Mathematical Aspects of Evolving Interfaces: Lectures given at the C.I.M.-C.I.M.E. joint Euro-Summer School held in Madeira Funchal, Portugal, July 3-9, … Mathematics / Fondazione C.I.M.E., Firenze)

Luigi Ambrosio, Klaus Deckelnick, Gerhard Dziuk, Masayasu Mimura, Vsvolod Solonnikov, Halil Mete Soner, Pierluigi Colli, José Francisco Rodrigues9783540140337, 3-540-14033-6

Interfaces are geometrical objects modelling free or moving boundaries and arise in a wide range of phase change problems in physical and biological sciences, particularly in material technology and in dynamics of patterns. Especially in the end of last century, the study of evolving interfaces in a number of applied fields becomes increasingly important, so that the possibility of describing their dynamics through suitable mathematical models became one of the most challenging and interdisciplinary problems in applied mathematics. The 2000 Madeira school reported on mathematical advances in some theoretical, modelling and numerical issues concerned with dynamics of interfaces and free boundaries. Specifically, the five courses dealt with an assessment of recent results on the optimal transportation problem, the numerical approximation of moving fronts evolving by mean curvature, the dynamics of patterns and interfaces in some reaction-diffusion systems with chemical-biological applications, evolutionary free boundary problems of parabolic type or for Navier-Stokes equations, and a variational approach to evolution problems for the Ginzburg-Landau functional.

---

Table of contents :
front-matter……Page 1
Introduction……Page 10
1 Some elementary examples……Page 12
2 Optimal transport plans: existence and regularity……Page 14
3 The one dimensional case……Page 23
4 The ODE version of the optimal transport problem……Page 25
5 The PDE version of the optimal transport problem and the $p$-laplacian approximation……Page 38
6 Existence of optimal transport maps……Page 41
7 Regularity and uniqueness of the transport density……Page 47
8 The Bouchitté–Buttazzo mass optimization problem……Page 50
9 Appendix: some measure theoretic results……Page 52
References……Page 59
1 Introduction……Page 62
2.1 The Differential Equation……Page 64
2.2 Some analysis for the problem……Page 65
2.3 Discretization……Page 67
2.4 Convergence of the fully discrete semi implicit scheme……Page 72
3.1 Viscosity Solutions……Page 84
3.2 Regularization……Page 85
3.3 Convergence of the numerical scheme for the level set problem……Page 87
3.4 Numerical Tests for the Level Set Algorithm……Page 90
References……Page 95
1.1 Discrete diffusion models……Page 97
1.2 Continuous models……Page 98
2 Paradox of diffusion……Page 99
3 Diffusive patterns and waves – bistable RD equations……Page 102
3.1 Scalar bistable RD equation……Page 103
3.2 Two component systems of bistable RD equations……Page 106
4.1 Bacterial colony model……Page 109
4.2 Grey-Scott model……Page 115
5 Competition-diffusion systems and singular limit procedures……Page 117
5.1 Spatial segregating limits of two competing species model……Page 119
5.2 Dynamics of triple junctions arising in three competing species model……Page 121
6 Chemotactic patterns – aggregation and colonies……Page 123
6.1 Aggregation of Dictyostelium discoideum……Page 124
6.2 New pattern arising in a chemotaxis-diffusion-growth system……Page 125
References……Page 127
Introduction……Page 130
1 Introduction……Page 131
2 The model problem……Page 135
3 Transformation of the problem (1)……Page 141
4 Proof of Theorem 3.1……Page 145
5 Introduction……Page 149
6 Linear and model problems……Page 153
7 Lagrangean coordinates and local existence theorems……Page 161
8 Proof of Theorem 5.3……Page 169
9 Scheme of the proof of Theorem 5.4……Page 173
References……Page 180
1 Introduction……Page 183
2.1 Degree and Jacobian……Page 189
2.2 Covering argument……Page 190
2.3 Main lower bound……Page 199
3.1 Jacobian estimate……Page 207
3.2 Compactness in two dimensions……Page 213
4 Gamma Limit……Page 219
4.1 Functions of $BnV$……Page 221
4.2 Gamma limit of $I^{varepsilon}$……Page 223
5 Compactness in Higher Dimensions……Page 228
6 Dynamic Problems: Evolution of Vortex Filaments……Page 231
6.1 Energy identities……Page 232
6.3 Convergence……Page 233
References……Page 237
back-matter……Page 240