  It is found the exact solution of the Poisson equation for the
multidimensional space with topology $M_{3+d}=\mathbb{R}^3\times T^d$. This
solution describes smooth transition from the newtonian behavior $1/r_3$ for
distances bigger than periods of tori (the extra dimension sizes) to
multidimensional behavior $1/r^{1+d}_{3+d}$ in opposite limit. In the case of
one extra dimension $d=1$, the gravitational potential is expressed via compact
and elegant formula. These exact solutions are applied to some practical
problems to get the gravitational potentials for considered configurations.
Found potentials are used to calculate the acceleration for point masses and
gravitational self-energy.It is proposed models where the test masses are
smeared over some (or all) extra dimensions. In 10-dimensional spacetime with 3
smeared extra dimensions, it is shown that the size of 3 rest extra dimensions
can be enlarged up to submillimeter for the case of 1TeV fundamental Planck
scale $M_{Pl(10)}$. In the models where all extra dimensions are smeared, the
gravitational potential exactly coincides with the newtonian one regardless of
size of the extra dimensions. Nevertheless, the hierarchy problem can be solved
in these models.
