Re: Existence of solution to Laplace's PDE

In article <rbisrael.20080606050319$3830@xxxxxxxxxxxxxxxxx>,
Robert Israel <israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

(throughout the discussion below x1,x2,x3 denote the components of a vector
x in R^3). Denote by V the punctured unit sphere in R^3,

V = {x in R^3:|x|<1, x1^2+x2^2>0}

(i.e. the unit sphere with the x3 axis removed). Consider the boundary
value problem

Laplacian(u) = 0 on V

with boundary conditions

u = 0 on |x|=1
u = 1-x3^2 on x1=x2=0

After some tinkering around with it, I believe I can show the PDE does not
admit a continuous solution, though my proof is quite inelegant, and not at
all general. Can anyone come up with a simple argument for the
nonexistence of a solution for this problem?

IIRC one argument uses Brownian motion. If X(t) is Brownian motion starting
at a point p in V and stopped on the boundary of V, and u(x,y,z) any function
continuous on the closure of V and harmonic on the interior of V, then
u(X(t)) is a martingale. In this case, since X(t) almost surely
never hits the x3 axis, lim_{t -> infty} u(X(t)) = 0 a.s., so u(p) = 0.

Then if someone wonders how we know that X(t) a.s. never hits the
x3 axis: Since the first two coordinates of Brownian motion in R^3
are Brownian motion in R^2, this follows from the fact that
Brownian motion in R^2 (started at a point other than the origin)
a.s. never hits the origin. _That_ follows from the fact that
the Dirichlet problem in a punctured disk can't be solved, which
follows from the fact that a single point is a removable singularity
for a bounded harmonic function, which follows (consider exp(u+iv))
from the same fact for bounded holomorphic functions.

Probably there are better arguments, I just think it's cute that
you can go back and forth from the BM side and the function-theory
side this way.

--
David C. Ullrich
.

Relevant Pages

  • Re: Existence of solution to Laplaces PDE
    ... (i.e. the unit sphere with the x3 axis removed). ... IIRC one argument uses Brownian motion. ... x3 axis: Since the first two coordinates of Brownian ... Measure zero is certainly not enough, for example BM started at the ...
    (sci.math)
  • Re: Existence of solution to Laplaces PDE
    ... Denote by V the punctured unit sphere in R^3, ... (i.e. the unit sphere with the x3 axis removed). ... admit a continuous solution, though my proof is quite inelegant, and not at ... IIRC one argument uses Brownian motion. ...
    (sci.math)
  • Re: brownian motion
    ... time when the brownian motion will touch the unit sphere, ... Your English had nothing to do with it, ... what the law of B_t is. ...
    (sci.math)
  • Re: Existence of solution to Laplaces PDE
    ... IIRC one argument uses Brownian motion. ... x3 axis: Since the first two coordinates of Brownian ... never hits the origin. ... for a bounded harmonic function ...
    (sci.math)
  • Re: brownian motion
    ... time when the brownian motion will touch the unit sphere, ... Your English had nothing to do with it, ... what the law of B_t is. ...
    (sci.math)
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