D. Trevisan)
- Strong Sard Conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3

(by A. Belotto da Silva, A. Figalli, A. Parusiński, L. Rifford) - Free boundary regularity in obstacle problems (by A. Figalli)
- The Monge-Ampère Equation (by C. Mooney)
- Regularity results for the equation u11u22=1 (by C. Mooney, O. Savin)
- Singularities of complex-valued solutions to linear parabolic equations (by C. Mooney)
- On the Monge-Ampère equation (by A. Figalli)
- On the Continuity of Center-Outward Distribution and Quantile Functions (by A. Figalli)

### 2017

- Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains (by M. Bonforte, A. Figalli, J. L. Vazquez)
- Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations (by M. Bonforte, A. Figalli, J. L. Vazquez)
- A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment (by A. Figalli, M-J. Kang)
- The Lagrangian structure of the Vlasov-Poisson system in domains with specular reflection (by X. Fernández-Real)
- On the fine structure of the free boundary for the classical obstacle problem (by A. Figalli, J. Serra)
- Stable s-minimal cones in ℝ3 are flat for s∼1 (by X. Cabre, E. Cinti, J. Serra)
- On the (In)stability of the Identity Map in Optimal Transportation (by Y. Jhaveri)
- Quantitative stability of the free boundary in the obstacle problem (by S. Serfaty, J. Serra)
- An obstacle problem for conical deformations of thin elastic sheets (by A. Figalli, C. Mooney)
- Improvement of flatness for nonlocal phase transitions (by S. Dipierro, J. Serra, E. Valdinoci)
- On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4+1 (by A. Figalli, J. Serra)
- Dimension of the minimum set for the real and complex Monge-Ampère equations in critical Sobolev spaces (by T. C. Collins, C. Mooney)
- Optimal regularity and structure of the free boundary for minimizers in cohesive zone models (by L. Caffarelli, F. Cagnetti, A. Figalli)
- Regularity theory for general stable operators: parabolic equations (by X. Fernández-Real, X. Ros-Oton)
- The obstacle problem for the fractional Laplacian with critical drift (by X. Fernández-Real, X. Ros-Oton)