# Point-instabilities, point-coercivity (meta-stability), and<br/><br/>point-calculus

Tuesday, July 22, 2008 - 2:45pm - 2:55pm

EE/CS 3-180

Evan Hohlfeld (University of California, Berkeley)

For general non-linear elliptic PDEs, e.g. non-linear rubber

elasticity, linear stability analysis is false. This is because of the

possibility of point-instabilities. A point-instability is a non-linear

instability with zero amplitude threshold that occurs while linear

stability still holds. Examples include cavitation, fracture, and the formation

of a crease, a self-contacting fold in an otherwise free surface.

Each of which represents a kind of topological change. For any such

PDE, a point-instability occurs whenever a certain auxiliary

scale-invariant problem has a non-trivial solution. E.g. when sufficient strain

is applied at infinity in a rubber (half-)space to support a

single, isolated crease, crack, cavity, etc. Owing to scale-invariance,

when one such solution exists, an infinite number or geometrically

similar solutions also exist, so the appearance of one particular

solution is the spontaneous breaking of scale-invariance. We then identify

this (half-)space with a point in a general domain. The condition

that no such solutions exist is called point-coercivity, and can be

formulated as non-linear eigenvalue problem that predicts the critical

stress for fracture, etc. And when point-coercivity fails for a system,

the system is susceptible to the nucleation and self-similar growth of

some kind of topological defect. Viewing fracture, etc. as symmetry breaking

processes explains their macroscopic robustness.

Point-coercivity is similar to, but more general than,

quasi-convexity, as it can be formulated for any elliptic PDE, not just

Euler-Lagrange systems (i.e. for out-of-equilibrium systems, and so defining

meta-stability in a general sense). Indeed, these are just two

examples of a host of point-conditions, the study of which might be

called point-calculus. Time allowing, I will show that for almost any

elliptic PDE, linear- and point-instabilities exhaust the possible kinds

of instabilities. The lessons learned from elliptic systems will

be just as valid for parabolic and hyperbolic systems since the underlying

reason linear analysis breaks down – taking certain limits in the

wrong order holds for these systems as well.

elasticity, linear stability analysis is false. This is because of the

possibility of point-instabilities. A point-instability is a non-linear

instability with zero amplitude threshold that occurs while linear

stability still holds. Examples include cavitation, fracture, and the formation

of a crease, a self-contacting fold in an otherwise free surface.

Each of which represents a kind of topological change. For any such

PDE, a point-instability occurs whenever a certain auxiliary

scale-invariant problem has a non-trivial solution. E.g. when sufficient strain

is applied at infinity in a rubber (half-)space to support a

single, isolated crease, crack, cavity, etc. Owing to scale-invariance,

when one such solution exists, an infinite number or geometrically

similar solutions also exist, so the appearance of one particular

solution is the spontaneous breaking of scale-invariance. We then identify

this (half-)space with a point in a general domain. The condition

that no such solutions exist is called point-coercivity, and can be

formulated as non-linear eigenvalue problem that predicts the critical

stress for fracture, etc. And when point-coercivity fails for a system,

the system is susceptible to the nucleation and self-similar growth of

some kind of topological defect. Viewing fracture, etc. as symmetry breaking

processes explains their macroscopic robustness.

Point-coercivity is similar to, but more general than,

quasi-convexity, as it can be formulated for any elliptic PDE, not just

Euler-Lagrange systems (i.e. for out-of-equilibrium systems, and so defining

meta-stability in a general sense). Indeed, these are just two

examples of a host of point-conditions, the study of which might be

called point-calculus. Time allowing, I will show that for almost any

elliptic PDE, linear- and point-instabilities exhaust the possible kinds

of instabilities. The lessons learned from elliptic systems will

be just as valid for parabolic and hyperbolic systems since the underlying

reason linear analysis breaks down – taking certain limits in the

wrong order holds for these systems as well.

MSC Code:

74B10

Keywords: