A nonsmooth extension of the Brezzi-Rappaz-Raviart approximation theorem via metric regularity techniques and applications to nonlinear PDEs

Francisco Jos\'e Silva (XLIM; IRMAR); Jules Berry (L2S); Olivier Ley (INSA Rennes; UNILIM)

arxiv: 2507.05774 · v2 · submitted 2025-07-08 · 🧮 math.NA · cs.NA

A nonsmooth extension of the Brezzi-Rappaz-Raviart approximation theorem via metric regularity techniques and applications to nonlinear PDEs

Jules Berry (L2S) , Olivier Ley (INSA Rennes , IRMAR) , Francisco Jos\'e Silva (XLIM , UNILIM) This is my paper

Pith reviewed 2026-05-19 06:32 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Brezzi-Rappaz-Raviart theoremmetric regularityLipschitz nonlinearityfinite element approximationviscous Hamilton-Jacobi equationsmean field gamesvariational analysisa priori error estimates

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The pith

The Brezzi-Rappaz-Raviart approximation theorem extends to nonlinear PDEs with merely Lipschitz nonlinearities by replacing differentiability with metric regularity of the associated mapping.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper generalizes the Brezzi-Rappaz-Raviart theorem, which originally gives existence and a priori error estimates for numerical solutions of certain nonlinear PDEs under differentiability assumptions on the nonlinearity. The generalization uses metric regularity techniques from variational analysis to handle nonlinearities that are only Lipschitz continuous. The result yields existence of discrete solutions and error bounds for finite element approximations. The authors then apply the extended theorem to viscous Hamilton-Jacobi equations and second-order mean field game systems, obtaining quasi-optimal convergence rates.

Core claim

The Brezzi-Rappaz-Raviart theorem continues to hold when the nonlinearity is only Lipschitz continuous, provided the nonlinear mapping that encodes the PDE satisfies a metric regularity condition at the exact solution. This condition replaces the classical differentiability hypothesis and directly implies both the existence of approximate solutions and the desired a priori error estimates between continuous and discrete solutions. The same framework is used to derive quasi-optimal error estimates for finite-element discretizations of viscous Hamilton-Jacobi equations and second-order mean-field games.

What carries the argument

Metric regularity of the nonlinear mapping that arises from the PDE, which supplies a quantitative stability property that substitutes for differentiability in the original Brezzi-Rappaz-Raviart argument.

If this is right

Existence of discrete solutions is guaranteed for a wider class of nonlinear PDEs without differentiability of the nonlinearity.
A priori error estimates between exact and approximate solutions follow directly from the metric regularity modulus.
Quasi-optimal convergence rates hold for finite-element approximations of viscous Hamilton-Jacobi equations.
Quasi-optimal convergence rates hold for finite-element approximations of second-order mean-field game systems.
The same proof pattern applies to other Galerkin-type discretizations whenever metric regularity can be verified.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach opens the door to numerical analysis of models whose nonlinear terms involve absolute values or max operations, which are common in optimal control and game theory.
Metric regularity checks can be performed via subdifferential calculus or coderivative conditions already available in the variational-analysis literature.
Numerical experiments on concrete nonsmooth Hamilton-Jacobi or mean-field problems could test whether the predicted error rates are observed in practice.

Load-bearing premise

The nonlinear mapping encoding the PDE must satisfy metric regularity at the solution.

What would settle it

An explicit Lipschitz nonlinearity for which the associated mapping fails to be metrically regular at its solution, yet a convergent finite-element approximation still exists with the claimed error rate.

read the original abstract

We generalize the Brezzi-Rappaz-Raviart approximation theorem, which allows to obtain existence and a priori error estimates for approximations of solutions to some nonlinear partial differential equations. Our contribution lies in the fact that we typically allow for nonlinearities having merely Lipschitz regularity, while previous results required some form of differentiability. This is achieved by making use of the theory of metrically regular mappings, developed in the context of variational analysis. We apply this generalization to derive quasi-optimal error estimates for finite element approximations to solutions of viscous Hamilton-Jacobi equations and second order mean field game systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper drops the differentiability assumption in the Brezzi-Rappaz-Raviart theorem by invoking metric regularity and then uses the result for a priori error bounds on finite-element approximations to viscous Hamilton-Jacobi and second-order mean-field game problems.

read the letter

The core advance is replacing the usual C1 or differentiable nonlinearity hypothesis in the classical BRR theorem with a metric-regularity condition on the nonlinear map at the solution. That change lets the argument run for merely Lipschitz nonlinearities, which is the stated novelty relative to the earlier literature they cite. They then apply the generalized theorem to obtain quasi-optimal error estimates for finite-element discretizations of viscous HJ equations and second-order MFG systems. If the metric-regularity modulus can be controlled in those settings, the existence of discrete solutions and the error bounds follow in a fairly clean way from the abstract result. That is useful technical work for people who need a priori estimates on problems that are not smooth enough for the old theorem. The paper does a reasonable job of laying out the variational-analysis background and sketching how the applications fit the framework. The main soft spot is that metric regularity is not automatic for Lipschitz maps; it requires a surjectivity condition on the coderivative (or limiting subdifferential) with a finite modulus at the exact solution. For the target HJ and MFG problems this may hold under the coercivity or monotonicity assumptions they impose, but the abstract does not show the verification, so the reader has to check the full argument to see whether the condition is verified explicitly or left as an assumption. If the verification is only sketched or relies on extra structure not stated up front, the practical reach of the result shrinks. This is the kind of paper that belongs in a numerical-analysis journal that handles variational-analysis tools. Readers who work on a priori error analysis for nonlinear PDEs or who already use metric regularity in their own work will find it directly relevant. It is coherent on its own terms and shows honest engagement with the literature, so it deserves a serious referee rather than a desk rejection. I would send it out for review and ask the referees to focus on whether the metric-regularity hypothesis is actually satisfied with an explicit constant in the two applications.

Referee Report

1 major / 0 minor

Summary. The manuscript generalizes the classical Brezzi-Rappaz-Raviart approximation theorem to nonlinear operators that are merely Lipschitz continuous (rather than differentiable) by replacing the usual linearization argument with a metric-regularity assumption on the nonlinear mapping F at the exact solution. The generalized abstract result is then applied to obtain existence of discrete solutions and quasi-optimal a priori error estimates for finite-element approximations of viscous Hamilton-Jacobi equations and second-order mean-field game systems.

