Scientists / Open questions

Pierre-Louis Lions

Nonlinear partial differential equations; mathematical physics

1994 Fields Medal

46 questions

Equivalence of the Crandall-Lions 0-centered and D(A)-centered formulations of viscosity solutions for non-self-adjoint maximal monotone linear operators in Hilbert spaces
Does the Euler-Lagrange condition for variance-optimal Malliavin Monte-Carlo weights force the minimizer to be a function of the payoff variable?
Regularity and structure of the representing matrix measure in Lions's identification of the dual cone of convex functions
Vanishing-regularization limit of the Catté–Lions–Morel–Coll selective-smoothing equation and well-posedness of the unregularized Perona–Malik equation
Moment finiteness at the critical correlation in power-law stochastic volatility models
Well-posedness of scalar conservation laws with semilinear multiplicative rough-path forcing (Lions–Perthame–Souganidis problem (9))
Rigorous foundations of the Lasry–Lions self-consistent volatility theory: N-investor limit and well-posedness
The convergence problem for finite-horizon mean field games: N-player Nash equilibria in the general (non-monotone) Lasry–Lions framework
Uniqueness classification and well-posedness of the infinite-horizon stochastic differential utility PDE in the superlinear regime (mu < -1) and under degenerate or unbounded coefficients
Thermodynamic limits for microscopic stochastic lattices under minimal hypotheses: point nuclei, weak geometric bounds, and increment-stationarity
Rigorous foundations for the Lasry–Lions (1999) partial-information stochastic control program in finance: controlled Zakai HJB equations and endogenous costly observation
Pathwise well-posedness of fully nonlinear stochastic Hamilton–Jacobi equations with spatially inhomogeneous noise and nonsmooth Hamiltonians
General influence laws and full asymptotics for the Lions–Lasry large-investor price-impact model
Shifted-center radial test functions in the definition of B-continuous viscosity solutions for Hamilton-Jacobi equations with unbounded linear terms in Hilbert spaces
Uniqueness without star-shapedness for the Crandall–Lions–Souganidis model problem $u_t-\Delta u+|\nabla u|^p=0$ with infinite initial data on a subdomain ( $1<p<2$ )
Discrete-to-continuum limits for random atomistic lattices: quantum (Thomas–Fermi-type) elastic energies and survival of lattice symmetry
Correctors, gradient-dependent diffusion, and quantitative rates for the stochastic homogenization of viscous Hamilton–Jacobi equations in stationary ergodic media
Rigorous long-time asymptotics for the optimal control of conditioned processes (the Achdou–Laurière–Lions ansatz)
Diffusive limit of the generalized Carleman system with superlinear density-dependent collision rate $ρ^α$ for $1 < α < 2$
Characterization of the convex-symmetrization comparison function for elliptic problems with non-smooth anisotropic gauge functions
Explicit characterization of the convex envelope of the minimum of the horizontal and vertical total variations
Uniqueness and selection of equilibria in the stationary mean field game system of Lasry and Lions beyond the monotonicity condition

Back to open problems

Equivalence of the Crandall-Lions 0-centered and D(A)-centered formulations of viscosity solutions for non-self-adjoint maximal monotone linear operators in Hilbert spaces

Does the Euler-Lagrange condition for variance-optimal Malliavin Monte-Carlo weights force the minimizer to be a function of the payoff variable?

Regularity and structure of the representing matrix measure in Lions's identification of the dual cone of convex functions

Vanishing-regularization limit of the Catté–Lions–Morel–Coll selective-smoothing equation and well-posedness of the unregularized Perona–Malik equation

Moment finiteness at the critical correlation in power-law stochastic volatility models

Well-posedness of scalar conservation laws with semilinear multiplicative rough-path forcing (Lions–Perthame–Souganidis problem (9))

Rigorous foundations of the Lasry–Lions self-consistent volatility theory: N-investor limit and well-posedness

The convergence problem for finite-horizon mean field games: N-player Nash equilibria in the general (non-monotone) Lasry–Lions framework

Uniqueness classification and well-posedness of the infinite-horizon stochastic differential utility PDE in the superlinear regime (mu < -1) and under degenerate or unbounded coefficients

Thermodynamic limits for microscopic stochastic lattices under minimal hypotheses: point nuclei, weak geometric bounds, and increment-stationarity

Rigorous foundations for the Lasry–Lions (1999) partial-information stochastic control program in finance: controlled Zakai HJB equations and endogenous costly observation

Pathwise well-posedness of fully nonlinear stochastic Hamilton–Jacobi equations with spatially inhomogeneous noise and nonsmooth Hamiltonians

General influence laws and full asymptotics for the Lions–Lasry large-investor price-impact model

Shifted-center radial test functions in the definition of B-continuous viscosity solutions for Hamilton-Jacobi equations with unbounded linear terms in Hilbert spaces

Uniqueness without star-shapedness for the Crandall–Lions–Souganidis model problem ut−Δu+∣∇u∣p=0u_t-\Delta u+|\nabla u|^p=0ut​−Δu+∣∇u∣p=0 with infinite initial data on a subdomain (1<p<21<p<21<p<2)

Discrete-to-continuum limits for random atomistic lattices: quantum (Thomas–Fermi-type) elastic energies and survival of lattice symmetry

Correctors, gradient-dependent diffusion, and quantitative rates for the stochastic homogenization of viscous Hamilton–Jacobi equations in stationary ergodic media

Rigorous long-time asymptotics for the optimal control of conditioned processes (the Achdou–Laurière–Lions ansatz)

Diffusive limit of the generalized Carleman system with superlinear density-dependent collision rate ραρ^αρα for 1<α<21 < α < 21<α<2

Characterization of the convex-symmetrization comparison function for elliptic problems with non-smooth anisotropic gauge functions

Explicit characterization of the convex envelope of the minimum of the horizontal and vertical total variations

Uniqueness and selection of equilibria in the stationary mean field game system of Lasry and Lions beyond the monotonicity condition

Uniqueness without star-shapedness for the Crandall–Lions–Souganidis model problem $u_t-\Delta u+|\nabla u|^p=0$ with infinite initial data on a subdomain ( $1<p<2$ )

Diffusive limit of the generalized Carleman system with superlinear density-dependent collision rate $ρ^α$ for $1 < α < 2$