Free boundary problems: the forefront of current and future developments

The term free boundary problem (FBP) refers, in the modern applied mathematical literature, to a problem in which one or several variables must be determined in different domains of the space, or space–time, for which each variable is governed in its domain by a set of state laws. If the domains

[1]  Henrik Shahgholian,et al.  An overview of unconstrained free boundary problems , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[2]  Juan Luis Vázquez,et al.  Nonlinear Diffusion with Fractional Laplacian Operators , 2012 .

[3]  Mathematisches Forschungsinstitut Oberwolfach,et al.  Hyperbolic Conservation Laws , 2004 .

[4]  E. N. Dancer,et al.  Spatial segregation limit of a competition–diffusion system , 1999, European Journal of Applied Mathematics.

[5]  C. M. Dafermos,et al.  Hyberbolic [i.e. Hyperbolic] conservation laws in continuum physics , 2005 .

[6]  Henrik Shahgholian,et al.  Regularity of free boundaries a heuristic retro , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[7]  J. Vázquez.  The Porous Medium Equation: Mathematical Theory , 2006 .

[8]  C. M. Elliott,et al.  Weak and variational methods for moving boundary problems , 1982 .

[9]  L. Caffarelli,et al.  A Geometric Approach to Free Boundary Problems , 2005 .

[10]  A. Petrosyan,et al.  Regularity of Free Boundaries in Obstacle-type Problems , 2012 .

[11]  Steve Shkoller,et al.  Global stability of steady states in the classical Stefan problem for general boundary shapes , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[12]  Marcos Antón Amayuelas.  The Stefan problem , 2015 .

[13]  Juan Luis Vázquez,et al.  Some free boundary problems involving non-local diffusion and aggregation , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[14]  D. Bucur,et al.  A free boundary approach to shape optimization problems , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[15]  J. Glimm,et al.  Euler equation existence, non-uniqueness and mesh converged statistics , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[16]  Mikhail Feldman,et al.  Global Solutions of Shock Reflection by Large-Angle Wedges for Potential Flow , 2007, 0708.2540.

[17]  A. M. Meirmanov,et al.  The Stefan Problem , 1992 .

[18]  L. Caffarelli,et al.  An Extension Problem Related to the Fractional Laplacian , 2006, math/0608640.

[19]  J. Vázquez.  The Porous Medium Equation , 2006 .

[20]  J. Vázquez,et al.  Nonlinear Porous Medium Flow with Fractional Potential Pressure , 2010, 1001.0410.

[21]  Charles Fefferman,et al.  Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves , 2011, 1102.1902.

[22]  C. M. Elliott,et al.  A Stefan problem on an evolving surface , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[23]  Avner Friedman,et al.  Free boundary problems in biology , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[24]  I I Danilyuk,et al.  On the Stefan problem , 1985 .

[25]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[26]  J. Stefan,et al.  Ueber die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere , 1891 .

[27]  Gui-Qiang G. Chen,et al.  Characteristic Discontinuities and Free Boundary Problems for Hyperbolic Conservation Laws , 2012 .

[28]  Mikhail Feldman,et al.  Free boundary problems in shock reflection/diffraction and related transonic flow problems , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[29]  Andrej Zlatoš,et al.  A note on stability shifting for the Muskat problem , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[30]  A. Friedman.  Variational principles and free-boundary problems , 1982 .

[31]  Nicolas Vauchelet,et al.  Incompressible limit of a mechanical model of tumour growth with viscosity , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[32]  A. Friedman,et al.  Free boundary problems in science and technology , 2000 .

[33]  Peter Constantin,et al.  Far-field perturbations of vortex patches , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[34]  Hiroshi Matano,et al.  Pattern Formation in Competition-Diffusion Systems in Nonconvex Domains , 1983 .

[35]  L. Caffarelli,et al.  Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation , 2006, math/0608447.

[36]  Susanna Terracini,et al.  A variational problem for the spatial segregation of reaction-diffusion systems , 2003 .

[37]  Lionel Levine,et al.  Internal DLA in Higher Dimensions , 2010 .

[38]  Enrique Otárola,et al.  Convergence rates for the classical, thin and fractional elliptic obstacle problems , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[39]  D E Apushkinskaya,et al.  Free boundaries in problems with hysteresis , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[40]  Luis A. Caffarelli,et al.  The regularity of free boundaries in higher dimensions , 1977 .