Significance. If the metric-regularity hypothesis can be verified for the target problems, the work would meaningfully enlarge the class of nonlinear PDEs for which rigorous approximation theory is available without differentiability. The applications to viscous HJ and MFG systems are of current interest, and the variational-analysis framework is a natural tool for nonsmooth settings.

major comments (1)

[Applications sections (viscous HJ and MFG)] The central claim replaces differentiability by metric regularity of F at the solution (invoked via the cited variational-analysis results). For the viscous Hamilton-Jacobi and second-order MFG applications, this surjectivity condition on the coderivative (or limiting subdifferential) with finite modulus is not automatic for merely Lipschitz nonlinearities; the manuscript must supply an explicit verification or proof that the condition holds in the function spaces used, as failure of this hypothesis would render the existence and quasi-optimal error conclusions inapplicable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the major comment below regarding the need for explicit verification of the metric regularity condition in the applications. We agree that this clarification is necessary and will revise the manuscript accordingly to include detailed proofs.

read point-by-point responses

Referee: [Applications sections (viscous HJ and MFG)] The central claim replaces differentiability by metric regularity of F at the solution (invoked via the cited variational-analysis results). For the viscous Hamilton-Jacobi and second-order MFG applications, this surjectivity condition on the coderivative (or limiting subdifferential) with finite modulus is not automatic for merely Lipschitz nonlinearities; the manuscript must supply an explicit verification or proof that the condition holds in the function spaces used, as failure of this hypothesis would render the existence and quasi-optimal error conclusions inapplicable.

Authors: We agree that the metric regularity hypothesis (in particular, surjectivity of the coderivative with finite modulus) is not automatic for merely Lipschitz nonlinearities and requires explicit verification in the specific function spaces. In the revised manuscript we will add dedicated subsections (one for the viscous Hamilton-Jacobi problem and one for the second-order MFG system) that supply the missing verification. The argument will combine the Lipschitz continuity of the nonlinearity with the strong monotonicity/coercivity induced by the viscous or diffusion terms, together with the structure of the limiting subdifferential, to establish metric regularity at the exact solution in the appropriate Sobolev spaces (e.g., W^{1,p} or H^1). These additions will directly invoke the cited variational-analysis results and will make the applicability of the abstract theorem fully transparent for both applications. revision: yes

Circularity Check

0 steps flagged

No significant circularity: generalization rests on external metric-regularity theory

full rationale

The paper's core contribution is an abstract extension of the classical Brezzi-Rappaz-Raviart theorem that replaces differentiability by a metric-regularity assumption on the nonlinear operator. This assumption is imported from the variational-analysis literature (Rockafellar, Dontchev-Rockafellar, etc.) and is not defined or fitted inside the present manuscript. The subsequent a-priori error estimates for viscous Hamilton-Jacobi and mean-field-game discretizations are direct consequences of the abstract theorem once the regularity condition is granted; they do not reduce to any self-referential fit, renormalization, or self-citation chain. No load-bearing step in the derivation is shown to be equivalent to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The generalization rests on the assumption that the nonlinear operator satisfies metric regularity at the solution; no free parameters or newly invented entities are introduced in the abstract.

axioms (1)

domain assumption The nonlinear mapping is metrically regular at the exact solution
This replaces the differentiability hypothesis of the classical theorem and is required for the existence and error-estimate conclusions.

pith-pipeline@v0.9.0 · 5652 in / 1367 out tokens · 44251 ms · 2026-05-19T06:32:46.239391+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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Relation between the paper passage and the cited Recognition theorem.

We generalize the Brezzi-Rappaz-Raviart approximation theorem... to nonlinearities having merely Lipschitz regularity... by making use of the theory of metrically regular mappings.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.9... generalized strict differentiability... sur(F; x̄) ≥ κ − μ

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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Works this paper leans on

63 extracted references · 63 canonical work pages

[1]

Finite Diﬀerence Methods for Mean Field Games

Y. Achdou. “Finite Diﬀerence Methods for Mean Field Games”. In: Y. Achdou, G. Bar- les, H. Ishii, and G. L. Litvinov. Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications. Vol. 2074. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013, pp. 1–47. /d.sc/o.sc/i.sc: 10.1007/978-3-642-36433-4_1 . 46

work page doi:10.1007/978-3-642-36433-4_1 2074
[2]

Mean ﬁeld games and applications: numerical aspects

Y. Achdou and M. Laurière. “Mean ﬁeld games and applications: numerical aspects”. En- glish. In: Mean ﬁeld games. Cetraro, Italy, June 10–14, 2019. Lecture no tes given at the summer school . Cham: Springer; Florence: Fondazione CIME, 2020, pp. 249–307 . /d.sc/o.sc/i.sc: 10.1007/978-3-030-59837-2_4

work page doi:10.1007/978-3-030-59837-2_4 2019
[3]

C. D. Aliprantis and K. C. Border. Inﬁnite dimensional analysis . Third. A hitchhiker’s guide. Springer, Berlin, 2006, pp. xxii+703

work page 2006
[4]

Appell and P

J. Appell and P. P. Zabrejko. Nonlinear superposition operators. Vol. 95. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1990, pp. viii+311. /d.sc/o.sc/i.sc: 10.1017/CBO9780511897450

work page doi:10.1017/cbo9780511897450 1990
[5]

Attal, A

S. Attal, A. Joye, and C.-A. Pillet, eds. Open quantum systems I. The Hamiltonian approach. Lecture notes of the summer school, Grenoble, France June 16 th – July 4th 2003. Vol. 1880. Lect. Notes Math. Berlin: Springer, 2006. /d.sc/o.sc/i.sc: 10.1007/b128449

work page doi:10.1007/b128449 2003
[6]

Aubin and A

J.-P. Aubin and A. Cellina. Diﬀerential Inclusions . Vol. 264. Grundlehren Der Mathema- tischen Wissenschaften [Fundamental Principles of Mathemat ical Sciences]. Springer- Verlag, Berlin, 1984. xiii+342. /d.sc/o.sc/i.sc: 10.1007/978-3-642-69512-4

work page doi:10.1007/978-3-642-69512-4 1984
[7]

Aubin and H

J.-P. Aubin and H. Frankowska. Set-Valued Analysis. Vol. 2. Systems & Control: Founda- tions & Applications. Birkhäuser Boston, Inc., Boston, MA, 199 0. xx+461

work page
[8]

H. H. Bauschke and P. L. Combettes. Convex analysis and monotone operator theory in Hilbert spaces . 2nd edition. CMS Books Math./Ouvrages Math. SMC. Cham: Sprin ger,

work page
[9]

/d.sc/o.sc/i.sc: 10.1007/978-3-319-48311-5

work page doi:10.1007/978-3-319-48311-5
[10]

Bensoussan, J

A. Bensoussan, J. Frehse, and P. Yam. Mean ﬁeld games and mean ﬁeld type control theory . SpringerBriefs Math. New York, NY: Springer, 2013. /d.sc/o.sc/i.sc: 10.1007/978-1-4614-8508-7

work page doi:10.1007/978-1-4614-8508-7 2013
[11]

