{"id":362,"date":"2025-11-16T14:51:34","date_gmt":"2025-11-16T17:51:34","guid":{"rendered":"https:\/\/eventos.cmm.uchile.cl\/ipphys2026\/?page_id=362"},"modified":"2025-11-16T18:09:33","modified_gmt":"2025-11-16T21:09:33","slug":"mini-course-1","status":"publish","type":"page","link":"https:\/\/eventos.cmm.uchile.cl\/ipphys2026\/program\/mini-course-1\/","title":{"rendered":"Perturbations of an elliptic PDE and applications to inverse problems and shape optimization."},"content":{"rendered":"<h3><strong>Eric Bonnetier<\/strong><\/h3>\n<h3>Universit\u00e9 de Grenoble-Alpes<\/h3>\n<p>Asymptotic expansions of the solution to an elliptic PDE in the presence<br \/>\nof inclusions of small size have found succesfull applications in inverse<br \/>\nproblems, as a means to detect inhomogeneities from boundary measurements<br \/>\nin a robust way.<\/p>\n<p>In the course, we revisit some of this work and the applications<br \/>\nit led to. We also consider perturbations on the boundary, such as<br \/>\nreplacing a Neumann boundary condition by a Dirichlet boundary condition<br \/>\n(or vice-versa). We derive the general form of an asymptotic expansion<br \/>\nbetween the solutions of the perturbed and unperturbed PDE&#8217;s, and show that the<br \/>\nconvergence relies on an argument similar to compensated compactness.<\/p>\n<p>We illustrate two of the applications of the resulting asymptotic formulas~:<br \/>\nWe consider shape optimization problems, in which the subset of the boundary of<br \/>\na domain, where a specific boundary condition is applied, is part of the design<br \/>\nvariables. We also discuss the problem of how such asymptotic expansions can<br \/>\nbe used to achieve approximate cloaking.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Eric Bonnetier Universit\u00e9 de Grenoble-Alpes Asymptotic expansions of the solution to an elliptic PDE in the presence of inclusions of small size have found succesfull applications in inverse problems, as a means to detect inhomogeneities from boundary measurements in a robust way. In the course, we revisit some of this work and the applications it &hellip; <a href=\"https:\/\/eventos.cmm.uchile.cl\/ipphys2026\/program\/mini-course-1\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Perturbations of an elliptic PDE and applications to inverse problems and shape optimization.<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":116,"featured_media":0,"parent":16,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-362","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/eventos.cmm.uchile.cl\/ipphys2026\/wp-json\/wp\/v2\/pages\/362","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/eventos.cmm.uchile.cl\/ipphys2026\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/eventos.cmm.uchile.cl\/ipphys2026\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/eventos.cmm.uchile.cl\/ipphys2026\/wp-json\/wp\/v2\/users\/116"}],"replies":[{"embeddable":true,"href":"https:\/\/eventos.cmm.uchile.cl\/ipphys2026\/wp-json\/wp\/v2\/comments?post=362"}],"version-history":[{"count":5,"href":"https:\/\/eventos.cmm.uchile.cl\/ipphys2026\/wp-json\/wp\/v2\/pages\/362\/revisions"}],"predecessor-version":[{"id":386,"href":"https:\/\/eventos.cmm.uchile.cl\/ipphys2026\/wp-json\/wp\/v2\/pages\/362\/revisions\/386"}],"up":[{"embeddable":true,"href":"https:\/\/eventos.cmm.uchile.cl\/ipphys2026\/wp-json\/wp\/v2\/pages\/16"}],"wp:attachment":[{"href":"https:\/\/eventos.cmm.uchile.cl\/ipphys2026\/wp-json\/wp\/v2\/media?parent=362"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}