tessel@um.bot
Nov4-06, 03:38 PM
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On Tue, 5 Sep 2006, Alan wrote:
> I am looking for (refs. to) explicit formulas for the heat kernel
> p(t,x,y) on the n-sphere S^n?
>
> Also, while I have found the formulas for the analogous known kernels on
> the hyperbolic spaces H^n, I am still looking for an "elementary
> derivation" of those.
Consider the following familiar statement about the heat equation on R:
Fact: The initial value problem
u_t = u_(xx)
u(0,x) = f(x) on -infty < x < infty
has the solution
u(x,t) = 1/sqrt(2 Pi)
int_-infty^infty f(xi) exp(-(x-xi)^2/(4 t))/sqrt(2 t) dxi
Here the "kernel" in the convolution, i.e. the function of t,x,xi which we
integrate against, is the function
exp(-(x-xi)^2/(4 t))/sqrt(2 t)
which is also called the "fundamental solution", or the "Green function"
for this problem. This is the solution which describes how an initial
"delta mass" concentration of some scalar field on R spreads out as time
increases, if its evolution is governed by the heat equation.
If I understand your question correctly, you are asking for the "kernels"
appearing in analogous statements for the heat equation
u_t = Lap(u)
where Lap is the Laplace-Beltrami operator on H^n or S^n, as expressed in
some convenient coordinate chart.
E.g. for S^2 we can use a stereographic chart
ds^2 = 1/(1+x^2+y^2)^2/4 (dx^2+dy^2),
-infty < x, y < infty
and then
Lap(u) = (1+x^2+y^2)^2/4 (u_(xx) + u_(yy))
so that the "S^2 heat equation" (written in the stereographic chart) is
u_t = k (1+x^2+y^2)^2/4 (u_(xx) + u_(yy))
And on H^2 we can use the upper half plane chart
ds^2 = (dx^2 + dy^2)/y^2,
-infty < x < infty, 0 < y < infty
and then the "H^2 heat equation" (written in the UHP chart) is
u_t = y^2 ( h_(xx) + h_(yy) )
And so on.
If so, IIRC, I asked and answered these very questions (for n= 2, 3) as an
exercise here when I was yakking about the symmetries of PDEs a few years
ago. I think I also discussed the wave equation
u_(tt) = Lap u
the vibration equation
u_(tt) = Lap Lap u
and the Calabi flow
u_t = -Lap Lap u
IIRC, I compared the symmetry method with the transform method (taking
Laplace transform wrt the time variable and then multivariable Fourier
transform wrt space variables, solving the resulting algebraic equation,
and inverting the transforms to obtain the desired convolution formula). I
would consider these elementary methods, although the first may be less
familiar to many students.
Here, needless to add, a "convenient" chart is one in which these methods
work out with a minimum of fuss!
Or if I do not recall correctly, this is similar to questions studied
detail in the book by Peter J. Olver, Applications of Lie groups to
Differential Equations, 2nd Ed., Springer, 1993. Note that this book
discusses the relationship of a fundamental solution, as per symmetry
method with the Green function, as per transform method.
HTH,
"T. Essel"
problem mentioned by a moderator, but for the nonce, if you quote, please
try to reformat any long lines in quoted text yourself, in order to save
work for the moderators! I happen to use a newsreader which makes this
task very easy, but apparently many other posters here have been unable or
unwilling to fix long lines in quoted text.]
On Tue, 5 Sep 2006, Alan wrote:
> I am looking for (refs. to) explicit formulas for the heat kernel
> p(t,x,y) on the n-sphere S^n?
>
> Also, while I have found the formulas for the analogous known kernels on
> the hyperbolic spaces H^n, I am still looking for an "elementary
> derivation" of those.
Consider the following familiar statement about the heat equation on R:
Fact: The initial value problem
u_t = u_(xx)
u(0,x) = f(x) on -infty < x < infty
has the solution
u(x,t) = 1/sqrt(2 Pi)
int_-infty^infty f(xi) exp(-(x-xi)^2/(4 t))/sqrt(2 t) dxi
Here the "kernel" in the convolution, i.e. the function of t,x,xi which we
integrate against, is the function
exp(-(x-xi)^2/(4 t))/sqrt(2 t)
which is also called the "fundamental solution", or the "Green function"
for this problem. This is the solution which describes how an initial
"delta mass" concentration of some scalar field on R spreads out as time
increases, if its evolution is governed by the heat equation.
If I understand your question correctly, you are asking for the "kernels"
appearing in analogous statements for the heat equation
u_t = Lap(u)
where Lap is the Laplace-Beltrami operator on H^n or S^n, as expressed in
some convenient coordinate chart.
E.g. for S^2 we can use a stereographic chart
ds^2 = 1/(1+x^2+y^2)^2/4 (dx^2+dy^2),
-infty < x, y < infty
and then
Lap(u) = (1+x^2+y^2)^2/4 (u_(xx) + u_(yy))
so that the "S^2 heat equation" (written in the stereographic chart) is
u_t = k (1+x^2+y^2)^2/4 (u_(xx) + u_(yy))
And on H^2 we can use the upper half plane chart
ds^2 = (dx^2 + dy^2)/y^2,
-infty < x < infty, 0 < y < infty
and then the "H^2 heat equation" (written in the UHP chart) is
u_t = y^2 ( h_(xx) + h_(yy) )
And so on.
If so, IIRC, I asked and answered these very questions (for n= 2, 3) as an
exercise here when I was yakking about the symmetries of PDEs a few years
ago. I think I also discussed the wave equation
u_(tt) = Lap u
the vibration equation
u_(tt) = Lap Lap u
and the Calabi flow
u_t = -Lap Lap u
IIRC, I compared the symmetry method with the transform method (taking
Laplace transform wrt the time variable and then multivariable Fourier
transform wrt space variables, solving the resulting algebraic equation,
and inverting the transforms to obtain the desired convolution formula). I
would consider these elementary methods, although the first may be less
familiar to many students.
Here, needless to add, a "convenient" chart is one in which these methods
work out with a minimum of fuss!
Or if I do not recall correctly, this is similar to questions studied
detail in the book by Peter J. Olver, Applications of Lie groups to
Differential Equations, 2nd Ed., Springer, 1993. Note that this book
discusses the relationship of a fundamental solution, as per symmetry
method with the Green function, as per transform method.
HTH,
"T. Essel"