## Introductory School: Free Boundary Problems and Related Topics

**Monday 6th January 2014 to Saturday 11th January 2014**

09:00 to 09:55 | Registration | ||

09:55 to 10:00 | Welcome from Christie Marr (INI Deputy Director) | INI 1 | |

10:00 to 11:00 |
Derivation of FBs for tumor growth - 1
Reaction-Diffusion systems - The Stefan problem - Latent heat
When used for biology and medicine, PDEs have to be used with care. Even though some are very classical, as front propagation for invading species, they are always questioned by comparison to observations or experiments. This course aims at showing some examples of free boundary problems motivated by biology and medicine, to concentrate on weak solutions, and to discuss their limitations and the need for further developments. Suggested reading [1] M. Belhadj, J.-F. Gerbeau and B. Perthame. A multiscale transport model of colloids with degenerate anisotropic diffusion. Asymptotic Analysis 34(1) (2003) 41--54. [2] G. Barles and P.E. Souganidis. Front propagation for reaction-diffusion equations arising in combustion theory. Asymptot. Anal., 14:277292, 1997. [3] A. Lorz, B. Perthame, P. Markowich, Bernoulli variational problem and beyond. To appear in ARMA (2014). http://hal.upmc.fr/hal-00881760 [4] B. Perthame, F. Quiros, J.-L. Vazquez, The Hele-Shaw asymptotics for mechanical models of tumor growth. To appear in ARMA (2014) http://hal.upmc.fr/hal-00831932 [5] Emeric Bouin, Vincent Calvez, Nicolas Meunier, Sepideh Mirrahimi, Benoît Perthame, Gael Raoul, and Raphaël Voituriez. Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration. C. R. Math. Acad. Sci. Paris, 350(15-16):761766, 2012. |
INI 1 | |

11:00 to 11:30 | Morning Coffee | ||

11:30 to 12:30 |
Numerical Methods for FBPs - 1
This tutorial is a tour from classical techniques to recent developments of numerical methods for free boundary problems. The emphasis is on ideas and methods rather than problems.
Lecture 1: Variational Inequalities I The classical obstacle problem. A priori error analysis in energy and maximum norm. The thin obstacle problem. The fractional obstacle problem. Lecture 2: Variational Inequalities II A priori rate of convergence for free boundaries. A posteriori error analysis in the maximum norm. A posteriori barrier sets. Lecture 3: Discrete Gradient Flows Evolution PDE: energy solutions, convexity, coercivity. Error analysis of time discretization. Error analysis of space discretization. Applications: parabolic variational inequalities, degenerate parabolic PDE, TV gradient flow. Lecture 4: Geometric Problems Shape differential calculus: examples. Geometric gradient flows: mean curvature, surface diffusion, Willmore flow. Parametric approach. Phase field approach. Level set approach. |
INI 1 | |

12:30 to 13:30 | Lunch at Wolfson Court | ||

13:30 to 14:30 |
Derivation of FBs for tumor growth - 2
The example of invasion fronts - The approach based on Hamilton-Jacobi equations - Accelerating fronts - The tail problem
When used for biology and medicine, PDEs have to be used with care. Even though some are very classical, as front propagation for invading species, they are always questioned by comparison to observations or experiments. This course aims at showing some examples of free boundary problems motivated by biology and medicine, to concentrate on weak solutions, and to discuss their limitations and the need for further developments. Suggested reading [1] M. Belhadj, J.-F. Gerbeau and B. Perthame. A multiscale transport model of colloids with degenerate anisotropic diffusion. Asymptotic Analysis 34(1) (2003) 41--54. [2] G. Barles and P.E. Souganidis. Front propagation for reaction-diffusion equations arising in combustion theory. Asymptot. Anal., 14:277292, 1997. [3] A. Lorz, B. Perthame, P. Markowich, Bernoulli variational problem and beyond. To appear in ARMA (2014). http://hal.upmc.fr/hal-00881760 [4] B. Perthame, F. Quiros, J.-L. Vazquez, The Hele-Shaw asymptotics for mechanical models of tumor growth. To appear in ARMA (2014) http://hal.upmc.fr/hal-00831932 [5] Emeric Bouin, Vincent Calvez, Nicolas Meunier, Sepideh Mirrahimi, Benoît Perthame, Gael Raoul, and Raphaël Voituriez. Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration. C. R. Math. Acad. Sci. Paris, 350(15-16):761766, 2012. |
INI 1 | |