Approximation and pertu rbations of stable solutions to a stationary mean ﬁeld game system

J. Berry, O. Ley, and F. J. Silva. “Approximation and pertu rbations of stable solutions to a stationary mean ﬁeld game system”. In: J. Math. Pures Appl. (9) 194 (2025). Id/No 103666, p. 28. /d.sc/o.sc/i.sc: 10.1016/j.matpur.2025.103666

work page doi:10.1016/j.matpur.2025.103666 2025
[12]

V. I. Bogachev. Measure theory. Vol. I, II . Springer-Verlag, Berlin, 2007, Vol. I: xviii+500 pp., Vol. II: xiv+575. /d.sc/o.sc/i.sc: 10.1007/978-3-540-34514-5

work page doi:10.1007/978-3-540-34514-5 2007
[13]

H. Brezis. Functional Analysis, Sobolev Spaces and Partial Diﬀerenti al Equations. Univer- sitext. Springer, New York, 2011. xiv+599

work page 2011
[14]

Finite-dimensi onal approximation of nonlinear problems. I. Branches of nonsingular solutions

F. Brezzi, J. Rappaz, and P.-A. Raviart. “Finite-dimensi onal approximation of nonlinear problems. I. Branches of nonsingular solutions”. In: Numer. Math. 36.1 (1980/81), pp. 1–

work page 1980
[15]

/d.sc/o.sc/i.sc: 10.1007/BF01395985

work page doi:10.1007/bf01395985
[16]

Finite-dimensi onal approximation of nonlinear problems. II. Limit points

F. Brezzi, J. Rappaz, and P.-A. Raviart. “Finite-dimensi onal approximation of nonlinear problems. II. Limit points”. In: Numer. Math.37.1 (1981), pp. 1–28. /d.sc/o.sc/i.sc: 10.1007/BF01396184

work page doi:10.1007/bf01396184 1981
[17]

Finite-dimensi onal approximation of nonlinear problems. III. Simple bifurcation points

F. Brezzi, J. Rappaz, and P.-A. Raviart. “Finite-dimensi onal approximation of nonlinear problems. III. Simple bifurcation points”. In: Numer. Math. 38.1 (1981/82), pp. 1–30. /d.sc/o.sc/i.sc: 10.1007/BF01395805

work page doi:10.1007/bf01395805 1981
[18]

Stable Solutions in Potenti al Mean Field Game Systems

A. Briani and P. Cardaliaguet. “Stable Solutions in Potenti al Mean Field Game Systems”. In: Nonlinear Diﬀerential Equations and Applications NoDEA25.1 (2018), p. 1./d.sc/o.sc/i.sc: 10.1007/s00030-017-0493-3 . 47

work page doi:10.1007/s00030-017-0493-3 2018
[19]

Numerical Analysis for Nonlinear a nd Bifurcation Problems

G. Caloz and J. Rappaz. “Numerical Analysis for Nonlinear a nd Bifurcation Problems”. In: Handbook of Numerical Analysis, Vol. V. Vol. V. Handb. Numer. Anal. North-Holland, Amsterdam, 1997, pp. 487–637. /d.sc/o.sc/i.sc: 10.1016/S1570-8659(97)80004-X

work page doi:10.1016/s1570-8659(97)80004-x 1997
[20]

Cardaliaguet and A

P. Cardaliaguet and A. Porretta, eds. Mean Field Games: Cetraro, Italy 2019. Vol. 2281. Lec- ture Notes in Mathematics. Cham: Springer International Publishing, 2020. /d.sc/o.sc/i.sc: 10.1007/978-3-030-59837-2

work page doi:10.1007/978-3-030-59837-2 2019
[21]

Error estimates for semidisc rete ﬁnite element approxi- mations of linear and semilinear parabolic equations under mini mal regularity assump- tions

K. Chrysaﬁnos and L. S. Hou. “Error estimates for semidisc rete ﬁnite element approxi- mations of linear and semilinear parabolic equations under mini mal regularity assump- tions”. In: SIAM J. Numer. Anal.40.1 (2002), pp. 282–306. /d.sc/o.sc/i.sc: 10.1137/S0036142900377991

work page doi:10.1137/s0036142900377991 2002
[22]

On Primal Regularity Estimates for Single-Valued Mappings

R. Cibulka, M. Fabian, and A. D. Ioﬀe. “On Primal Regularity Estimates for Single-Valued Mappings”. In: Journal of Fixed Point Theory and Applications 17.1 (2015), pp. 187–208. /d.sc/o.sc/i.sc: 10.1007/s11784-015-0240-5

work page doi:10.1007/s11784-015-0240-5 2015
[23]

F. H. Clarke. Optimization and Nonsmooth Analysis . Second. Vol. 5. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics ( SIAM), Philadelphia, PA,

work page
[24]

Classics in Applied Mathe- matics, vol

xii+308. /d.sc/o.sc/i.sc: 10.1137/1.9781611971309

work page doi:10.1137/1.9781611971309
[25]

Crouzeix and J

M. Crouzeix and J. Rappaz. On Numerical Approximation in Bifurcation Theory . Vol. 13. Recherches En Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris; Springer-Verlag, Berlin, 1990. x+165

work page 1990
[26]

Approximation of solution bra nches of nonlinear equations

J. Descloux and J. Rappaz. “Approximation of solution bra nches of nonlinear equations”. In: RAIRO Anal. Numér. 16.4 (1982), pp. 319–349

work page 1982
[27]

Hitchhiker's guide to the fractional Sobolev spaces , journal =

E. Di Nezza, G. Palatucci, and E. Valdinoci. “Hitchhiker’s guide to the fractional Sobolev spaces”. In: Bull. Sci. Math. 136.5 (2012), pp. 521–573. /d.sc/o.sc/i.sc: 10.1016/j.bulsci.2011.12.004

work page doi:10.1016/j.bulsci.2011.12.004 2012
[28]

Ljusternik’s theorem and the theory of the extremum

A. V. Dmitruk, A. A. Milyutin, and N. P. Osmolovski ˘ı. “Ljusternik’s theorem and the theory of the extremum”. In: Uspekhi Mat. Nauk 35.6(216) (1980), pp. 11–46, 215

work page 1980
[29]

A. L. Dontchev and R. T. Rockafellar. Implicit functions and solution mappings . Second. Springer Series in Operations Research and Financial Engin eering. A view from varia- tional analysis. Springer, New York, 2014, pp. xxviii+466./d.sc/o.sc/i.sc: 10.1007/978-1-4939-1037-3

work page doi:10.1007/978-1-4939-1037-3 2014
[30]

