Scientists / Open questions

皮埃尔-路易·利翁

Pierre-Louis Lions

非线性偏微分方程、数学物理

1994 年菲尔兹奖

46 个问题

Comparison and uniqueness for viscosity solutions of first-order Hamilton-Jacobi equations in Banach spaces without smooth norms: the Crandall-Lions problem for $L^1$ , $L^\infty$ , spaces of continuous functions, and spaces of measures
The unresolved core of the Crandall-Lions existence theory for viscosity solutions of Hamilton-Jacobi equations in infinite dimensions: convergent finite-dimensional approximation, non-smooth Banach spaces, boundary value problems, and growth at infinity
Well-posedness theory, microstructure-faithful extensions, and empirical validation of the mean-field-game model of price formation with high-frequency participants
Renormalized solutions without defect measure for the non-cutoff Boltzmann equation: the soft-potential regime
Lions’ relaxed-positivity conjecture for the real Monge-Ampère Dirichlet problem: $W^{2,\infty}$ solutions in dimensions $N\ge 3$ under smooth degeneracy of the right-hand side
Extending the Lions–Souganidis weak-convergence proof of stochastic homogenization to fully nonlinear second-order equations with convex nonlinearities, and resolving the degenerate invariant-measure difficulty for its variational formula
Infinitesimal generators of the Alvarez–Guichard–Lions–Morel axiomatic scale spaces: existence without a regularity axiom, and how far it can be weakened
Global weak existence for the Doi rod model outside the free-energy dissipation regime $b>-\frac{N}{N-1}\nu$
Comparison and well-posedness for fully nonlinear stochastic PDEs with general spatially dependent nonlinear noise (the Lions-Souganidis program)
Is the coercivity-type condition (1.13) of Lions–Souganidis (2005) necessary for homogenization of almost periodic degenerate quasilinear equations when $N>1$ ?
Open core of the stochastic regularization program for pathwise Hamilton–Jacobi equations: the law of the asymptotic random constant $u∞$ , $C^{1,1}$ regularization in $d>1$ , and exceptional time sets beyond Brownian driving paths
Global convergence of the eductive forward-relaxation dynamics for stationary mean field game systems
Quantitative error analysis for Crandall–Lions monotone approximations of mean curvature motion: sharp rates, practical parameter couplings, and the nonlocal equation
Open problems from Lions's 1985 memoir 'Sur les équations de Monge-Ampère': the Monge–Ampère obstacle problem, non-convex domains, viscosity solutions, first eigenvalue, uniqueness, and boundary regularity
Boundedness and infinite-time blow-up of global solutions of the superlinear heat equation at and above the critical Sobolev exponent
Necessary and sufficient conditions for convexity preservation beyond $F(x,D^2u)$ : the Lions–Musiela extension program for parabolic equations (general dependence, obstacle, elliptic, and nonlocal equations, boundary conditions, partial convexity, and joint state–strike convexity)
Variance-optimal Malliavin weights and localization for Monte Carlo Greeks in general diffusion models
Well-posedness of the local-profile (defect corrector) equation for a periodic elliptic coefficient perturbed by a defect that merely vanishes at infinity: the $r=\infty$ case
Precise local regularity and asymptotic profile of multimeron solutions at boundary sign-change points of the Brezis–Lions degenerate elliptic equation
Uniqueness of the stationary equilibrium in the continuous-time Aiyagari-Bewley-Huggett model under conditions weaker than IES >= 1
Comparison and uniqueness of pathwise viscosity solutions for fully nonlinear stochastic PDEs with non-smooth (difference-of-convex) Hamiltonians
Well-posedness and completion of the Hamilton–Jacobi theory for the Achdou–Giraud–Lasry–Lions long-term mining model
Optimal regularity of velocity averages for the stationary transport equation with general velocity measures: sharp conditions on the measure, endpoint behavior, and compactness classification (open questions of Golse-Lions-Perthame-Sentis, 1988)
Rigorous convergence of molecular models to continuum mechanics beyond perfect periodic zero-temperature crystals: defects, free-surface boundary layers, and temperature

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Comparison and uniqueness for viscosity solutions of first-order Hamilton-Jacobi equations in Banach spaces without smooth norms: the Crandall-Lions problem for L1L^1L1, L∞L^\inftyL∞, spaces of continuous functions, and spaces of measures

The unresolved core of the Crandall-Lions existence theory for viscosity solutions of Hamilton-Jacobi equations in infinite dimensions: convergent finite-dimensional approximation, non-smooth Banach spaces, boundary value problems, and growth at infinity

Well-posedness theory, microstructure-faithful extensions, and empirical validation of the mean-field-game model of price formation with high-frequency participants

Renormalized solutions without defect measure for the non-cutoff Boltzmann equation: the soft-potential regime

