Emerging Developments in Interfaces and Free Boundaries

The field of the mathematical and numerical analysis of systems of nonlinear partial differential equations involving interfaces and free boundaries is a well established and flourishing area of research. This workshop focused on recent developments and emerging new themes. By bringing together experts in these fields we achieved progress in open questions and developed novel research directions in mathematics related to interfaces and free boundaries. This interdisciplinary workshop brought together researchers from distinct mathematical fields such as analysis, computation, optimisation and modelling to discuss emerging challenges. Mathematics Subject Classification (2010): 35-XX, 49-XX, 65-XX. Introduction by the Organisers The workshop Emerging Developments in Interfaces and Free Boundaries, organised by Charles M. Elliott (Warwick), Yoshikazu Giga (Tokyo), Michael Hinze (Hamburg) and Vanessa Styles (Brighton) was attended by 54 participants from Austria, France, Germany, Great Britain, Japan, and the United States, with expertise from three main areas: optimal control of partial differential equations, modelling involving free boundary problems and mathematical and numerical analysis of free boundary problems. Apart from discussing current problems, techniques and issues across the differing communities the focus of the workshop was set on 268 Oberwolfach Report 6/2017 (1) Diffuse interface methods – modelling, analysis, and optimization, (2) Analysis and approximation of PDEs on evolving domains, (3) Evolution of interfaces and free boundaries with applications. The presentations of Abels, Aland, Blank, Deckelnick, Kahle, Lam, Salgado, Stoll and Wirth concerned modelling, analytical, numerical and optimization approaches to interfaces and free boundaries using diffuse interface methods, while Antil, Bartels, Chambolle, Fritz, van Gennip, Mi-Ho Giga, Hamamuki, Kimura, Lehrenfeld, Nochetto, Ohtsuka, Olshanski, Schulz, Shirakawa and Yamamoto presented talks related to analysis and numerical methods for problems related to evolving shapes and PDEs on evolving domains. Gräser, Kenmochi, King, Liu, Luckhaus, Ranner and Ren gave talks on mathematical modelling of applications related to interfaces and free boundaries. To offer young researchers a stage for presenting their research, a session for young researchers was organized on Wednesday evening where Tatsu-Hiko Miura, Tatsuya Miura and Siebenborn together with the Oberwolfach Leibniz Graduate Students Alphonse and Djurdjevac took this opportunity and gave talks on their current research results. Surveys and articles concerning mathematical and numerical approaches to interfaces and free boundary problems may be found in the conference and symposium proceedings [3, 6, 11, 15]. The books [2, 10, 13] contain models, numerical methods and analysis for variational and level set approaches. Modern mathematical concepts of control and optimization with partial differential equation constraints are developed in [14, 16]. Shape optimisation and problems of optimal design are surveyed in [4, 5]. Analysis and numerics for surface PDEs are presented in [1, 9, 17]. Also we mention surveys of models and numerical methods for interface evolution involving curvature, [8, 12]. Finally we remark that many recent references concerning the issues of the workshop are provided at the end of each extended abstract. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Moreover, the MFO and the workshop organizers would like to thank the Simons Foundation for supporting Professor Ricardo Nochetto in the “Simons Visiting Professors” program at the MFO.