Ern and J.-L

A. Ern and J.-L. Guermond. Theory and Practice of Finite Elements. Vol. 159. Applied Math- ematical Sciences. Springer-Verlag, New York, 2004. xiv+524. /d.sc/o.sc/i.sc: 10.1007/978-1-4757-4355-5

work page doi:10.1007/978-1-4757-4355-5 2004
[31]

L. C. Evans and R. F. Gariepy. Measure theory and ﬁne properties of functions . Revised. Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015, pp . xiv+299

work page 2015
[32]

Gilbarg and N

D. Gilbarg and N. S. Trudinger. Elliptic partial diﬀerential equations of second order . Clas- sics in Mathematics. Reprint of the 1998 edition. Springer-Verlag, Berlin, 2001, pp. xiv+517

work page 1998
[33]

Girault and P.-A

V. Girault and P.-A. Raviart. Finite Element Methods for Navier-Stokes Equations . Vol. 5. Springer Series in Computational Mathematics. Springer-V erlag, Berlin, 1986. x+374. /d.sc/o.sc/i.sc: 10.1007/978-3-642-61623-5

work page doi:10.1007/978-3-642-61623-5 1986
[34]

Mean Field Games Models—A Brief Sur vey

D. A. Gomes and J. Saúde. “Mean Field Games Models—A Brief Sur vey”. In: Dynamic Games and Applications 4.2 (2014), pp. 110–154. /d.sc/o.sc/i.sc: 10.1007/s13235-013-0099-2

work page doi:10.1007/s13235-013-0099-2 2014
[35]

Some mapping theorems

L. M. Graves. “Some mapping theorems”. In: Duke Math. J. 17 (1950), pp. 111–114. 48

work page 1950
[36]

Grisvard

P. Grisvard. Elliptic problems in nonsmooth domains . Vol. 24. Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, 1985, pp. xiv+410

work page 1985
[37]

Large-Populatio n Cost-Coupled LQG Prob- lems With Nonuniform Agents: Individual-Mass Behavior and Decen tralized ε-Nash Equilibria

M. Huang, P. E. Caines, and R. P. Malhame. “Large-Populatio n Cost-Coupled LQG Prob- lems With Nonuniform Agents: Individual-Mass Behavior and Decen tralized ε-Nash Equilibria”. In: IEEE Transactions on Automatic Control 52.9 (2007), pp. 1560–1571. /d.sc/o.sc/i.sc: 10.1109/TAC.2007.904450

work page doi:10.1109/tac.2007.904450 2007
[38]

Large population st ochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equiv alence principle

M. Huang, R. Malhamé, and P. E. Caines. “Large population st ochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equiv alence principle”. In: Commun. Inf. Syst. 6.3 (2006), pp. 221–252. /d.sc/o.sc/i.sc: 10.4310/CIS.2006.v6.n3.a5

work page doi:10.4310/cis.2006.v6.n3.a5 2006
[39]

Support functions of the Clarke generalized Ja cobian and of its plenary hull

C. Imbert. “Support functions of the Clarke generalized Ja cobian and of its plenary hull”. In: Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Metho ds 49.8 (2002), pp. 1111–

work page 2002
[40]

/d.sc/o.sc/i.sc: 10.1016/S0362-546X(01)00730-1

work page doi:10.1016/s0362-546x(01)00730-1
[41]

A. D. Ioﬀe. Variational Analysis of Regular Mappings . Springer Monographs in Mathe- matics. Springer, Cham, 2017. xxi+495. /d.sc/o.sc/i.sc: 10.1007/978-3-319-64277-2

work page doi:10.1007/978-3-319-64277-2 2017
[42]

On the convergence of ﬁnite eleme nt methods for Hamilton- Jacobi-Bellman equations

M. Jensen and I. Smears. “On the convergence of ﬁnite eleme nt methods for Hamilton- Jacobi-Bellman equations”. In: SIAM J. Numer. Anal.51.1 (2013), pp. 137–162. /d.sc/o.sc/i.sc: 10.1137/110856198

work page doi:10.1137/110856198 2013
[43]

A priori and a posteriori analysis o f mixed ﬁnite element methods for nonlinear elliptic equations

D. Kim and E.-J. Park. “A priori and a posteriori analysis o f mixed ﬁnite element methods for nonlinear elliptic equations”. In: SIAM J. Numer. Anal.48.3 (2010), pp. 1186–1207. /d.sc/o.sc/i.sc: 10.1137/090747002

work page doi:10.1137/090747002 2010
[44]

Jeux à champ moyen. I – Le cas stati onnaire

J.-M. Lasry and P.-L. Lions. “Jeux à champ moyen. I – Le cas stati onnaire”. In: Comptes Rendus Mathematique 343.9 (2006), pp. 619–625. /d.sc/o.sc/i.sc: 10.1016/j.crma.2006.09.019

work page doi:10.1016/j.crma.2006.09.019 2006
[45]

Jeux à champ moyen. II – Horizon ﬁn i et contrôle optimal

J.-M. Lasry and P.-L. Lions. “Jeux à champ moyen. II – Horizon ﬁn i et contrôle optimal”. In: Comptes Rendus Mathematique 343.10 (2006), pp. 679–684. /d.sc/o.sc/i.sc: 10.1016/j.crma.2006.09.018

work page doi:10.1016/j.crma.2006.09.018 2006
[46]

Mean Field Games

J.-M. Lasry and P.-L. Lions. “Mean Field Games”. In: Japanese Journal of Mathematics 2.1 (2007), pp. 229–260. /d.sc/o.sc/i.sc: 10.1007/s11537-007-0657-8

work page doi:10.1007/s11537-007-0657-8 2007
[47]

A. Lunardi. Interpolation theory. 3rd edition. Vol. 16. Appunti, Sc. Norm. Super. Pisa (N.S.) Pisa: Edizioni della Normale, 2018. /d.sc/o.sc/i.sc: 10.1007/978-88-7642-638-4

work page doi:10.1007/978-88-7642-638-4 2018
[48]

Approximation of Burgers’ e quation by pseudospectral methods

Y. Maday and A. Quarteroni. “Approximation of Burgers’ e quation by pseudospectral methods”. In: RAIRO Anal. Numér.16.4 (1982), pp. 375–404. /d.sc/o.sc/i.sc: 10.1051/m2an/1982160403751

work page doi:10.1051/m2an/1982160403751 1982
[49]

B. S. Mordukhovich. Variational analysis and generalized diﬀerentiation. I: Basic theory. II: Applications. Vol. 330/331. Grundlehren Math. Wiss. Berlin: Springer, 2005./d.sc/o.sc/i.sc: 10.1007/3-540-31247-1

work page doi:10.1007/3-540-31247-1 2005
[50]

Near and full quasi-optimality o f ﬁnite element approxi- mations of stationary second-order mean ﬁeld games

