{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T17:58:18Z","timestamp":1776707898522,"version":"3.51.2"},"reference-count":23,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The present article is concerned with solving Bernoulli\u2019s exterior\nfree boundary problem in the case of an interior boundary that is\nrandom. We provide a new regularity result on the map that sends\na parametrization of the inner boundary to a parametrization of the\nouter boundary. Moreover, assuming that the interior boundary\nis convex, also the exterior boundary is convex, which enables to\nidentify the boundaries with support functions and to determine their\nexpectations. We in particular construct a confidence region for the\nouter boundary based on Aumann\u2019s expectation and provide a numerical\nmethod to compute it.<\/jats:p>","DOI":"10.1515\/cmam-2022-0038","type":"journal-article","created":{"date-parts":[[2022,11,23]],"date-time":"2022-11-23T10:39:04Z","timestamp":1669199944000},"page":"333-352","source":"Crossref","is-referenced-by-count":3,"title":["Bernoulli Free Boundary Problems Under Uncertainty: The Convex Case"],"prefix":"10.1515","volume":"23","author":[{"given":"Marc","family":"Dambrine","sequence":"first","affiliation":[{"name":"Laboratoire de Math\u00e9matiques et de leurs Applications de Pau \u2013 UMR CNRS 5142 , Universit\u00e9 de Pau et des Pays de l\u2019Adour , Pau , France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0093-5706","authenticated-orcid":false,"given":"Helmut","family":"Harbrecht","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science , University of Basel , Basel , Switzerland"}]},{"given":"Benedicte","family":"Puig","sequence":"additional","affiliation":[{"name":"Laboratoire de Math\u00e9matiques et de leurs Applications de Pau \u2013 UMR CNRS 5142 , Universit\u00e9 de Pau et des Pays de l\u2019Adour , Pau , France"}]}],"member":"374","published-online":{"date-parts":[[2022,11,24]]},"reference":[{"key":"2023033113095740664_j_cmam-2022-0038_ref_001","doi-asserted-by":"crossref","unstructured":"G.  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Seri,\nBootstrap confidence sets for the Aumann mean of a random closed set,\nComput. Statist. Data Anal. 71 (2014), 803\u2013817.","DOI":"10.1016\/j.csda.2012.10.015"},{"key":"2023033113095740664_j_cmam-2022-0038_ref_005","doi-asserted-by":"crossref","unstructured":"M.  Dambrine, C.  Dapogny and H.  Harbrecht,\nShape optimization for quadratic functionals and states with random right-hand sides,\nSIAM J. Control Optim. 53 (2015), no. 5, 3081\u20133103.","DOI":"10.1137\/15M1017041"},{"key":"2023033113095740664_j_cmam-2022-0038_ref_006","doi-asserted-by":"crossref","unstructured":"M.  Dambrine, H.  Harbrecht, M. D.  Peters and B.  Puig,\nOn Bernoulli\u2019s free boundary problem with a random boundary,\nInt. J. Uncertain. Quantif. 7 (2017), no. 4, 335\u2013353.","DOI":"10.1615\/Int.J.UncertaintyQuantification.2017019550"},{"key":"2023033113095740664_j_cmam-2022-0038_ref_007","doi-asserted-by":"crossref","unstructured":"M.  Dambrine, H.  Harbrecht and B.  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