{"id":{"repo_id":"trento","oai_identifier":"oai:iris.unitn.it:11572/417050"},"canonical_url":"https://search.dev.ndltd.org/etd/trento/oai:iris.unitn.it:11572/417050","repository":{"repo_id":"trento","name":"Università degli Studi di Trento","base_url":"https://iris.unitn.it/oai/request"},"display":{"title":"Boundary Properties for Almost-Minimizers of the Relative Perimeter","abstract":"Let A be an Euclidean open Lipschitz set. This dissertation aims to discuss some results concerning the boundary regularity for almost-minimizers of the relative perimeter in A. An almost-minimizer of the relative perimeter in A is a measurable set E that minimizes the perimeter functional P(E;A), roughly speaking the (n-1)-area of the boundary of E in A, among local competitors for E. Important examples of almost-minimizers in A are given, for instance, by the solutions to relative isoperimetric problems like min { P(E;A) : E is contained in A and |E| = m}. While when A is smooth the theory of the boundary regularity for almost-minimizers is well-established, little is known when the boundary of A contains singular points such as edges, vertices, cusps, etc. In particular, we prove a boundary Monotonicity Formula, holding under a so-called visibility condition on A at a point x on the boundary of A, and a Vertex-skipping Theorem, valid when n = 3 and A is convex. This latter result establishes that the closure of the boundary of an almost-minimizer of the relative perimeter in a 3-dimensional open, convex set A cannot contain vertex-type singularities of the boundary of A. The optimality of the dimensional restriction n = 3 is also examined in the thesis.","abstract_html":"Let A be an Euclidean open Lipschitz set. This dissertation aims to discuss some results concerning the boundary regularity for almost-minimizers of the relative perimeter in A. An almost-minimizer of the relative perimeter in A is a measurable set E that minimizes the perimeter functional P(E;A), roughly speaking the (n-1)-area of the boundary of E in A, among local competitors for E. Important examples of almost-minimizers in A are given, for instance, by the solutions to relative isoperimetric problems like min { P(E;A) : E is contained in A and |E| = m}. While when A is smooth the theory of the boundary regularity for almost-minimizers is well-established, little is known when the boundary of A contains singular points such as edges, vertices, cusps, etc. In particular, we prove a boundary Monotonicity Formula, holding under a so-called visibility condition on A at a point x on the boundary of A, and a Vertex-skipping Theorem, valid when n = 3 and A is convex. This latter result establishes that the closure of the boundary of an almost-minimizer of the relative perimeter in a 3-dimensional open, convex set A cannot contain vertex-type singularities of the boundary of A. The optimality of the dimensional restriction n = 3 is also examined in the thesis.","abstract_has_math":false,"creators":["Vianello, Giacomo"],"institution":"Università degli studi di Trento","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Leonardi, Gian Paolo"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-07-08","date_published":"2024-07-08","updated_at":"2026-07-24T05:04:45Z","subjects":["Boundary regularity, monotonicity, almost-minimizers, capillarity, singularities","Settore MAT/05 - Analisi Matematica"],"languages":["eng"],"rights":["info:eu-repo/semantics/openAccess","license:Tutti i diritti riservati (All rights reserved)","license uri:iris.PRI01"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["http://dx.doi.org/10.15168/11572_417050","10.15168/11572_417050"],"render_values":[{"text":"http://dx.doi.org/10.15168/11572_417050","href":"http://dx.doi.org/10.15168/11572_417050","code":true},{"text":"10.15168/11572_417050","href":"https://doi.org/10.15168/11572_417050","code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/11572/417050","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Vianello, Giacomo","Leonardi, Gian Paolo"]},{"key":"dc:creator","label":"Author","values":["Vianello, Giacomo"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2024-07-08"]},{"key":"dc:publisher","label":"Institution","values":["Università degli studi di Trento","place:TRENTO"]},{"key":"dc:relation","label":"Dc Relation","values":["firstpage:1","lastpage:102","numberofpages:102"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/doctoralThesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Boundary regularity, monotonicity, almost-minimizers, capillarity, singularities","Settore MAT/05 - Analisi Matematica"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess","license:Tutti i diritti riservati (All rights reserved)","license uri:iris.PRI01"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/11572/417050","http://dx.doi.org/10.15168/11572_417050","10.15168/11572_417050"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let A be an Euclidean open Lipschitz set. This dissertation aims to discuss some results concerning the boundary regularity for almost-minimizers of the relative perimeter in A. An almost-minimizer of the relative perimeter in A is a measurable set E that minimizes the perimeter functional P(E;A), roughly speaking the (n-1)-area of the boundary of E in A, among local competitors for E. Important examples of almost-minimizers in A are given, for instance, by the solutions to relative isoperimetric problems like min { P(E;A) : E is contained in A and |E| = m}. While when A is smooth the theory of the boundary regularity for almost-minimizers is well-established, little is known when the boundary of A contains singular points such as edges, vertices, cusps, etc. In particular, we prove a boundary Monotonicity Formula, holding under a so-called visibility condition on A at a point x on the boundary of A, and a Vertex-skipping Theorem, valid when n = 3 and A is convex. This latter result establishes that the closure of the boundary of an almost-minimizer of the relative perimeter in a 3-dimensional open, convex set A cannot contain vertex-type singularities of the boundary of A. The optimality of the dimensional restriction n = 3 is also examined in the thesis."]},{"key":"dc:title","label":"Title","values":["Boundary Properties for Almost-Minimizers of the Relative Perimeter"]}]}],"canonical_facts":{"dc:contributor":["Vianello, Giacomo","Leonardi, Gian Paolo"],"dc:creator":["Vianello, Giacomo"],"dc:date":["2024-07-08"],"dc:description":["Let A be an Euclidean open Lipschitz set. This dissertation aims to discuss some results concerning the boundary regularity for almost-minimizers of the relative perimeter in A. An almost-minimizer of the relative perimeter in A is a measurable set E that minimizes the perimeter functional P(E;A), roughly speaking the (n-1)-area of the boundary of E in A, among local competitors for E. Important examples of almost-minimizers in A are given, for instance, by the solutions to relative isoperimetric problems like min { P(E;A) : E is contained in A and |E| = m}. While when A is smooth the theory of the boundary regularity for almost-minimizers is well-established, little is known when the boundary of A contains singular points such as edges, vertices, cusps, etc. In particular, we prove a boundary Monotonicity Formula, holding under a so-called visibility condition on A at a point x on the boundary of A, and a Vertex-skipping Theorem, valid when n = 3 and A is convex. This latter result establishes that the closure of the boundary of an almost-minimizer of the relative perimeter in a 3-dimensional open, convex set A cannot contain vertex-type singularities of the boundary of A. The optimality of the dimensional restriction n = 3 is also examined in the thesis."],"dc:identifier":["https://hdl.handle.net/11572/417050","http://dx.doi.org/10.15168/11572_417050","10.15168/11572_417050"],"dc:language":["eng"],"dc:publisher":["Università degli studi di Trento","place:TRENTO"],"dc:relation":["firstpage:1","lastpage:102","numberofpages:102"],"dc:rights":["info:eu-repo/semantics/openAccess","license:Tutti i diritti riservati (All rights reserved)","license uri:iris.PRI01"],"dc:subject":["Boundary regularity, monotonicity, almost-minimizers, capillarity, singularities","Settore MAT/05 - Analisi Matematica"],"dc:title":["Boundary Properties for Almost-Minimizers of the Relative Perimeter"],"dc:type":["info:eu-repo/semantics/doctoralThesis"]},"updated_at":"2026-07-24T05:04:45Z"}