{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T19:19:06Z","timestamp":1774379946980,"version":"3.50.1"},"reference-count":40,"publisher":"American Mathematical Society (AMS)","issue":"342","license":[{"start":{"date-parts":[[2024,2,6]],"date-time":"2024-02-06T00:00:00Z","timestamp":1707177600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>This paper proposes a regularization of the Monge\u2013Amp\u00e8re equation in planar convex domains through uniformly elliptic Hamilton\u2013Jacobi\u2013Bellman equations. The regularized problem possesses a unique strong solution <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u Subscript epsilon\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>u<\/mml:mi>\n      <mml:mi>\u03b5<\/mml:mi>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">u_\\varepsilon<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and is accessible to the discretization with finite elements. This work establishes uniform convergence of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u Subscript epsilon\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>u<\/mml:mi>\n      <mml:mi>\u03b5<\/mml:mi>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">u_\\varepsilon<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> to the convex Alexandrov solution <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u\">\n  <mml:semantics>\n    <mml:mi>u<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">u<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> to the Monge\u2013Amp\u00e8re equation as the regularization parameter <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon\">\n  <mml:semantics>\n    <mml:mi>\u03b5<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">\\varepsilon<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> approaches <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0\">\n  <mml:semantics>\n    <mml:mn>0<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">0<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. A mixed finite element method for the approximation of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u Subscript epsilon\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>u<\/mml:mi>\n      <mml:mi>\u03b5<\/mml:mi>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">u_\\varepsilon<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is proposed, and the regularized finite element scheme is shown to be uniformly convergent. The class of admissible right-hand sides are the functions that can be approximated from below by positive continuous functions in the <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript 1\">\n  <mml:semantics>\n    <mml:msup>\n      <mml:mi>L<\/mml:mi>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:msup>\n    <mml:annotation encoding=\"application\/x-tex\">L^1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> norm. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u\">\n  <mml:semantics>\n    <mml:mi>u<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">u<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>.<\/p>","DOI":"10.1090\/mcom\/3794","type":"journal-article","created":{"date-parts":[[2023,2,6]],"date-time":"2023-02-06T15:54:56Z","timestamp":1675698896000},"page":"1467-1490","source":"Crossref","is-referenced-by-count":7,"title":["Convergence of a regularized finite element discretization of the two-dimensional Monge\u2013Amp\u00e8re equation"],"prefix":"10.1090","volume":"92","author":[{"given":"Dietmar","family":"Gallistl","sequence":"first","affiliation":[]},{"given":"Ngoc Tien","family":"Tran","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,2,6]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics (Amsterdam)","isbn-type":"print","volume-title":"Sobolev spaces","volume":"140","author":"Adams, Robert A.","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0120441438","edition":"2"},{"issue":"1","key":"2","first-page":"5","article-title":"Dirichlet\u2019s problem for the equation \ud835\udc37\ud835\udc52\ud835\udc61||\ud835\udc67\u1d62\u2c7c||=\ud835\udf11(\ud835\udc67\u2081,\u22ef,\ud835\udc67_{\ud835\udc5b},\ud835\udc67,\ud835\udc65\u2081,\u22ef,\ud835\udc65_{\ud835\udc5b}). 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