Y. Osborne and I. Smears. “Near and full quasi-optimality o f ﬁnite element approxi- mations of stationary second-order mean ﬁeld games”. In: Mathematics of Computation (2025). /d.sc/o.sc/i.sc: 10.1090/mcom/4080

work page doi:10.1090/mcom/4080 2025
[51]

Analysis and Numerical App roximation of Stationary Second-Order Mean Field Game Partial Diﬀerential Inclusions

Y. A. P. Osborne and I. Smears. “Analysis and Numerical App roximation of Stationary Second-Order Mean Field Game Partial Diﬀerential Inclusions”. In : SIAM J. Numer. Anal. 62.1 (2024), pp. 138–166. /d.sc/o.sc/i.sc: 10.1137/22M1519274. 49

work page doi:10.1137/22m1519274 2024
[52]

Finite element approximat ion of time-dependent mean ﬁeld games with nondiﬀerentiable Hamiltonians

Y. A. P. Osborne and I. Smears. “Finite element approximat ion of time-dependent mean ﬁeld games with nondiﬀerentiable Hamiltonians”. In: Numer. Math.157.1 (2025), pp. 165–

work page 2025
[53]

/d.sc/o.sc/i.sc: 10.1007/s00211-024-01447-2

work page doi:10.1007/s00211-024-01447-2
[54]

Y. A. P. Osborne and I. Smears. Rates of convergence of ﬁnite element approximations of second-order mean ﬁeld games with nondiﬀerentiable Hamiltonians. Preprint, arXiv:2506.03039 [math.NA]. 2025

work page arXiv 2025
[55]

Y. A. P. Osborne, I. Smears, and H. Wells. A posteriori error bounds for ﬁnite element approximations of steady-state mean ﬁeld games . Preprint, arXiv:2502.14687 [math.NA]. 2025

work page arXiv 2025
[56]

Approximation of double bifurcat ion points for nonlinear eigenvalue problems

J. Rappaz and G. Raugel. “Approximation of double bifurcat ion points for nonlinear eigenvalue problems”. In: The mathematics of ﬁnite elements and applications, IV . Aca- demic Press, London-New York, 1982, pp. 453–461

work page 1982
[57]

Approximation of a nondiﬀerentiable nonlinea r problem related to MHD equilibria

J. Rappaz. “Approximation of a nondiﬀerentiable nonlinea r problem related to MHD equilibria”. In: Numer. Math. 45.1 (1984), pp. 117–133. /d.sc/o.sc/i.sc: 10.1007/BF01379665

work page doi:10.1007/bf01379665 1984
[58]

R. T. Rockafellar and R. J.-B. Wets. Variational analysis. Vol. 317. Grundlehren der mathe- matischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer- Verlag, Berlin, 1998, pp. xiv+733. /d.sc/o.sc/i.sc: 10.1007/978-3-642-02431-3

work page doi:10.1007/978-3-642-02431-3 1998
[59]

Discontinuous Galerkin ﬁnite elemen t approximation of Hamilton- Jacobi-Bellman equations with Cordes coeﬃcients

I. Smears and E. Süli. “Discontinuous Galerkin ﬁnite elemen t approximation of Hamilton- Jacobi-Bellman equations with Cordes coeﬃcients”. English. In: SIAM J. Numer. Anal. 52.2 (2014), pp. 993–1016. /d.sc/o.sc/i.sc: 10.1137/130909536

work page doi:10.1137/130909536 2014
[60]

Thibault

L. Thibault. Unilateral variational analysis in Banach spaces. Part I: g eneral theory. Part II: special classes of functions and sets. In 2 volumes . Singapore: World Scientiﬁc, 2023. /d.sc/o.sc/i.sc: 10.1142/12797

work page doi:10.1142/12797 2023
[61]

Semismooth Newton methods for operator equa tions in function spaces

M. Ulbrich. “Semismooth Newton methods for operator equa tions in function spaces”. In: SIAM J. Optim. 13.3 (2002), pp. 805–842. /d.sc/o.sc/i.sc: 10.1137/S1052623400371569

work page doi:10.1137/s1052623400371569 2002
[62]

M. Ulbrich. Semismooth Newton methods for variational inequalities an d constrained opti- mization problems in function spaces. Vol. 11. MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; M athematical Opti- mization Society, Philadelphia, PA, 2011, pp. xiv+308. /d.sc/o.sc/i.sc: 10.1137/1.9781611970692

work page doi:10.1137/1.9781611970692 2011
[63]

Fat homeomorphisms and unbounded derivate con tainers

J. Warga. “Fat homeomorphisms and unbounded derivate con tainers”. In: J. Math. Anal. Appl. 81.2 (1981), pp. 545–560. /d.sc/o.sc/i.sc: 10.1016/0022-247X(81)90081-0. 50

work page doi:10.1016/0022-247x(81)90081-0 1981

[1] [1]

Finite Diﬀerence Methods for Mean Field Games

Y. Achdou. “Finite Diﬀerence Methods for Mean Field Games”. In: Y. Achdou, G. Bar- les, H. Ishii, and G. L. Litvinov. Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications. Vol. 2074. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013, pp. 1–47. /d.sc/o.sc/i.sc: 10.1007/978-3-642-36433-4_1 . 46

work page doi:10.1007/978-3-642-36433-4_1 2074

[2] [2]

Mean ﬁeld games and applications: numerical aspects

Y. Achdou and M. Laurière. “Mean ﬁeld games and applications: numerical aspects”. En- glish. In: Mean ﬁeld games. Cetraro, Italy, June 10–14, 2019. Lecture no tes given at the summer school . Cham: Springer; Florence: Fondazione CIME, 2020, pp. 249–307 . /d.sc/o.sc/i.sc: 10.1007/978-3-030-59837-2_4

work page doi:10.1007/978-3-030-59837-2_4 2019

[3] [3]

C. D. Aliprantis and K. C. Border. Inﬁnite dimensional analysis . Third. A hitchhiker’s guide. Springer, Berlin, 2006, pp. xxii+703

work page 2006

[4] [4]

Appell and P

J. Appell and P. P. Zabrejko. Nonlinear superposition operators. Vol. 95. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1990, pp. viii+311. /d.sc/o.sc/i.sc: 10.1017/CBO9780511897450

work page doi:10.1017/cbo9780511897450 1990

[5] [5]

Attal, A

S. Attal, A. Joye, and C.-A. Pillet, eds. Open quantum systems I. The Hamiltonian approach. Lecture notes of the summer school, Grenoble, France June 16 th – July 4th 2003. Vol. 1880. Lect. Notes Math. Berlin: Springer, 2006. /d.sc/o.sc/i.sc: 10.1007/b128449

work page doi:10.1007/b128449 2003

[6] [6]