14:30 to 15:00 | Afternoon Tea | ||

15:00 to 16:00 |
Numerical Methods for FBPs - 2
This tutorial is a tour from classical techniques to recent developments of numerical methods for free boundary problems. The emphasis is on ideas and methods rather than problems.
Lecture 1: Variational Inequalities I The classical obstacle problem. A priori error analysis in energy and maximum norm. The thin obstacle problem. The fractional obstacle problem. Lecture 2: Variational Inequalities II A priori rate of convergence for free boundaries. A posteriori error analysis in the maximum norm. A posteriori barrier sets. Lecture 3: Discrete Gradient Flows Evolution PDE: energy solutions, convexity, coercivity. Error analysis of time discretization. Error analysis of space discretization. Applications: parabolic variational inequalities, degenerate parabolic PDE, TV gradient flow. Lecture 4: Geometric Problems Shape differential calculus: examples. Geometric gradient flows: mean curvature, surface diffusion, Willmore flow. Parametric approach. Phase field approach. Level set approach. |
INI 1 | |

16:00 to 17:00 | Perspectives on Free Boundaries | INI 1 | |

17:00 to 18:00 | Welcome Wine Reception |

10:00 to 11:00 |
Derivation of FBs for tumor growth - 3
Mechanical models of tumor growth - boundary stability and instability - variants
When used for biology and medicine, PDEs have to be used with care. Even though some are very classical, as front propagation for invading species, they are always questioned by comparison to observations or experiments. This course aims at showing some examples of free boundary problems motivated by biology and medicine, to concentrate on weak solutions, and to discuss their limitations and the need for further developments. Suggested reading [1] M. Belhadj, J.-F. Gerbeau and B. Perthame. A multiscale transport model of colloids with degenerate anisotropic diffusion. Asymptotic Analysis 34(1) (2003) 41--54. [2] G. Barles and P.E. Souganidis. Front propagation for reaction-diffusion equations arising in combustion theory. Asymptot. Anal., 14:277292, 1997. [3] A. Lorz, B. Perthame, P. Markowich, Bernoulli variational problem and beyond. To appear in ARMA (2014). http://hal.upmc.fr/hal-00881760 [4] B. Perthame, F. Quiros, J.-L. Vazquez, The Hele-Shaw asymptotics for mechanical models of tumor growth. To appear in ARMA (2014) http://hal.upmc.fr/hal-00831932 [5] Emeric Bouin, Vincent Calvez, Nicolas Meunier, Sepideh Mirrahimi, Benoît Perthame, Gael Raoul, and Raphaël Voituriez. Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration. C. R. Math. Acad. Sci. Paris, 350(15-16):761766, 2012. |
INI 1 | |

11:00 to 11:30 | Morning Coffee | ||

11:30 to 12:30 |
Numerical Methods for FBPs - 3
This tutorial is a tour from classical techniques to recent developments of numerical methods for free boundary problems. The emphasis is on ideas and methods rather than problems.
Lecture 1: Variational Inequalities I The classical obstacle problem. A priori error analysis in energy and maximum norm. The thin obstacle problem. The fractional obstacle problem. Lecture 2: Variational Inequalities II A priori rate of convergence for free boundaries. A posteriori error analysis in the maximum norm. A posteriori barrier sets. Lecture 3: Discrete Gradient Flows Evolution PDE: energy solutions, convexity, coercivity. Error analysis of time discretization. Error analysis of space discretization. Applications: parabolic variational inequalities, degenerate parabolic PDE, TV gradient flow. Lecture 4: Geometric Problems Shape differential calculus: examples. Geometric gradient flows: mean curvature, surface diffusion, Willmore flow. Parametric approach. Phase field approach. Level set approach. |
INI 1 | |

