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We consider possibly degenerate and singular elliptic equations in a possiblyanisotropic medium.  We obtain monotonicity results for the energy density, rigidity results forthe solutions and classification results for thesingularity/degeneracy/anisotropy allowed.  As far as we know, these results are new even in the case of non-singular andnon-degenerate anisotropic equations.

Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs