{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:34:04Z","timestamp":1772120044335,"version":"3.50.1"},"reference-count":19,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,10,4]],"date-time":"2024-10-04T00:00:00Z","timestamp":1728000000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,10,4]],"date-time":"2024-10-04T00:00:00Z","timestamp":1728000000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Prin 2022, \u201cOptimal control problems: analysis, approximation\u201d","award":["2022238YY5"],"award-info":[{"award-number":["2022238YY5"]}]},{"name":"INdAM-GNCS Project \u201dMetodi numerici per l\u2019imaging: dal 2D al 3D\u201d","award":["E55F22000270001"],"award-info":[{"award-number":["E55F22000270001"]}]},{"name":"INdAM-GNCS Project\u201cModelli e metodi avanzati in Computer Vision\u201d","award":["E53C22001930001"],"award-info":[{"award-number":["E53C22001930001"]}]},{"name":"Project \u201cMOdelli e MEtodi Numerici per il Trattamento delle Immagini","award":["E53C23001670001"],"award-info":[{"award-number":["E53C23001670001"]}]},{"name":"Universit\u00e0 degli Studi di Roma La Sapienza"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Dyn Games Appl"],"published-print":{"date-parts":[[2025,5]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We propose a new semi-Lagrangian scheme for the game\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\infty $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u221e<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -Laplacian. We demonstrate the convergence of the scheme to the viscosity solution of the given problem, showing its consistency, monotonicity, and stability. The proof of this result is established following the Barles-Souganidis analysis. This analysis assumes convergence at the boundary in a strong sense and is applied to our proposed scheme, augmented with an artificial viscosity term.\n                  <\/jats:p>","DOI":"10.1007\/s13235-024-00596-1","type":"journal-article","created":{"date-parts":[[2024,10,4]],"date-time":"2024-10-04T13:01:43Z","timestamp":1728046903000},"page":"406-416","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Convergent Semi-Lagrangian Scheme for the Game $$\\infty $$-Laplacian"],"prefix":"10.1007","volume":"15","author":[{"given":"Elisabetta","family":"Carlini","sequence":"first","affiliation":[]},{"given":"Silvia","family":"Tozza","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,10,4]]},"reference":[{"issue":"4","key":"596_CR1","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1090\/S0273-0979-04-01035-3","volume":"41","author":"G Aronsson","year":"2004","unstructured":"Aronsson G, Crandall MG, Juutinen P (2004) A tour of the theory of absolutely minimizing functions. Bull Am Math Soc 41(4):439\u2013505","journal-title":"Bull Am Math Soc"},{"issue":"11\u201312","key":"596_CR2","doi-asserted-by":"publisher","first-page":"2323","DOI":"10.1081\/PDE-100107824","volume":"26","author":"G Barles","year":"2001","unstructured":"Barles G, Busca J (2001) Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Commun Partial Differ Equ 26(11\u201312):2323\u20132337","journal-title":"Commun Partial Differ Equ"},{"key":"596_CR3","first-page":"271","volume":"4","author":"G Barles","year":"1991","unstructured":"Barles G, Souganidis PE (1991) Convergence of approximation schemes for fully nonlinear second-order equations. Asymp Anal 4:271\u2013283","journal-title":"Asymp Anal"},{"key":"596_CR4","doi-asserted-by":"publisher","first-page":"409","DOI":"10.4171\/ifb\/240","volume":"12","author":"E Carlini","year":"2011","unstructured":"Carlini E, Falcone M, Ferretti R (2011) Convergence of a large time step scheme for mean curvature motion. Interfaces Free Bound 12:409\u2013441","journal-title":"Interfaces Free Bound"},{"key":"596_CR5","volume":"461","author":"E Carlini","year":"2024","unstructured":"Carlini E, Tozza S (2024) A scheme for the game $$p$$-Laplacian and its application to image inpainting. Appl Math Comput 461:128299","journal-title":"Appl Math Comput"},{"issue":"3","key":"596_CR6","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1109\/83.661188","volume":"7","author":"V Caselles","year":"1998","unstructured":"Caselles V, Morel J-M, Sbert C (1998) An axiomatic approach to image interpolation. IEEE Trans Image Process 7(3):376\u2013386","journal-title":"IEEE Trans Image Process"},{"key":"596_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S0273-0979-1992-00266-5","volume":"27","author":"MG Crandall","year":"1992","unstructured":"Crandall MG, Ishii H, Lions PL (1992) User\u2019s guide to viscosity solutions of second order partial differential equations. Bull Am Math Soc 27:1\u201367","journal-title":"Bull Am Math Soc"},{"key":"596_CR8","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/s002110050228","volume":"75","author":"MG Crandall","year":"1996","unstructured":"Crandall MG, Lions PL (1996) Convergent difference schemes for nonlinear parabolic equations and mean curvature motion. Numer Math 75:17\u201341","journal-title":"Numer Math"},{"key":"596_CR9","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1016\/j.apnum.2012.11.006","volume":"73","author":"M Falcone","year":"2013","unstructured":"Falcone M, Finzi Vita S, Giorgi T, Smits RG (2013) A semi-Lagrangian scheme for the game p-Laplacian via p-averaging. Appl Numer Math 73:63\u201380","journal-title":"Appl Numer Math"},{"key":"596_CR10","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/BF00386368","volume":"123","author":"R Jensen","year":"1993","unstructured":"Jensen R (1993) Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch Rational Mech Anal 123:51\u201374","journal-title":"Arch Rational Mech Anal"},{"key":"596_CR11","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1137\/S0036141000372179","volume":"33","author":"P Juutinen","year":"2001","unstructured":"Juutinen P, Lindqvist P, Manfredi J (2001) On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. SIAM J Math Anal 33:699\u2013717","journal-title":"SIAM J Math Anal"},{"key":"596_CR12","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1007\/s002050050157","volume":"148","author":"P Juutinen","year":"1999","unstructured":"Juutinen P, Lindqvist P, Manfredi J (1999) The $$\\infty $$-eigenvalue problem. S Arch Ration Mech Anal 148:89\u2013105","journal-title":"S Arch Ration Mech Anal"},{"key":"596_CR13","first-page":"473","volume":"103","author":"J Manfredi","year":"1988","unstructured":"Manfredi J (1988) p-harmonic functions in the plane. Proc Am Math Soc 103:473\u2013479","journal-title":"Proc Am Math Soc"},{"key":"596_CR14","doi-asserted-by":"publisher","first-page":"1217","DOI":"10.1090\/S0025-5718-04-01688-6","volume":"74","author":"AM Oberman","year":"2004","unstructured":"Oberman AM (2004) A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions. Math Comput 74:1217\u20131230","journal-title":"Math Comput"},{"key":"596_CR15","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/j.cam.2012.11.023","volume":"254","author":"AM Oberman","year":"2013","unstructured":"Oberman AM (2013) Finite difference methods for the infinity Laplace and p-Laplace equations. J Comput Appl Math 254:65\u201380","journal-title":"J Comput Appl Math"},{"key":"596_CR16","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1215\/00127094-2008-048","volume":"145","author":"Y Peres","year":"2008","unstructured":"Peres Y, Sheffield S (2008) Tug-of-war with noise: a game-theoretic view of the p-Laplacian. Duke Math J 145:91\u2013120","journal-title":"Duke Math J"},{"key":"596_CR17","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1090\/S0894-0347-08-00606-1","volume":"22","author":"Y Peres","year":"2009","unstructured":"Peres Y, Schramm O, Sheffield S, Wilson DB (2009) Tug-of-war and the infinity Laplacian. J Am Math Soc 22:167\u2013210","journal-title":"J Am Math Soc"},{"key":"596_CR18","doi-asserted-by":"crossref","unstructured":"Precup R (2002) The Leray\u2013Schauder principle and applications. In: Methods in Nonlinear Integral Equations. Springer, Dordrecht","DOI":"10.1201\/9781420022209"},{"key":"596_CR19","unstructured":"Rossi JD The infinity Laplacian and tug-of-war games (games that PDE people like to play). https:\/\/crm.sns.it\/media\/course\/3204\/Julio%20Rossi%20Lectures.pdf"}],"container-title":["Dynamic Games and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-024-00596-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s13235-024-00596-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-024-00596-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,19]],"date-time":"2025-05-19T03:10:50Z","timestamp":1747624250000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s13235-024-00596-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,4]]},"references-count":19,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,5]]}},"alternative-id":["596"],"URL":"https:\/\/doi.org\/10.1007\/s13235-024-00596-1","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-3914722\/v1","asserted-by":"object"}]},"ISSN":["2153-0785","2153-0793"],"issn-type":[{"value":"2153-0785","type":"print"},{"value":"2153-0793","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,4]]},"assertion":[{"value":"19 September 2024","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 October 2024","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no conflict of interest to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"No human or animal participants were involved in this study; hence, ethical approval was not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}}]}}