Random thoughts

Well-posedness of boundary initial value problems (BIVP) under perturbations of the boundary (2 September 2011)

A boundary initial value problem consists of an evolution PDE, a set of initial conditions, and a boundary data. For example, the (1+1 dimensional) heat equation for u(t,x) with (t,x) both nonegative, with initial data u(0,x) and boundary data u(t,0). A standard question is to determine the well-posedness of this problem, i.e. to determine if (i) there exist solutions to this problem, (ii) if they are unique, and (iii) if the solution (in an appropriate function space Y) depends continuously on the boundary-initial data (in its own function space X).

In the study of initial value problems a lot is known. The examples I'm most familiar with out (nonlinear or linear) wave or Schrodinger equations. In such cases, one typically takes data in a Sobolev space X in the x variable and then looks for solutions which varies continuously in t in that space, i.e. in $C^0(X)$. Much is known about the limits of the level of regularity and how this depends on the nonlinearity. Many other equations can be studied in this framework, Benjamin-Ono, KdV, Navier-Stokes, et c. I'm only vaguelly aware of boundary initial value problems, but a similar type of analysis should apply. A problem where I am not aware of any study is where one specifies boundary data on a curve (or hypersurface) y(t), specifies initial data u(0,x)=f(x) and boundary data u(y(t),t))=g(t) and then looks for continuous dependence of the solution as a function of y.

(This problem arose in a mathematical finance talk by Borovkov. Essentially the Black-Scholes equation is a (backwards) heat equation with the terminal (instead of initial) data given by the pay-off of the option and the solution of the PDE giving the current value of the option. A "barrier option" has the additional constraint that the option becomes worthless if the price ever goes outside a (possibly time dependent) window. Thus, this is a (Dirichlet) boundary initial value problem. A question then arises, if one approximates the boundary by a piecewise continuous approximation, how much does this approximation distort the solution?)

For me, natural questions might be

  1. For NLS or NLW (with your favourite nonlinearity) is there well-posedness for the BIVP with continuous dependence on the boundary?
  2. If so, is there some heuristic, such as scaling, which suggests the natural function space for the boundary?
  3. For NLW, what happens with data specified on a rough hypersurface (with timelike normal a.e.)?
  4. Does NLS make sense on a curved hypersurface? If so, what happens?
  5. What happens with the nonlinear heat equation?
  6. For geometric PDE (such as Einstein's equation and Ricci flow), does varying the initial surface make sense, or is this merely a perturbation of the geometric variables (induced metric, second fundamental form, et c.) which describe the initial conditions?