Free Boundary Problems 2026

Scientific Programme

Thematically, the conference will cover a broad range of topics from modeling, analysis, optimization to numerical realization. These include aspects such as uncertainties or stochastics, data-driven intelligent numerical solution techniques (e.g. utilizing techniques of machine learning), and applications.

Plenary Talks

PL

Contributed Talks

CS

Minisymposia

Parallel minisymposia at FBP.

• Degenerate PDEs and Free Boundary Problems

  MS-1

  Degenerate and singular partial differential equations arise naturally in the mathematical modeling of diffusion, phase transitions, geometric phenomena, fluid flows, and biological processes. The lack of uniform ellipticity or parabolicity in these equations often gives rise to free boundaries and interfaces whose structure encodes subtle analytical and geometric features of the underlying models. In recent years, significant progress has been made in understanding the regularity, stability, and geometric properties of solutions and free boundaries associated with nonlinear, nonlocal, and fully nonlinear degenerate equations. These developments have revealed deep connections among free boundary theory, nonlinear potential theory, the calculus of variations, and geometric analysis.

  This minisymposium will bring together researchers working on a broad range of problems at the interface of degenerate PDEs and free boundary theory. The talks will highlight recent advances in obstacle-type and Bernoulli-type problems, nonlinear diffusion equations, nonlocal and fully nonlinear operators, singular perturbation methods, and the regularity theory of solutions and interfaces. By gathering experts with complementary perspectives, the minisymposium aims to showcase current developments, foster interactions across different research communities, and stimulate new directions in the analysis of nonlinear PDEs and free boundary phenomena.

• Geometric Flows and Evolving Free Boundaries

  MS-2

  This minisymposium is focused on recent advances in the analysis of geometric flows and evolving free boundaries. A particular emphasis lies on the underlying gradient flow structure of such problems and how one capitalizes on it for rigorous analysis. The minisymposium brings together an international group of researchers, new and established, to discuss topics covering a representative range of mathematical questions and state-of-the-art techniques for geometric flows. These include, but are not limited to, weak and strong solution theories, quantitative relaxation to equilibrium, and effective geometric motion through random obstacles.

• Numerical Methods for Geometric PDEs

  MS-3

  Geometric partial differential equations arise in a broad range of free-boundary and interface-evolution problems, where the geometry and the governing physical processes are strongly coupled. Their numerical approximation presents substantial challenges in achieving accuracy, stability, efficiency, and reliable treatment of the evolving geometry. This minisymposium will provide a forum for recent advances in the design, implementation, and mathematical analysis of computational methods for geometric PDEs. It will also highlight applications in areas such as differential geometry, biological sciences, and engineering.

• Optimization and Free Boundaries

  MS-4

  This minisymposium explores the intersection of mathematical optimization and free boundary problems, highlighting recent advancements in both theoretical frameworks and applied methodologies. The program features investigations into robust numerical algorithms, ranging from proximal point methods for variational inequalities to rigorous convergence analyses for shape optimization schemes in infinite-dimensional spaces. We also examine versatile modeling strategies, including phase-field approaches for microstructure evolution and reduced-order computational fluid dynamics for industrial reactor design. Furthermore, the sessions address the control-theoretic aspects of interface dynamics, specifically focusing on the manipulation of boundaries within obstacle-type problems. Together, these contributions illustrate the synergistic relationship between advanced algorithmic development and practical optimization across diverse scientific and engineering disciplines.

• Free Boundaries in Shape Optimization

  MS-5

  Free boundary problems play a central role in many shape optimization models, where the geometry of the optimal domain has to be determined in order to optimize a given criterion and must be obtained together with the solution of the governing equations. Such problems arise in a broad range of applications, including fluid mechanics, materials science, phase transitions, and optimal design, and present significant analytical and computational challenges.
  This minisymposium will bring together researchers working on theoretical, numerical, and applied aspects of free boundaries in shape optimization. Topics of interest include existence, regularity, and qualitative properties of optimal shapes; variational and PDE-based methods; geometric and topological optimization; numerical algorithms for free boundary computation; and applications motivated by science and engineering. The goal is to provide a forum for the exchange of recent advances, novel methodologies, and emerging perspectives, while promoting interactions between experts in analysis, optimization, and scientific computing.

• Hydrodynamic Models with Moving Contact Lines

  MS-6

  Fluid flows over substrates lead to free boundary problems whose mathematical properties depend much on the modelling assumptions at small scales. Where the free surface meets a wall, the no-slip condition is usually incompatible with a moving contact line, and the available regularisations produce genuinely different evolution problems. Moving contact lines can elevate that difficulty: dissipation and degeneracy concentrate at the free boundary, and the regularity of solutions there decides which models admit a rigorous theory and can be discretised robustly.

  The minisymposium covers the full hydrodynamic setting, that is free boundary problems for the Stokes and Navier-Stokes equations, alongside reduced thin-film descriptions in the form of degenerate parabolic equations and systems. Contributions concern the analysis of the resulting equations, study of qualitative properties of solutions, modeling questions and model derivation and derivation of numerical discretisation strategies for hydrodynamic models with and without moving contact lines.

