Scientists / Open questions

阿莱西奥·菲加利

Alessio Figalli

最优输运、偏微分方程、变分法

2018 年菲尔兹奖

28 个问题

Minimal kernel assumptions and the local limit $s\to\tfrac12^-$ for the sharp stability theory of hypersurfaces with almost-constant nonlocal mean curvature
Criticality of $C^1$ -topology minimax levels for the Tonelli action and integer-coefficient multiplicity for superlinear Hamiltonians with conormal boundary conditions
Mather measures, Aubry-set structure, and convergence rates for the weak KAM theory of deterministic optimal switching
Optimal stability of the critical two-dimensional Gagliardo-Nirenberg-Sobolev inequality in density metrics, and sharp convergence rates for the critical-mass Keller-Segel equation
Failure of semiconvexity of the squared sub-Riemannian distance at the cut locus in the presence of abnormal minimizing geodesics
Sharp conditions for proper blow-up and global existence of maximal regular flows for non-smooth vector fields
Geometric lower bounds on Grad's number in spatial dimensions four and higher
Well-posedness of the continuity equation in the class $L^\infty_+([0,T];L^1\cap L^\infty(\mathbb{R}^d))$ for vector fields beyond the Sobolev, BV, and repulsive-Hamiltonian classes
Quantitative dynamics of multi-bubble regimes for the Sobolev-critical fast diffusion equation and Yamabe-type flows
Fine regularity and the singular set of optimal transport maps for the squared-distance cost on Riemannian manifolds
Removing the finite-perimeter hypothesis from the Figalli-Jerison convexity-from-marginals theorem
Global $W^{2,p}$ regularity for optimal transport maps with costs far from quadratic, and the structure of the singular set in the partial $W^{2,p}$ regularity theory
Constructive derivation of the porosity threshold ε(n) and of the explicit higher-integrability constants for Mumford–Shah minimizers
Removing the center-pinning condition $u(0)=0$ in the $\Gamma$ -convergence derivation of the d-cone obstacle problem for thin elastic sheets
Villani's conjecture: convexity of injectivity domains on compact Riemannian manifolds satisfying the Ma–Trudinger–Wang condition
Structure of global minimizers of the interaction energy for mildly repulsive power-law potentials in the intermediate regime below the simplex threshold
Fine structure of the free boundary at vanishing-gradient points in the p-Laplacian obstacle problem
Continuity of optimal transport maps on Riemannian manifolds: the (CI)+(MTW) characterization beyond dimension two, transport continuity on products of small perturbations of the 2-sphere, and the (MTW) versus $(\mathrm{MTW}^{\not\perp})$ hierarchy
Sharp form of the dependence of interior $C^{2,α}$ Monge–Ampère estimates on $α$ and on the $C^α$ norm of the right-hand side
Up-to-boundary regularity of the convex envelope: minimal hypotheses without uniform convexity or high-order boundary smoothness, and the boundary-regularity versus semiconcavity trade-off for general moduli
Fine structure of the singular set of optimal transport maps for general costs and on Riemannian manifolds
Is the center-outward median set a single point in dimension $d \ge 3$ ? (Removability of the origin singularity of the optimal-transport quantile map)
Fine-scale regularity of constraint maps with free boundaries in the obstacle case: the geometric approximation property in dimensions $n\ge3$ , optimal regularity of the image map, vectorial fine structure, analyticity, and leaky maps
Semiclassical limit of quantum dynamics with rough Born-Oppenheimer potentials: regularity of molecular potential surfaces for general molecules and deterministic initial data

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Minimal kernel assumptions and the local limit s→12−s\to\tfrac12^-s→21​− for the sharp stability theory of hypersurfaces with almost-constant nonlocal mean curvature

Criticality of C1C^1C1-topology minimax levels for the Tonelli action and integer-coefficient multiplicity for superlinear Hamiltonians with conormal boundary conditions

Mather measures, Aubry-set structure, and convergence rates for the weak KAM theory of deterministic optimal switching

Optimal stability of the critical two-dimensional Gagliardo-Nirenberg-Sobolev inequality in density metrics, and sharp convergence rates for the critical-mass Keller-Segel equation

Failure of semiconvexity of the squared sub-Riemannian distance at the cut locus in the presence of abnormal minimizing geodesics

Sharp conditions for proper blow-up and global existence of maximal regular flows for non-smooth vector fields

