Stripe patterns and a projection-valued formulation of the eikonal equation.

نویسندگان

  • Mark A Peletier
  • Marco Veneroni
چکیده

We describe recent work on striped patterns in a system of block copolymers. A by-product of the characterization of such patterns is a new formulation of the eikonal equation. In this formulation, the unknown is a field of projection matrices of the form P=e⊗e, where e is a unit vector field. We describe how this formulation is better adapted to the description of striped patterns than the classical eikonal equation, and illustrate this with examples.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Stripe Patterns and the Eikonal Equation

We study a new formulation for the eikonal equation |∇u| = 1 on a bounded subset of R. Considering a field P of orthogonal projections onto 1-dimensional subspaces, with divP ∈ L, we prove existence and uniqueness for solutions of the equation P divP = 0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular n...

متن کامل

Partial Differential Equations Non-oriented Solutions of the Eikonal Equation

We study a new formulation for the eikonal equation |∇u| = 1 on a bounded subset of R. Instead of a vector field ∇u, we consider a field P of orthogonal projections on 1-dimensional subspaces, with div P ∈ L. We prove that solutions of this equation propagate direction as in the classical eikonal equation. We also show that solutions exist if and only if the domain is a tubular neighbourhood of...

متن کامل

Stripe patterns in a model for block copolymers

We consider a pattern-forming system in two space dimensions defined by an energy Gε. The functional Gε models strong phase separation in AB diblock copolymer melts, and patterns are represented by {0, 1}-valued functions; the values 0 and 1 correspond to the A and B phases. The parameter ε is the ratio between the intrinsic, material length scale and the scale of the domain Ω. We show that in ...

متن کامل

Non-oriented solutions of the eikonal equation

We study a new formulation for the eikonal equation |∇u| = 1 on a bounded subset of R. Instead of a vector field ∇u, we consider a field P of orthogonal projections on 1-dimensional subspaces, with divP ∈ L. We prove existence and uniqueness for solutions of the equation P divP = 0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only ...

متن کامل

A Paraxial Formulation for the Viscosity Solution of Quasi-P Eikonal Equations

Stationary quasi-P eikonal equations, stationary Hamilton-Jacobi equations, arise from the asymptotic approximation of anisotropic wave propagation. A paraxial formulation of the quasi-P eikonal equation results in a paraxial quasi-P eikonal equation, an evolution Hamilton-Jacobi equation in a preferred direction, which provides a fast and efficient way for computing viscosity solutions of quas...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • Philosophical transactions. Series A, Mathematical, physical, and engineering sciences

دوره 370 1965  شماره 

صفحات  -

تاریخ انتشار 2012