12:30 to 13:30 | Lunch at Wolfson Court | ||

13:30 to 14:30 |
E Varvaruca (University of Reading)Geometric approaches to water waves and free surface flows - 1
These lectures aim to present a new geometric approach to the asymptotic behaviour near singularities in some classical free-boundary problems in fluid dynamics. We start by introducing the problems and providing an outline of the methods that have been used to prove existence of solutions. We then present a modern proof, using monotonicity formulas and frequency formulas, of the famous Stokes conjecture from 1880, which asserts that at any stagnation point on the free surface of a two-dimensional steady irrotational gravity water wave, the wave profile necessarily has lateral tangents enclosing a symmetric angle of 120 degrees. (This result was first proved in the 1980s under restrictive assumptions and by somewhat ad-hoc methods.) We then explain how the methods extend to the case of two-dimensional steady gravity water waves with vorticity. Finally, we show how the same methods can be adapted to describe the asymptotic behaviour near singularities in the problem of steady three-dimensional axisymmetric free surface flows with gravity.
References: [1] Buffoni, B.; Toland, J. F. Analytic theory of global bifurcation. An introduction. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, N.J., 2003. [2] Constantin, A. Nonlinear water waves with applications to wave-current interactions and tsunamis. CBMS-NSF Regional Conference Series in Applied Mathematics, 81. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, P.A., 2011. [3] Constantin, A.; Strauss, W. Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57 (2004), no. 4, 481--527. [4] Varvaruca, E. On the existence of extreme waves and the Stokes conjecture with vorticity. J. Differential Equations 246 (2009), no. 10, 4043--4076. [5] Varvaruca, E.; Weiss, G. S. A geometric approach to generalized Stokes conjectures. Acta Math. 206 (2011), no. 2, 363--403. [6] Varvaruca, E.; Weiss, G. S. The Stokes conjecture for waves with vorticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 6, 861--885. [7] Varvaruca, E.; Weiss, G. S. Singularities of steady axisymmetric free surface flows with gravity, to appear in Comm. Pure Appl. Math., http://arxiv.org/abs/1210.3682. |
INI 1 | |

14:30 to 15:00 | Afternoon Tea | ||

15:00 to 16:00 |
Numerical Methods for FBPs - 4 Lecture 1: Variational Inequalities I Lecture 2: Variational Inequalities II Lecture 3: Discrete Gradient Flows Lecture 4: Geometric Problems |
INI 1 |

10:00 to 11:00 |
Derivation of FBs for tumor growth - 4
The limit of stiff pressure law - Weak solutions of the Hele-Shaw free boundary problem
Suggested reading |
INI 1 | |

11:00 to 11:30 | Morning Coffee | ||

11:30 to 12:30 |
E Varvaruca (University of Reading)Geometric approaches to water waves and free surface flows - 2
These lectures aim to present a new geometric approach to the asymptotic behaviour near singularities in some classical free-boundary problems in fluid dynamics. We start by introducing the problems and providing an outline of the methods that have been used to prove existence of solutions. We then present a modern proof, using monotonicity formulas and frequency formulas, of the famous Stokes conjecture from 1880, which asserts that at any stagnation point on the free surface of a two-dimensional steady irrotational gravity water wave, the wave profile necessarily has lateral tangents enclosing a symmetric angle of 120 degrees. (This result was first proved in the 1980s under restrictive assumptions and by somewhat ad-hoc methods.) We then explain how the methods extend to the case of two-dimensional steady gravity water waves with vorticity. Finally, we show how the same methods can be adapted to describe the asymptotic behaviour near singularities in the problem of steady three-dimensional axisymmetric free surface flows with gravity.
References: [1] Buffoni, B.; Toland, J. F. Analytic theory of global bifurcation. An introduction. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, N.J., 2003. [2] Constantin, A. Nonlinear water waves with applications to wave-current interactions and tsunamis. CBMS-NSF Regional Conference Series in Applied Mathematics, 81. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, P.A., 2011. [3] Constantin, A.; Strauss, W. Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57 (2004), no. 4, 481--527. [4] Varvaruca, E. On the existence of extreme waves and the Stokes conjecture with vorticity. J. Differential Equations 246 (2009), no. 10, 4043--4076. [5] Varvaruca, E.; Weiss, G. S. A geometric approach to generalized Stokes conjectures. Acta Math. 206 (2011), no. 2, 363--403. [6] Varvaruca, E.; Weiss, G. S. The Stokes conjecture for waves with vorticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 6, 861--885. [7] Varvaruca, E.; Weiss, G. S. Singularities of steady axisymmetric free surface flows with gravity, to appear in Comm. Pure Appl. Math., http://arxiv.org/abs/1210.3682. |
INI 1 | |

