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

## Papers

### 2022

1. Uniqueness of solutions to nonlinear Schrödinger equations from their zeros (by Ch. Kehle, J. P. G. Ramos)
2. On Gaussian decay rates of harmonic oscillators and equivalences of related Fourier uncertainty principles (by A. Kulikov, L. Oliveira, J. P. G. Ramos)
3. The Multilinear Spherical Maximal Function in one dimension (by G. Dosidis, J. P. G. Ramos)
4. A Faber-Krahn inequality for wavelet transforms (by J. P. G. Ramos, P. Tilli)
5. Local existence of smooth solutions for the semigeostrophic equations on curved domains (by L. Silini)
6. Approaching the isoperimetric problem in Hmℂ via the hyperbolic log-convex density conjecture (by L. Silini)
7. On the prescribed negative Gauss curvature problem for graphs (by A. Figalli, Ch. Kehle)
8. Complete classification of global solutions to the obstacle problem (by S. Eberle, A. Figalli, G. S. Weiss)
9. A quantitative stability result for the Prékopa-Leindler inequality for arbitrary measurable functions (by K. J. Böröczky, A. Figalli, J. P. G. Ramos)

### 2021

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

### 2020

1. Sharp quantitative stability for isoperimetric inequalities with homogeneous weights (by E. Cinti, F. Glaudo, A. Pratelli, X. Ros-Oton, J. Serra)
2. Lipschitz Regularity in Vectorial Linear Transmission Problems (by A. Figalli, S. Kim, H. Shahgholian)
3. The power of quantum neural networks (by A. Abbas, D. Sutter, Ch. Zoufal, A. Lucchi, A. Figalli, S. Woerner)
4. A scale-dependent notion of effective dimension (by O. Berezniuk, A. Figalli, R. Ghigliazza, K. Musaelian)
5. The Thin Obstacle Problem: A Survey (by X. Fernández-Real)
6. Stable cones in the thin one-phase problem (by X. Fernández-Real, X. Ros-Oton)
7. Singular solutions for fractional parabolic boundary value problems (by H. Chan, D. Gómez-Castro, J. L. Vázquez)
8. Lamellar phase solutions for diblock copolymers with nonlocal diffusions (by H. Chan, M. Jamshid Nejad, J. Wei)
9. A note on discrete Heisenberg uniqueness pairs for the parabola (by F. Gonçalves, J.P.G. Ramos)
10. ODE-methods in non-local equations (by W. Ao, H. Chan, A. DelaTorre, M. A. Fontelos, M. Del Mar González, J. Wei)
11. Bounds for the Lonely Runner Problem via Linear Programming (by F. Gonçalves, J.P.G. Ramos)
12. 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)
13. Ahlfors reflection theorem for p-morphisms (by P. Koskela, P. Pankka, Y. Zhang)
14. Blow-up phenomena in nonlocal eigenvalue problems: when theories of L1 and L2 meet (by H. Chan, D. Gómez-Castro, J. L. Vázquez)
15. An analytic construction of singular solutions related to a critical Yamabe problem (by H. Chan, A. DelaTorre)
16. Sharp gradient stability for the Sobolev inequality by A. Figalli, Y. Zhang)
17. Optimal regularity and structure of the free boundary for minimizers in cohesive zone models (by L. Caffarelli, F. Cagnetti, A. Figalli)
18. Uniqueness of entire ground states for the fractional plasma problem (by H. Chan, M. del Mar González, Y. Huang, E. Mainini, B. Volzone)
19. Duality of capacities and Sobolev extendability in the plane (by Y. Zhang)

### 2019

1. Generic regularity of free boundaries for the obstacle problem (by A. Figalli, X. Ros-Oton, J. Serra)
2. Stable solutions to semilinear elliptic equations are smooth up to dimension 9 (by X. Cabré, A. Figalli, X. Ros-Oton, J. Serra)
3. On the obstacle problem for the 1D wave equation (by X. Fernández-Real, A. Figalli)
4. Free boundary regularity for almost every solution to the Signorini problem (by X. Fernández-Real, X. Ros-Oton)
5. 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)
6. Regularity of minimal surfaces with lower dimensional obstacles (by X. Fernández-Real, J. Serra)
7. Strong stability for the Wulff inequality with a crystalline norm (by A. Figalli, Y. Zhang)
8. Symmetry results for critical anisotropic p-Laplacian equations in convex cones (by G. Ciraolo, A. Figalli, A. Roncoroni)
9. On the sharp stability of critical points of the Sobolev inequality (by A. Figalli, F. Glaudo)
10. On the optimal map in the 2-dimensional random matching problem (by L. Ambrosio, F. Glaudo, D. Trevisan)
11. Regularity of monotone transport maps between unbounded domains (by D. Cordero-Erausquin, A. Figalli)
12. Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains (by M. Bonforte, A. Figalli)
13. Bound state solutions for the supercritical fractional Schrödinger equation (by W. Ao, H. Chan, M. del Mar Gonzalez, J. Wei)

### 2018

1. On global solutions to semilinear elliptic equations related to the one-phase free boundary problem (by X. Fernández-Real, X. Ros-Oton)
2. Partial Regularity of Solutions to the Second Boundary Value Problem for Generated Jacobian Equations (by Y. Jhaveri)
3. Finer estimates on the 2-dimensional matching problem (by L. Ambrosio, F. Glaudo, D. Trevisan)
4. 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)
5. A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment (by A. Figalli, M-J. Kang)
6. The Lagrangian structure of the Vlasov-Poisson system in domains with specular reflection (by X. Fernández-Real)
7. Free boundary regularity in obstacle problems (by A. Figalli)
8. The Monge-Ampère Equation (by C. Mooney)
9. Regularity results for the equation u11u22=1 (by C. Mooney, O. Savin)
10. Singularities of complex-valued solutions to linear parabolic equations (by C. Mooney)
11. On the Monge-Ampère equation (by A. Figalli)
12. On the Continuity of Center-Outward Distribution and Quantile Functions (by A. Figalli)
13. Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations (by M. Bonforte, A. Figalli, J. L. Vazquez)

### 2017

1. Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains (by M. Bonforte, A. Figalli, J. L. Vazquez)
2. On the fine structure of the free boundary for the classical obstacle problem (by A. Figalli, J. Serra)
3. Stable s-minimal cones in ℝ3 are flat for s∼1 (by X. Cabre, E. Cinti, J. Serra)
4. On the (In)stability of the Identity Map in Optimal Transportation (by Y. Jhaveri)
5. Quantitative stability of the free boundary in the obstacle problem (by S. Serfaty, J. Serra)
6. An obstacle problem for conical deformations of thin elastic sheets (by A. Figalli, C. Mooney)
7. Improvement of flatness for nonlocal phase transitions (by S. Dipierro, J. Serra, E. Valdinoci)
8. On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4+1 (by A. Figalli, J. Serra)
9. Dimension of the minimum set for the real and complex Monge-Ampère equations in critical Sobolev spaces (by T. C. Collins, C. Mooney)
10. Regularity theory for general stable operators: parabolic equations (by X. Fernández-Real, X. Ros-Oton)
11. The obstacle problem for the fractional Laplacian with critical drift (by X. Fernández-Real, X. Ros-Oton)