Re: Laplaces equation and measures

craig <ctcowan@xxxxxxxxxxx> writes:

Suppose Omega is some open bounded set in R^n (n
=3)

Let - \Delta E = \mu in Omega

where \mu is some measure with compact support in
Omega.

I am interested in when I can say that E is finite
on the support of \mu.


Example 1
Suppose \mu is the surface measure asscociated to
the unit ball (which we
assume is contained in Omega). Then a direct
calculation shows that E is
finite on the surface of the unit ball.

Example 2
If \mu is a Dirac mass then we know E is not finite
on the support.


So I would assume if \mu is supported on some n-1
dimeansional manifold
then E is finite on the support.

What about the sum (surface measure on a ball) +
(Dirac mass at a point on the
surface)? The support is still the surface of the
ball, but E is no longer
finite on the support.

I guess by above i meant to add a condition so the measure behaved like a surface measure; but as your example below shows it doesn't matter.

Without looking at examples one can even see what you point out below. Say n big enough and p small enough such that
there is a u in W^{2,p} \ L^\infty and then take

f:= - \Delta u \in L^p.

Thanks for the counter example.

NEW QUESTION:

Let A \subset \subset Omega.

Let \mu be the Hausdorf measure associated with A ie. if dim(A)=a then \mu(E):=H^{a}(A \cap E) where I am assuming that H^{a} (A)<infty.

I still think that if a \ge n-1 then we have desired result but ?

If the above true then is it true for some a < n-1?








You could also find a finite measure that is
absolutely continuous wrt
Lebesgue measure on R^n but where E is not finite on
the support.
Consider the spherically symmetric case, and take the
measure mu with
density r^(p-n-2) where n > p > 2. Note that this is
integrable.
If I'm not mistaken, you get a solution E(r) =
r^(p-n)/((p-2)(n-p))
with goes to infinity as r -> 0.
--
Robert Israel
israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics
http://www.math.ubc.ca/~israel
University of British Columbia Vancouver,
BC, Canada
.

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