(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.