(2) The action of the Laplace operator
on the Riesz fractional integral is
An interior Neumann function for the Laplace operator
is the solution of the following boundary value problem for the potential [N.sup.i](r, [r.sub.s]):
Denote by [DELTA] = [[summation].sup.m.sub.j=1]([[partial derivative].sup.2]/[partial derivative][x.sup.2.sub.j]) the Laplace operator
, with domain D([DELTA]) = [W.sup.1,2.sub.0](Y) [intersection] [W.sup.2,2.sub.0](Y), which generates a strongly continuous semigroup [mathematical expression not reproducible], where [W.sup.1,2.sub.0](Y) and [W.sup.2,2.sub.0](Y) are the Sobolev spaces with compactly supported sets.
requires corresponding interpretation of the Laplace operator
in the Landau --Ginzburg equation as Laplace--Beltrami operator generated by the fibre structure of the manifold M.
Thus, [v.sub.j] belongs to the Dirichlet realization of the Laplace operator
on [R.sub.+], and this concludes the proof of the second statement of the theorem.
where k, l are natural numbers, v is velocity, f is force, s is Laplace operator
, and [d.sub.l], ..., [d.sub.0], [c.sub.k], ..., [c.sub.0] are real numbers.
In this paper, a controllability problems for co-operative parabolic linear system involving Laplace operator
with boundary Dirichlet control and distributed or boundary observations are considered.
Applying the Laplace operator
on both sides of the above equation, we have the following ordinary differential equation
(1), [bar.r] is position vector, [[nabla].sup.2] represents the Laplace operator
, and k denotes wavenumber.
Defining the differential operator [DELTA]* := [DELTA] + [lambda]+[mu]/[mu] grad div, where [DELTA] is the Laplace operator
and [lambda] and [mu] are the Lame elastic constants with u > 0 and [lambda] + 2[mu] > 0.
We'll consider perturbation of Laplace Operator
-[?] with a singular potential q.
The Laplace operator
is very much sensitive to noise.