Aubin and A

J.-P. Aubin and A. Cellina. Diﬀerential Inclusions . Vol. 264. Grundlehren Der Mathema- tischen Wissenschaften [Fundamental Principles of Mathemat ical Sciences]. Springer- Verlag, Berlin, 1984. xiii+342. /d.sc/o.sc/i.sc: 10.1007/978-3-642-69512-4

work page doi:10.1007/978-3-642-69512-4 1984

[7] [7]

Aubin and H

J.-P. Aubin and H. Frankowska. Set-Valued Analysis. Vol. 2. Systems & Control: Founda- tions & Applications. Birkhäuser Boston, Inc., Boston, MA, 199 0. xx+461

work page

[8] [8]

H. H. Bauschke and P. L. Combettes. Convex analysis and monotone operator theory in Hilbert spaces . 2nd edition. CMS Books Math./Ouvrages Math. SMC. Cham: Sprin ger,

work page

[9] [9]

/d.sc/o.sc/i.sc: 10.1007/978-3-319-48311-5

work page doi:10.1007/978-3-319-48311-5

[10] [10]

Bensoussan, J

A. Bensoussan, J. Frehse, and P. Yam. Mean ﬁeld games and mean ﬁeld type control theory . SpringerBriefs Math. New York, NY: Springer, 2013. /d.sc/o.sc/i.sc: 10.1007/978-1-4614-8508-7

work page doi:10.1007/978-1-4614-8508-7 2013

[11] [11]

Approximation and pertu rbations of stable solutions to a stationary mean ﬁeld game system

J. Berry, O. Ley, and F. J. Silva. “Approximation and pertu rbations of stable solutions to a stationary mean ﬁeld game system”. In: J. Math. Pures Appl. (9) 194 (2025). Id/No 103666, p. 28. /d.sc/o.sc/i.sc: 10.1016/j.matpur.2025.103666

work page doi:10.1016/j.matpur.2025.103666 2025

[12] [12]

V. I. Bogachev. Measure theory. Vol. I, II . Springer-Verlag, Berlin, 2007, Vol. I: xviii+500 pp., Vol. II: xiv+575. /d.sc/o.sc/i.sc: 10.1007/978-3-540-34514-5

work page doi:10.1007/978-3-540-34514-5 2007

[13] [13]

H. Brezis. Functional Analysis, Sobolev Spaces and Partial Diﬀerenti al Equations. Univer- sitext. Springer, New York, 2011. xiv+599

work page 2011

[14] [14]

Finite-dimensi onal approximation of nonlinear problems. I. Branches of nonsingular solutions

F. Brezzi, J. Rappaz, and P.-A. Raviart. “Finite-dimensi onal approximation of nonlinear problems. I. Branches of nonsingular solutions”. In: Numer. Math. 36.1 (1980/81), pp. 1–

work page 1980

[15] [15]

/d.sc/o.sc/i.sc: 10.1007/BF01395985

work page doi:10.1007/bf01395985

[16] [16]

Finite-dimensi onal approximation of nonlinear problems. II. Limit points

F. Brezzi, J. Rappaz, and P.-A. Raviart. “Finite-dimensi onal approximation of nonlinear problems. II. Limit points”. In: Numer. Math.37.1 (1981), pp. 1–28. /d.sc/o.sc/i.sc: 10.1007/BF01396184

work page doi:10.1007/bf01396184 1981

[17] [17]

Finite-dimensi onal approximation of nonlinear problems. III. Simple bifurcation points

F. Brezzi, J. Rappaz, and P.-A. Raviart. “Finite-dimensi onal approximation of nonlinear problems. III. Simple bifurcation points”. In: Numer. Math. 38.1 (1981/82), pp. 1–30. /d.sc/o.sc/i.sc: 10.1007/BF01395805

work page doi:10.1007/bf01395805 1981

[18] [18]

Stable Solutions in Potenti al Mean Field Game Systems

A. Briani and P. Cardaliaguet. “Stable Solutions in Potenti al Mean Field Game Systems”. In: Nonlinear Diﬀerential Equations and Applications NoDEA25.1 (2018), p. 1./d.sc/o.sc/i.sc: 10.1007/s00030-017-0493-3 . 47

work page doi:10.1007/s00030-017-0493-3 2018

[19] [19]

Numerical Analysis for Nonlinear a nd Bifurcation Problems

G. Caloz and J. Rappaz. “Numerical Analysis for Nonlinear a nd Bifurcation Problems”. In: Handbook of Numerical Analysis, Vol. V. Vol. V. Handb. Numer. Anal. North-Holland, Amsterdam, 1997, pp. 487–637. /d.sc/o.sc/i.sc: 10.1016/S1570-8659(97)80004-X

work page doi:10.1016/s1570-8659(97)80004-x 1997

[20] [20]

Cardaliaguet and A

P. Cardaliaguet and A. Porretta, eds. Mean Field Games: Cetraro, Italy 2019. Vol. 2281. Lec- ture Notes in Mathematics. Cham: Springer International Publishing, 2020. /d.sc/o.sc/i.sc: 10.1007/978-3-030-59837-2

work page doi:10.1007/978-3-030-59837-2 2019

[21] [21]

Error estimates for semidisc rete ﬁnite element approxi- mations of linear and semilinear parabolic equations under mini mal regularity assump- tions

K. Chrysaﬁnos and L. S. Hou. “Error estimates for semidisc rete ﬁnite element approxi- mations of linear and semilinear parabolic equations under mini mal regularity assump- tions”. In: SIAM J. Numer. Anal.40.1 (2002), pp. 282–306. /d.sc/o.sc/i.sc: 10.1137/S0036142900377991

work page doi:10.1137/s0036142900377991 2002

[22] [22]

On Primal Regularity Estimates for Single-Valued Mappings

R. Cibulka, M. Fabian, and A. D. Ioﬀe. “On Primal Regularity Estimates for Single-Valued Mappings”. In: Journal of Fixed Point Theory and Applications 17.1 (2015), pp. 187–208. /d.sc/o.sc/i.sc: 10.1007/s11784-015-0240-5

work page doi:10.1007/s11784-015-0240-5 2015

[23] [23]

F. H. Clarke. Optimization and Nonsmooth Analysis . Second. Vol. 5. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics ( SIAM), Philadelphia, PA,

work page

[24] [24]

Classics in Applied Mathe- matics, vol

xii+308. /d.sc/o.sc/i.sc: 10.1137/1.9781611971309

work page doi:10.1137/1.9781611971309

[25] [25]

Crouzeix and J

M. Crouzeix and J. Rappaz. On Numerical Approximation in Bifurcation Theory . Vol. 13. Recherches En Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris; Springer-Verlag, Berlin, 1990. x+165

work page 1990

[26] [26]

