  Pattern formation in reaction-diffusion systems where the diffusion terms
correspond to a Sturm-Liouville problem are studied. These correspond to a
problem where the diffusion coefficient depends on the spatial variable:
$\nabla \cdot \left( \mathcal{D} ( {\mathbf x} ) \nabla {\mathbf u} \right)$.
We found that the conditions for Turing instability are the same as in the case
of homogeneous diffusion but the nonlinear analysis must be generalized to
consider general orthogonal eigenfunctions instead of the standard Fourier
approach. The particular case $\mathcal{D} (x)= 1-x^2$, where solutions are
linear combinations of Legendre polynomials, is studied in detail. From the
developed general nonlinear analysis, conditions for producing stripes and
spots are obtained, which are numerically verified using the Schaneknberg
system. Unlike to the case with homogeneous diffusion, and due to the
properties of the Legendre polynomials, stripped and spotted patterns with
variable wavelength are produced, and a change from stripes to spots is
predicted when the wavelength increases. The patterns obtained can model
biological systems where stripes or spots accumulate close to the boundaries
and the theory developed here can be applied to study Turing patterns
associated to other eigenfunctions related with Sturm-Liouville problems.