• Phase Field Methods in Real-World Applications

  MS-7

  Phase field methods have become an important mathematical tool for modeling interfacial phenomena in a wide range of scientific and engineering applications. Their diffuse-interface nature makes them particularly suitable for problems involving evolving geometries, topological changes, and coupled multiphysics effects, as they arise for instance in materials science, fracture mechanics, geophysics, multiphase flow, biological systems, medical application, electrochemistry, and additive manufacturing.
  This mini-symposium is devoted to the mathematical analysis and application-driven development of phase field models. Topics of interest include well-posedness, regularity, stability, thermodynamic consistency, and the analysis of numerical approximations. Special attention will be given to models motivated by real-world problems and to the mathematical challenges arising from coupling mechanisms, nonlinear effects, complex geometries, and multiscale phenomena.
  The mini-symposium aims to provide a forum for researchers working on rigorous analytical foundations, numerical analysis, and realistic applications of phase field methods, with the goal of strengthening the connection between mathematical analysis and Real-World Applications.

• Phase Transitions and Pattern Formation

  MS-8

  Phase-field models offer a versatile framework for the description of evolving interfaces, phase transitions, and pattern formation in complex systems. Beyond classical phase separation, such models arise in diverse applications ranging from multiphase fluid dynamics to mathematical biology, where pattern formation plays a key role in the organization of structures at cellular surfaces. This minisymposium brings together recent developments in the analysis, modeling and numerical treatment of phase-field models and related nonlinear evolution equations. Topics include diffuse-interface models for multiphase flows, pattern-forming systems, and coupled phase-field equations with transport mechanisms or fluid dynamics.

• Fluid-structure Interactions

  MS-9

  Fluid‑structure interaction (FSI) problems form a central class of coupled partial differential equation systems where the evolution of a viscous/inviscid incompressible/compressible fluid is directly linked to the dynamics of an elastic or viscoelastic solid and the interface separating them. These interactions pose fundamental analytical challenges due to nonlinear coupling across moving domains, free boundary motion, and multiphysics effects such as porous media, thermal coupling, and contact formation. This minisymposium brings together leading contributions that address these challenges from the perspectives of existence, uniqueness, regularity, stability, and singular limits for FSI models, as well as numerical and computational frameworks faithful to the underlying continuum mechanics.

• Free Boundaries in Biology

  MS-10

  In this minisymposium we are interested in how boundaries and interfaces evolve in biological systems, especially in situations where those boundaries are not fixed, but move and change over time. These so-called free boundary problems show up in many biological processes, including tumor growth, cell-cell adhesion, tissue invasion, and other important biological phenomena. Our aim is to bring together experts in analytical, computational, and modeling aspects, in order to better understand the structure of biological free boundary problems. A special focus will also be to understand how they may emerge from diffuse or compressible approximations. For example, sharp-interface models can often be seen as limits of phase-field systems, while incompressible free boundary problems may arise from compressible models.

• Thin Material Structures

  MS-11

  Thin material structures, elastic films or surfaces come in a great variety: they range from fluid to solid, can be passive or active, within linear response or strongly nonlinear, relaxed or pre-stressed, etc. They display fascinating and often unexpected mechanical and rheological properties. The peculiar properties of these structures have a geometric origin. Within this minisymposium we aim to uncover the fundamental geometric interactions driving the distinctive behavior. We mainly focus on two applications: fluid deformable surfaces and solid-state
  dewetting. We address modelling issues, numerical analysis and scientific computing problems.

• Free Boundaries in Active Matter

  MS-12

  This mini-symposium is on mathematical modeling, analysis, and numerical simulation of active matter systems, which arise in material science (e.g. new technological devices) and in biology (e.g. bacterial swarms). They are characterized by non-linear and anisotropic material responses, and many of these systems exhibit moving boundaries or geometric motion. The talks in this mini-symposium cover a range of topics and applications with new insight into numerical analysis and mathematical modeling.

• Free Boundary Problems in Data Science and Machine Learning

  MS-13

  This minisymposium gathers researchers from different career levels interested in synergies between free boundary problems and machine learning. On one hand, this encompasses the use of machine learning tools like operator learning for deriving efficient and versatile numerical schemes to approximate the solution of free boundary problems like the mean curvature flow. On the other hand, it involves the mathematical analysis of geometric problems arising in machine learning algorithms. For example, this kind of analysis can provide mathematical justification for graph clustering or robust classification algorithms using neural networks. The aim of the minisymposium is to present recent developments to a mathematically diverse audience, connect and exchange ideas with the free boundary community, and spark discussions about future collaborations.?

• Domain Walls and Patterns in Local and Non-local Geometric Variational Problems

  MS-14

  This minisymposium focuses on recent mathematical advances at the intersection of the Calculus of Variations, Geometric Measure Theory, and Nonlinear Partial Differential Equations, with a strong emphasis on free boundary problems and interface dynamics. The session aims to bring together different analytical perspectives to discuss cutting-edge techniques for modeling and understanding these phenomena. Key topics include the regularity of free boundaries and energy minimizers, and the geometric analysis of local and non-local variational models for pattern formation.