12:30 to 13:30 | Lunch at Wolfson Court | ||

13:30 to 14:30 | Poster Session | ||

14:30 to 15:00 | Afternoon Tea | ||

15:00 to 16:00 |
Regularity of Free Boundaries in Obstacle Type Problems - 1
The aim of these lectures is to give an introduction to the regularity theory of free boundaries related to the obstacle problem. Besides the classical obstacle problem, we will consider the problem on harmonic continuation of Newtonian potentials, the thin obstacle problem, and their parabolic counterparts (as much as the time permits).
Lecture 1. In this lecture, we will introduce the problems we will be working on and discuss initial regularity results for the solutions. Lecture 2. In this lecture, we will discuss the optimal regularity of solutions and give proofs by using monotonicity formulas. Lecture 3. In this lecture, we will consider the blowups of the solutions at free boundary points. We will then classify the blowups and thereby classify the free boundary points. Lecture 4. In this lecture, we will show how to prove the regularity of the "regular set" and obtain a structural theorem on the singular set. Suggested reading: [1] Petrosyan, Arshak ; Shahgholian, Henrik; Uraltseva, Nina . Regularity of free boundaries in obstacle-type problems. Graduate Studies in Mathematics, 136. American Mathematical Society, Providence, RI, 2012. x+221 pp. ISBN: 978-0-8218-8794-3 [2] Caffarelli, L. A. The obstacle problem revisited. J. Fourier Anal. Appl. 4 (1998), no. 4-5, 383--402. [3] Weiss, Georg S. A homogeneity improvement approach to the obstacle problem. Invent. Math. 138 (1999), no. 1, 23--50. [4] Caffarelli, Luis A. ; Karp, Lavi ; Shahgholian, Henrik . Regularity of a free boundary with application to the Pompeiu problem. Ann. of Math. (2) 151 (2000), no. 1, 269--292. [5] Caffarelli, Luis ; Petrosyan, Arshak ; Shahgholian, Henrik . Regularity of a free boundary in parabolic potential theory. J. Amer. Math. Soc. 17 (2004), no. 4, 827--869. [6] Garofalo, Nicola ; Petrosyan, Arshak . Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent. Math. 177 (2009), no. 2, 415461. [7] Danielli, Donatella ; Garofalo, Nicola ; Petrosyan, Arshak ; To, Tung . Optimal regularity and the free boundary in the parabolic Signorini problem. arXiv:1306.5213 |
INI 1 | |

19:30 to 22:00 | Conference Dinner at Gonville and Caius College |

10:00 to 11:00 |
Regularity of Free Boundaries in Obstacle Type Problems - 2
The aim of these lectures is to give an introduction to the regularity theory of free boundaries related to the obstacle problem. Besides the classical obstacle problem, we will consider the problem on harmonic continuation of Newtonian potentials, the thin obstacle problem, and their parabolic counterparts (as much as the time permits).
Lecture 1. In this lecture, we will introduce the problems we will be working on and discuss initial regularity results for the solutions. Lecture 2. In this lecture, we will discuss the optimal regularity of solutions and give proofs by using monotonicity formulas. Lecture 3. In this lecture, we will consider the blowups of the solutions at free boundary points. We will then classify the blowups and thereby classify the free boundary points. Lecture 4. In this lecture, we will show how to prove the regularity of the "regular set" and obtain a structural theorem on the singular set. Suggested reading: [1] Petrosyan, Arshak ; Shahgholian, Henrik; Uraltseva, Nina . Regularity of free boundaries in obstacle-type problems. Graduate Studies in Mathematics, 136. American Mathematical Society, Providence, RI, 2012. x+221 pp. ISBN: 978-0-8218-8794-3 [2] Caffarelli, L. A. The obstacle problem revisited. J. Fourier Anal. Appl. 4 (1998), no. 4-5, 383--402. [3] Weiss, Georg S. A homogeneity improvement approach to the obstacle problem. Invent. Math. 138 (1999), no. 1, 23--50. [4] Caffarelli, Luis A. ; Karp, Lavi ; Shahgholian, Henrik . Regularity of a free boundary with application to the Pompeiu problem. Ann. of Math. (2) 151 (2000), no. 1, 269--292. [5] Caffarelli, Luis ; Petrosyan, Arshak ; Shahgholian, Henrik . Regularity of a free boundary in parabolic potential theory. J. Amer. Math. Soc. 17 (2004), no. 4, 827--869. [6] Garofalo, Nicola ; Petrosyan, Arshak . Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent. Math. 177 (2009), no. 2, 415461. [7] Danielli, Donatella ; Garofalo, Nicola ; Petrosyan, Arshak ; To, Tung . Optimal regularity and the free boundary in the parabolic Signorini problem. arXiv:1306.5213 |
INI 1 | |