Approximation of solution bra nches of nonlinear equations

J. Descloux and J. Rappaz. “Approximation of solution bra nches of nonlinear equations”. In: RAIRO Anal. Numér. 16.4 (1982), pp. 319–349

work page 1982

[27] [27]

Hitchhiker's guide to the fractional Sobolev spaces , journal =

E. Di Nezza, G. Palatucci, and E. Valdinoci. “Hitchhiker’s guide to the fractional Sobolev spaces”. In: Bull. Sci. Math. 136.5 (2012), pp. 521–573. /d.sc/o.sc/i.sc: 10.1016/j.bulsci.2011.12.004

work page doi:10.1016/j.bulsci.2011.12.004 2012

[28] [28]

Ljusternik’s theorem and the theory of the extremum

A. V. Dmitruk, A. A. Milyutin, and N. P. Osmolovski ˘ı. “Ljusternik’s theorem and the theory of the extremum”. In: Uspekhi Mat. Nauk 35.6(216) (1980), pp. 11–46, 215

work page 1980

[29] [29]

A. L. Dontchev and R. T. Rockafellar. Implicit functions and solution mappings . Second. Springer Series in Operations Research and Financial Engin eering. A view from varia- tional analysis. Springer, New York, 2014, pp. xxviii+466./d.sc/o.sc/i.sc: 10.1007/978-1-4939-1037-3

work page doi:10.1007/978-1-4939-1037-3 2014

[30] [30]

Ern and J.-L

A. Ern and J.-L. Guermond. Theory and Practice of Finite Elements. Vol. 159. Applied Math- ematical Sciences. Springer-Verlag, New York, 2004. xiv+524. /d.sc/o.sc/i.sc: 10.1007/978-1-4757-4355-5

work page doi:10.1007/978-1-4757-4355-5 2004

[31] [31]

L. C. Evans and R. F. Gariepy. Measure theory and ﬁne properties of functions . Revised. Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015, pp . xiv+299

work page 2015

[32] [32]

Gilbarg and N

D. Gilbarg and N. S. Trudinger. Elliptic partial diﬀerential equations of second order . Clas- sics in Mathematics. Reprint of the 1998 edition. Springer-Verlag, Berlin, 2001, pp. xiv+517

work page 1998

[33] [33]

Girault and P.-A

V. Girault and P.-A. Raviart. Finite Element Methods for Navier-Stokes Equations . Vol. 5. Springer Series in Computational Mathematics. Springer-V erlag, Berlin, 1986. x+374. /d.sc/o.sc/i.sc: 10.1007/978-3-642-61623-5

work page doi:10.1007/978-3-642-61623-5 1986

[34] [34]

Mean Field Games Models—A Brief Sur vey

D. A. Gomes and J. Saúde. “Mean Field Games Models—A Brief Sur vey”. In: Dynamic Games and Applications 4.2 (2014), pp. 110–154. /d.sc/o.sc/i.sc: 10.1007/s13235-013-0099-2

work page doi:10.1007/s13235-013-0099-2 2014

[35] [35]

Some mapping theorems

L. M. Graves. “Some mapping theorems”. In: Duke Math. J. 17 (1950), pp. 111–114. 48

work page 1950

[36] [36]

Grisvard

P. Grisvard. Elliptic problems in nonsmooth domains . Vol. 24. Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, 1985, pp. xiv+410

work page 1985

[37] [37]

Large-Populatio n Cost-Coupled LQG Prob- lems With Nonuniform Agents: Individual-Mass Behavior and Decen tralized ε-Nash Equilibria

M. Huang, P. E. Caines, and R. P. Malhame. “Large-Populatio n Cost-Coupled LQG Prob- lems With Nonuniform Agents: Individual-Mass Behavior and Decen tralized ε-Nash Equilibria”. In: IEEE Transactions on Automatic Control 52.9 (2007), pp. 1560–1571. /d.sc/o.sc/i.sc: 10.1109/TAC.2007.904450

work page doi:10.1109/tac.2007.904450 2007

[38] [38]

Large population st ochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equiv alence principle

M. Huang, R. Malhamé, and P. E. Caines. “Large population st ochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equiv alence principle”. In: Commun. Inf. Syst. 6.3 (2006), pp. 221–252. /d.sc/o.sc/i.sc: 10.4310/CIS.2006.v6.n3.a5

work page doi:10.4310/cis.2006.v6.n3.a5 2006

[39] [39]

Support functions of the Clarke generalized Ja cobian and of its plenary hull

C. Imbert. “Support functions of the Clarke generalized Ja cobian and of its plenary hull”. In: Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Metho ds 49.8 (2002), pp. 1111–

work page 2002

[40] [40]

/d.sc/o.sc/i.sc: 10.1016/S0362-546X(01)00730-1

work page doi:10.1016/s0362-546x(01)00730-1

[41] [41]

A. D. Ioﬀe. Variational Analysis of Regular Mappings . Springer Monographs in Mathe- matics. Springer, Cham, 2017. xxi+495. /d.sc/o.sc/i.sc: 10.1007/978-3-319-64277-2

work page doi:10.1007/978-3-319-64277-2 2017

[42] [42]

On the convergence of ﬁnite eleme nt methods for Hamilton- Jacobi-Bellman equations

M. Jensen and I. Smears. “On the convergence of ﬁnite eleme nt methods for Hamilton- Jacobi-Bellman equations”. In: SIAM J. Numer. Anal.51.1 (2013), pp. 137–162. /d.sc/o.sc/i.sc: 10.1137/110856198

work page doi:10.1137/110856198 2013

[43] [43]

A priori and a posteriori analysis o f mixed ﬁnite element methods for nonlinear elliptic equations

D. Kim and E.-J. Park. “A priori and a posteriori analysis o f mixed ﬁnite element methods for nonlinear elliptic equations”. In: SIAM J. Numer. Anal.48.3 (2010), pp. 1186–1207. /d.sc/o.sc/i.sc: 10.1137/090747002

work page doi:10.1137/090747002 2010

[44] [44]

Jeux à champ moyen. I – Le cas stati onnaire

J.-M. Lasry and P.-L. Lions. “Jeux à champ moyen. I – Le cas stati onnaire”. In: Comptes Rendus Mathematique 343.9 (2006), pp. 619–625. /d.sc/o.sc/i.sc: 10.1016/j.crma.2006.09.019

work page doi:10.1016/j.crma.2006.09.019 2006

[45] [45]