Geometric lower bounds on Grad's number in spatial dimensions four and higher

Well-posedness of the continuity equation in the class L+∞([0,T];L1∩L∞(Rd))L^\infty_+([0,T];L^1\cap L^\infty(\mathbb{R}^d))L+∞​([0,T];L1∩L∞(Rd)) for vector fields beyond the Sobolev, BV, and repulsive-Hamiltonian classes

Quantitative dynamics of multi-bubble regimes for the Sobolev-critical fast diffusion equation and Yamabe-type flows

Fine regularity and the singular set of optimal transport maps for the squared-distance cost on Riemannian manifolds

Removing the finite-perimeter hypothesis from the Figalli-Jerison convexity-from-marginals theorem

Global W2,pW^{2,p}W2,p regularity for optimal transport maps with costs far from quadratic, and the structure of the singular set in the partial W2,pW^{2,p}W2,p regularity theory

Constructive derivation of the porosity threshold ε(n) and of the explicit higher-integrability constants for Mumford–Shah minimizers

Removing the center-pinning condition u(0)=0u(0)=0u(0)=0 in the Γ\GammaΓ-convergence derivation of the d-cone obstacle problem for thin elastic sheets

Villani's conjecture: convexity of injectivity domains on compact Riemannian manifolds satisfying the Ma–Trudinger–Wang condition

Structure of global minimizers of the interaction energy for mildly repulsive power-law potentials in the intermediate regime below the simplex threshold

Fine structure of the free boundary at vanishing-gradient points in the p-Laplacian obstacle problem

Continuity of optimal transport maps on Riemannian manifolds: the (CI)+(MTW) characterization beyond dimension two, transport continuity on products of small perturbations of the 2-sphere, and the (MTW) versus (MTW⊥̸)(\mathrm{MTW}^{\not\perp})(MTW⊥) hierarchy

Sharp form of the dependence of interior C2,αC^{2,α}C2,α Monge–Ampère estimates on ααα and on the CαC^αCα norm of the right-hand side

Up-to-boundary regularity of the convex envelope: minimal hypotheses without uniform convexity or high-order boundary smoothness, and the boundary-regularity versus semiconcavity trade-off for general moduli

Fine structure of the singular set of optimal transport maps for general costs and on Riemannian manifolds

Is the center-outward median set a single point in dimension d≥3d \ge 3d≥3? (Removability of the origin singularity of the optimal-transport quantile map)

Fine-scale regularity of constraint maps with free boundaries in the obstacle case: the geometric approximation property in dimensions n≥3n\ge3n≥3, optimal regularity of the image map, vectorial fine structure, analyticity, and leaky maps

Semiclassical limit of quantum dynamics with rough Born-Oppenheimer potentials: regularity of molecular potential surfaces for general molecules and deterministic initial data

Minimal kernel assumptions and the local limit $s\to\tfrac12^-$ for the sharp stability theory of hypersurfaces with almost-constant nonlocal mean curvature

Criticality of $C^1$ -topology minimax levels for the Tonelli action and integer-coefficient multiplicity for superlinear Hamiltonians with conormal boundary conditions

Well-posedness of the continuity equation in the class $L^\infty_+([0,T];L^1\cap L^\infty(\mathbb{R}^d))$ for vector fields beyond the Sobolev, BV, and repulsive-Hamiltonian classes

Global $W^{2,p}$ regularity for optimal transport maps with costs far from quadratic, and the structure of the singular set in the partial $W^{2,p}$ regularity theory

Removing the center-pinning condition $u(0)=0$ in the $\Gamma$ -convergence derivation of the d-cone obstacle problem for thin elastic sheets

Continuity of optimal transport maps on Riemannian manifolds: the (CI)+(MTW) characterization beyond dimension two, transport continuity on products of small perturbations of the 2-sphere, and the (MTW) versus $(\mathrm{MTW}^{\not\perp})$ hierarchy

Sharp form of the dependence of interior $C^{2,α}$ Monge–Ampère estimates on $α$ and on the $C^α$ norm of the right-hand side

Is the center-outward median set a single point in dimension $d \ge 3$ ? (Removability of the origin singularity of the optimal-transport quantile map)

Fine-scale regularity of constraint maps with free boundaries in the obstacle case: the geometric approximation property in dimensions $n\ge3$ , optimal regularity of the image map, vectorial fine structure, analyticity, and leaky maps