Ros, XavierDomingo Pasarin, Joan2024-12-032024-12-032024-06-27https://hdl.handle.net/2445/216888Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Curs: 2023-2024. Director: Xavier RosIn this work we study the regularity of Lipschitz free boundaries in the Alt-Caffarelli problem. We prove that Lipschitz free boundaries are $C^{1, \alpha}$ by exploiting the rescaling invariance of the problem and the initial Lipschitz regularity of the boundary. Moreover, we also show that $C^{1, \alpha}$ boundaries are smooth, which combined with the previous result implies that Lipschitz free boundaries are smooth.39 p.application/pdfengcc by-nc-nd (c) Joan Domingo Pasarin, 2024http://creativecommons.org/licenses/by-nc-nd/3.0/es/Funcions harmòniquesEquacions en derivades parcialsTreballs de fi de màsterProblemes de contornHarmonic functionsPartial differential equationsMaster's thesisBoundary value problemsinfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/openAccess