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Introduction to Linear Algebra with Mathematica

Glossary

The Shapiro–Lopatinskii condition has a natural parabolic analogue, often called the parabolic Lopatinskii–Shapiro condition, the parabolic complementing condition, or a parameter-dependent Lopatinskii condition. The essential difference from the elliptic case is that the time variable introduces an additional complex parameter, usually obtained by a Laplace transform in time.

Shapiro--Lopatinskii condition

Let us consider a parabolic initial-boundary value problem of the form

\[ \partial_t u + A\left( x, \texttt{D}_x \right) u = f \]
in a spatial domain Ω, together with boundary conditions
\[ B_j \left( x, \texttt{D}_x \right) u = g_j \qquad\mbox{on } \partial\Omega . \]
Suppose that x⁰ ∈ ∂Ω. Near x⁰, one first flatten the boundary and chooses local coordinates
\[ x = \left( x' , x_n \right) , \qquad x' = \left( x_1 , x_2 , \ldots , x_{n-1} \right) , \]
where xₙ is the normal variable and x′ consists of the tangential variables.

Next, freeze the principal coefficients at x⁰. Then take

The principal equation is thereby reduced to an ordinary differential equation in the normal variable xₙ:

\[ \left[ \lambda + A^{(0)} \left( x^0 , \xi' , \texttt{D}_n \right) \right] v\left( x_n \right) = 0 . \qquad \texttt{D}_n = \partial_{x_n} = \frac{\partial}{\partial x_n} . \]
The boundary conditions become
\[ B^{(0)}_j \left( x^0 , \xi' , \texttt(ED)_n \right) v(+0) = 0 , \qquad \texttt{D}_n = \partial_{x_n} . \]
The parabolic Shapiro–Lopatinskii condition says, roughly, that for each admissible pair (λ, ξ′) ≠ (0, 0), the only solution of the transformed homogeneous equation that decays into the interior and satisfies all homogeneous boundary conditions is the trivial solution v ≡ 0.

Typically one considers Re λ ≥ 0, or more generally λ in an appropriate sector of the complex plane, depending on the convention used for the parabolic operator.    

Example 1: Let us take the heat equation in the half-line
\[ u_t - u_{xx} = 0 , \qquad x > 0 , \]
with the homogeneous Dirichlet boundary condition
\[ u(0, t) = 0 . \]
There are no spatial tangential variables in one dimension, so the only transform parameter comes from time. Taking the Laplace transform in t gives
\[ \lambda\,v - v'' = 0 , \]
or equivalently
\[ v'' - \lambda\,v = 0 . \]
The normal characteristic equation is
\[ r^2 = \lambda , \]
with roots
\[ r_{\pm} = \pm \sqrt{\lambda} . \]
For ℜ  > 0, choose the branch of √λ with positive real part. The mode that decays as x → +∞ is
\[ v(x) = C\,s^{-\sqrt{\lambda}\,x} . \]
Applying the transformed Dirichlet boundary condition gives
\[ v(0) = C = 0 . \]
Hence, the only decaying solution satisfying the homogeneous boundary condition is the zero solution. Therefore, the Dirichlet heat problem satisfies the parabolic Lopatinskii–Shapiro condition.

The same idea works for the homogeneous Neumann condition ux(0, t) = 0:

\[ v'(0) = -C\,\sqrt{\lambda} = 0 . \]
which again forces C = 0 for λ ≠ 0.    ■
End of Example 1

For an elliptic operator, the corresponding frozen normal equation has the form
\[ A^{(0)} \left( x^0 , \xi' , \texttt{D}_n \right) = 0 . \]
For a parabolic operator, the time-Laplace parameter appears as part of the principal symbol:
\[ \boxed{A^{(0)} \left( x^0 , \xi' , \texttt{D}_n \right) \,\to\, \lambda + A^{(0)} \left( x^0 , \xi' , \texttt{D}_n \right) } \]
Thus, the parabolic condition may be viewed as a parameter-dependent version of the elliptic Shapiro–Lopatinskii condition. If the spatial differential operator has order 2m, the natural parabolic scaling is
\[ \lambda \sim |\xi |^{2m} \qquad \iff \qquad (\lambda , \xi ) \mapsto \left( r^{2m}\lambda , r\xi \right) . \]
This anisotropic scaling is one of the main distinctions between elliptic and parabolic boundary theory.    
Example 2: For the heat equation on
\[ Q = \left\{ (x, t) \ : \ x> 0 , \ t > 0 \right\} , \]
there are two geometrically visible edges, but they play different analytic roles:
  • x = 0 : spatial boundary, where the parabolic Lopatinskii condition is tested,
  • t = 0 : initial-time surface, where initial data are prescribed.
At their intersection (0, 0) one imposes compatibility conditions between the initial and boundary data. For example, if
\[ u(+0, t) = g(t) , \qquad u(x, +0) = u_0 (x) , \]
then the lowest-order compatibility condition is
\[ g(+0) = u_0 (+0) . \]
For smoother solutions, higher-order compatibility conditions follow by repeatedly using the PDE. It is therefore important not to treat the x- and t-edges as two equivalent spatial boundaries. The heat equation is anisotropic: its natural scaling is
\[ x \mapsto r\,x , \qquad t \mapsto r^2 t . \]
For the heat equation on the quarter-plane, the correct framework is
\[ \begin{array}{c} \mbox{parabolic Lopatinskii condition at } x = 0 \\ + \ \mbox{initial condition at } t = 0 \\ + \ \mbox{compatibility at } (0,0) . \end{array} \]
   ■
End of Example 2

 

  1. Krainer, T., (2005) Elliptic boundary problems on manifolds with polycylindrical ends, arXiv:math/0508516 [math.AP]
  2. Krupchyk, K. & Tuomela, J., (2006) The Shapiro–Lopatinskij Condition for Elliptic Boundary Value Problems, London Mathematical Society.
  3. Lopatinskii, Ya.B., (1953) On a method of reducing boundary problems for a system of differential equations of elliptic type to regular integral equations Journal: Ukraine. Mat. Zh., 5 (1953), 123–151.
  5. Shapiro, Z.Ya,, (1953) On general boundary problems for equations of elliptic type, Journal: Izvestiya Akad. Nauk SSSR. Ser. Mat., 17 (1953), 539–562.

 

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