11:00 to 11:30 | Morning Coffee | ||

11:30 to 12:30 |
E Varvaruca (University of Reading)Geometric approaches to water waves and free surface flows - 3
These lectures aim to present a new geometric approach to the asymptotic behaviour near singularities in some classical free-boundary problems in fluid dynamics. We start by introducing the problems and providing an outline of the methods that have been used to prove existence of solutions. We then present a modern proof, using monotonicity formulas and frequency formulas, of the famous Stokes conjecture from 1880, which asserts that at any stagnation point on the free surface of a two-dimensional steady irrotational gravity water wave, the wave profile necessarily has lateral tangents enclosing a symmetric angle of 120 degrees. (This result was first proved in the 1980s under restrictive assumptions and by somewhat ad-hoc methods.) We then explain how the methods extend to the case of two-dimensional steady gravity water waves with vorticity. Finally, we show how the same methods can be adapted to describe the asymptotic behaviour near singularities in the problem of steady three-dimensional axisymmetric free surface flows with gravity.
References: [1] Buffoni, B.; Toland, J. F. Analytic theory of global bifurcation. An introduction. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, N.J., 2003. [2] Constantin, A. Nonlinear water waves with applications to wave-current interactions and tsunamis. CBMS-NSF Regional Conference Series in Applied Mathematics, 81. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, P.A., 2011. [3] Constantin, A.; Strauss, W. Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57 (2004), no. 4, 481--527. [4] Varvaruca, E. On the existence of extreme waves and the Stokes conjecture with vorticity. J. Differential Equations 246 (2009), no. 10, 4043--4076. [5] Varvaruca, E.; Weiss, G. S. A geometric approach to generalized Stokes conjectures. Acta Math. 206 (2011), no. 2, 363--403. [6] Varvaruca, E.; Weiss, G. S. The Stokes conjecture for waves with vorticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 6, 861--885. [7] Varvaruca, E.; Weiss, G. S. Singularities of steady axisymmetric free surface flows with gravity, to appear in Comm. Pure Appl. Math., http://arxiv.org/abs/1210.3682. |
INI 1 | |

12:30 to 13:30 | Lunch at Wolfson Court | ||

13:30 to 14:30 |
Regularity of Free Boundaries in Obstacle Type Problems - 3
The aim of these lectures is to give an introduction to the regularity theory of free boundaries related to the obstacle problem. Besides the classical obstacle problem, we will consider the problem on harmonic continuation of Newtonian potentials, the thin obstacle problem, and their parabolic counterparts (as much as the time permits).
Lecture 1. In this lecture, we will introduce the problems we will be working on and discuss initial regularity results for the solutions. Lecture 2. In this lecture, we will discuss the optimal regularity of solutions and give proofs by using monotonicity formulas. Lecture 3. In this lecture, we will consider the blowups of the solutions at free boundary points. We will then classify the blowups and thereby classify the free boundary points. Lecture 4. In this lecture, we will show how to prove the regularity of the "regular set" and obtain a structural theorem on the singular set. Suggested reading: [1] Petrosyan, Arshak ; Shahgholian, Henrik; Uraltseva, Nina . Regularity of free boundaries in obstacle-type problems. Graduate Studies in Mathematics, 136. American Mathematical Society, Providence, RI, 2012. x+221 pp. ISBN: 978-0-8218-8794-3 [2] Caffarelli, L. A. The obstacle problem revisited. J. Fourier Anal. Appl. 4 (1998), no. 4-5, 383--402. [3] Weiss, Georg S. A homogeneity improvement approach to the obstacle problem. Invent. Math. 138 (1999), no. 1, 23--50. [4] Caffarelli, Luis A. ; Karp, Lavi ; Shahgholian, Henrik . Regularity of a free boundary with application to the Pompeiu problem. Ann. of Math. (2) 151 (2000), no. 1, 269--292. [5] Caffarelli, Luis ; Petrosyan, Arshak ; Shahgholian, Henrik . Regularity of a free boundary in parabolic potential theory. J. Amer. Math. Soc. 17 (2004), no. 4, 827--869. [6] Garofalo, Nicola ; Petrosyan, Arshak . Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent. Math. 177 (2009), no. 2, 415461. [7] Danielli, Donatella ; Garofalo, Nicola ; Petrosyan, Arshak ; To, Tung . Optimal regularity and the free boundary in the parabolic Signorini problem. arXiv:1306.5213 |
INI 1 | |

14:30 to 15:00 | Afternoon Tea | ||

15:00 to 16:00 | Open problem session | INI 1 |

09:00 to 10:00 |
N Garofalo (Università degli Studi di Padova)Regularity of Free Boundaries in Obstacle Type Problems - 4 Suggested reading: |
INI 1 | |

10:00 to 11:00 |
E Varvaruca (University of Reading)Geometric approaches to water waves and free surface flows - 4 References: |
INI 1 | |

11:00 to 11:30 | Morning Coffee | ||

11:30 to 12:30 | Closing session | INI 1 | |

12:30 to 13:30 | Lunch at Wolfson Court |