Jeux à champ moyen. II – Horizon ﬁn i et contrôle optimal

J.-M. Lasry and P.-L. Lions. “Jeux à champ moyen. II – Horizon ﬁn i et contrôle optimal”. In: Comptes Rendus Mathematique 343.10 (2006), pp. 679–684. /d.sc/o.sc/i.sc: 10.1016/j.crma.2006.09.018

work page doi:10.1016/j.crma.2006.09.018 2006

[46] [46]

Mean Field Games

J.-M. Lasry and P.-L. Lions. “Mean Field Games”. In: Japanese Journal of Mathematics 2.1 (2007), pp. 229–260. /d.sc/o.sc/i.sc: 10.1007/s11537-007-0657-8

work page doi:10.1007/s11537-007-0657-8 2007

[47] [47]

A. Lunardi. Interpolation theory. 3rd edition. Vol. 16. Appunti, Sc. Norm. Super. Pisa (N.S.) Pisa: Edizioni della Normale, 2018. /d.sc/o.sc/i.sc: 10.1007/978-88-7642-638-4

work page doi:10.1007/978-88-7642-638-4 2018

[48] [48]

Approximation of Burgers’ e quation by pseudospectral methods

Y. Maday and A. Quarteroni. “Approximation of Burgers’ e quation by pseudospectral methods”. In: RAIRO Anal. Numér.16.4 (1982), pp. 375–404. /d.sc/o.sc/i.sc: 10.1051/m2an/1982160403751

work page doi:10.1051/m2an/1982160403751 1982

[49] [49]

B. S. Mordukhovich. Variational analysis and generalized diﬀerentiation. I: Basic theory. II: Applications. Vol. 330/331. Grundlehren Math. Wiss. Berlin: Springer, 2005./d.sc/o.sc/i.sc: 10.1007/3-540-31247-1

work page doi:10.1007/3-540-31247-1 2005

[50] [50]

Near and full quasi-optimality o f ﬁnite element approxi- mations of stationary second-order mean ﬁeld games

Y. Osborne and I. Smears. “Near and full quasi-optimality o f ﬁnite element approxi- mations of stationary second-order mean ﬁeld games”. In: Mathematics of Computation (2025). /d.sc/o.sc/i.sc: 10.1090/mcom/4080

work page doi:10.1090/mcom/4080 2025

[51] [51]

Analysis and Numerical App roximation of Stationary Second-Order Mean Field Game Partial Diﬀerential Inclusions

Y. A. P. Osborne and I. Smears. “Analysis and Numerical App roximation of Stationary Second-Order Mean Field Game Partial Diﬀerential Inclusions”. In : SIAM J. Numer. Anal. 62.1 (2024), pp. 138–166. /d.sc/o.sc/i.sc: 10.1137/22M1519274. 49

work page doi:10.1137/22m1519274 2024

[52] [52]

Finite element approximat ion of time-dependent mean ﬁeld games with nondiﬀerentiable Hamiltonians

Y. A. P. Osborne and I. Smears. “Finite element approximat ion of time-dependent mean ﬁeld games with nondiﬀerentiable Hamiltonians”. In: Numer. Math.157.1 (2025), pp. 165–

work page 2025

[53] [53]

/d.sc/o.sc/i.sc: 10.1007/s00211-024-01447-2

work page doi:10.1007/s00211-024-01447-2

[54] [54]

Y. A. P. Osborne and I. Smears. Rates of convergence of ﬁnite element approximations of second-order mean ﬁeld games with nondiﬀerentiable Hamiltonians. Preprint, arXiv:2506.03039 [math.NA]. 2025

work page arXiv 2025

[55] [55]

Y. A. P. Osborne, I. Smears, and H. Wells. A posteriori error bounds for ﬁnite element approximations of steady-state mean ﬁeld games . Preprint, arXiv:2502.14687 [math.NA]. 2025

work page arXiv 2025

[56] [56]

Approximation of double bifurcat ion points for nonlinear eigenvalue problems

J. Rappaz and G. Raugel. “Approximation of double bifurcat ion points for nonlinear eigenvalue problems”. In: The mathematics of ﬁnite elements and applications, IV . Aca- demic Press, London-New York, 1982, pp. 453–461

work page 1982

[57] [57]

Approximation of a nondiﬀerentiable nonlinea r problem related to MHD equilibria

J. Rappaz. “Approximation of a nondiﬀerentiable nonlinea r problem related to MHD equilibria”. In: Numer. Math. 45.1 (1984), pp. 117–133. /d.sc/o.sc/i.sc: 10.1007/BF01379665

work page doi:10.1007/bf01379665 1984

[58] [58]

R. T. Rockafellar and R. J.-B. Wets. Variational analysis. Vol. 317. Grundlehren der mathe- matischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer- Verlag, Berlin, 1998, pp. xiv+733. /d.sc/o.sc/i.sc: 10.1007/978-3-642-02431-3

work page doi:10.1007/978-3-642-02431-3 1998

[59] [59]

Discontinuous Galerkin ﬁnite elemen t approximation of Hamilton- Jacobi-Bellman equations with Cordes coeﬃcients

I. Smears and E. Süli. “Discontinuous Galerkin ﬁnite elemen t approximation of Hamilton- Jacobi-Bellman equations with Cordes coeﬃcients”. English. In: SIAM J. Numer. Anal. 52.2 (2014), pp. 993–1016. /d.sc/o.sc/i.sc: 10.1137/130909536

work page doi:10.1137/130909536 2014

[60] [60]

Thibault

L. Thibault. Unilateral variational analysis in Banach spaces. Part I: g eneral theory. Part II: special classes of functions and sets. In 2 volumes . Singapore: World Scientiﬁc, 2023. /d.sc/o.sc/i.sc: 10.1142/12797

work page doi:10.1142/12797 2023

[61] [61]

Semismooth Newton methods for operator equa tions in function spaces

M. Ulbrich. “Semismooth Newton methods for operator equa tions in function spaces”. In: SIAM J. Optim. 13.3 (2002), pp. 805–842. /d.sc/o.sc/i.sc: 10.1137/S1052623400371569

work page doi:10.1137/s1052623400371569 2002

[62] [62]

M. Ulbrich. Semismooth Newton methods for variational inequalities an d constrained opti- mization problems in function spaces. Vol. 11. MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; M athematical Opti- mization Society, Philadelphia, PA, 2011, pp. xiv+308. /d.sc/o.sc/i.sc: 10.1137/1.9781611970692

work page doi:10.1137/1.9781611970692 2011

[63] [63]

Fat homeomorphisms and unbounded derivate con tainers

J. Warga. “Fat homeomorphisms and unbounded derivate con tainers”. In: J. Math. Anal. Appl. 81.2 (1981), pp. 545–560. /d.sc/o.sc/i.sc: 10.1016/0022-247X(81)90081-0. 50

work page doi:10.1016/0022-247x(81)90081-0 1981