Lions’ relaxed-positivity conjecture for the real Monge-Ampère Dirichlet problem: W2,∞W^{2,\infty}W2,∞ solutions in dimensions N≥3N\ge 3N≥3 under smooth degeneracy of the right-hand side

Extending the Lions–Souganidis weak-convergence proof of stochastic homogenization to fully nonlinear second-order equations with convex nonlinearities, and resolving the degenerate invariant-measure difficulty for its variational formula

Infinitesimal generators of the Alvarez–Guichard–Lions–Morel axiomatic scale spaces: existence without a regularity axiom, and how far it can be weakened

Global weak existence for the Doi rod model outside the free-energy dissipation regime b>−NN−1νb>-\frac{N}{N-1}\nub>−N−1N​ν

Comparison and well-posedness for fully nonlinear stochastic PDEs with general spatially dependent nonlinear noise (the Lions-Souganidis program)

Is the coercivity-type condition (1.13) of Lions–Souganidis (2005) necessary for homogenization of almost periodic degenerate quasilinear equations when N>1N>1N>1?

Open core of the stochastic regularization program for pathwise Hamilton–Jacobi equations: the law of the asymptotic random constant u∞u∞u∞, C1,1C^{1,1}C1,1 regularization in d>1d>1d>1, and exceptional time sets beyond Brownian driving paths

Global convergence of the eductive forward-relaxation dynamics for stationary mean field game systems

Quantitative error analysis for Crandall–Lions monotone approximations of mean curvature motion: sharp rates, practical parameter couplings, and the nonlocal equation

Open problems from Lions's 1985 memoir 'Sur les équations de Monge-Ampère': the Monge–Ampère obstacle problem, non-convex domains, viscosity solutions, first eigenvalue, uniqueness, and boundary regularity

Boundedness and infinite-time blow-up of global solutions of the superlinear heat equation at and above the critical Sobolev exponent

Necessary and sufficient conditions for convexity preservation beyond F(x,D2u)F(x,D^2u)F(x,D2u): the Lions–Musiela extension program for parabolic equations (general dependence, obstacle, elliptic, and nonlocal equations, boundary conditions, partial convexity, and joint state–strike convexity)

Variance-optimal Malliavin weights and localization for Monte Carlo Greeks in general diffusion models

Well-posedness of the local-profile (defect corrector) equation for a periodic elliptic coefficient perturbed by a defect that merely vanishes at infinity: the r=∞r=\inftyr=∞ case

Precise local regularity and asymptotic profile of multimeron solutions at boundary sign-change points of the Brezis–Lions degenerate elliptic equation

Uniqueness of the stationary equilibrium in the continuous-time Aiyagari-Bewley-Huggett model under conditions weaker than IES >= 1

Comparison and uniqueness of pathwise viscosity solutions for fully nonlinear stochastic PDEs with non-smooth (difference-of-convex) Hamiltonians

Well-posedness and completion of the Hamilton–Jacobi theory for the Achdou–Giraud–Lasry–Lions long-term mining model

Optimal regularity of velocity averages for the stationary transport equation with general velocity measures: sharp conditions on the measure, endpoint behavior, and compactness classification (open questions of Golse-Lions-Perthame-Sentis, 1988)

Rigorous convergence of molecular models to continuum mechanics beyond perfect periodic zero-temperature crystals: defects, free-surface boundary layers, and temperature

Comparison and uniqueness for viscosity solutions of first-order Hamilton-Jacobi equations in Banach spaces without smooth norms: the Crandall-Lions problem for $L^1$ , $L^\infty$ , spaces of continuous functions, and spaces of measures

Lions’ relaxed-positivity conjecture for the real Monge-Ampère Dirichlet problem: $W^{2,\infty}$ solutions in dimensions $N\ge 3$ under smooth degeneracy of the right-hand side

Global weak existence for the Doi rod model outside the free-energy dissipation regime $b>-\frac{N}{N-1}\nu$

Is the coercivity-type condition (1.13) of Lions–Souganidis (2005) necessary for homogenization of almost periodic degenerate quasilinear equations when $N>1$ ?

Open core of the stochastic regularization program for pathwise Hamilton–Jacobi equations: the law of the asymptotic random constant $u∞$ , $C^{1,1}$ regularization in $d>1$ , and exceptional time sets beyond Brownian driving paths

Necessary and sufficient conditions for convexity preservation beyond $F(x,D^2u)$ : the Lions–Musiela extension program for parabolic equations (general dependence, obstacle, elliptic, and nonlocal equations, boundary conditions, partial convexity, and joint state–strike convexity)

Well-posedness of the local-profile (defect corrector) equation for a periodic elliptic coefficient perturbed by a defect that merely vanishes at infinity: the $r=\infty$ case