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\leftline{Functional analytic methods in differential equations}
\leftline{Dubrovnik, Croatia, September 15-20, 1997}

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\centerline{{\bf Nonlocal effects induced by homogenisation}}
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\centerline{Nenad Antoni\'c}
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Thirty years ago Sergio Spagnolo showed that the weak limit of a sequence of
inverses of second order elliptic operators is the inverse of such an operator as well.
On the other hand, such a limit of operators of the form $-\dv(a\nabla\cdot),$
where $a$ is a measurable nonnegative function, corresponding to an isotropic material
for stationary heat conduction, in general is of the form $-\dv({\bf A}\nabla\cdot),$
with ${\bf A}$ taking values among symmetric positive matrices, which corresponds 
to an aniso\-tropic material.

The classes of physical laws stable by the homogenisation procedure (corresponding
to weak limits) are the candidates for good descriptors of continuum mechanics.

Some examples of unstable linear equations will be given, in each case leading
to convolution operators; as any linear operator commuting with translations
has to be of that form. 
In particular, a sequence of semigroups of operators with a limit
not being a semigroup will be included above.
The research in progress regarding more complicated linear
equations, like the Maxwell system, as well as nonlinear equations, will be discussed,
including some open problems.

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