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A Saint-Venant type principle for Dirichlet forms on discontinuous media. (English) Zbl 0851.31008
The authors of this very interesting paper consider families of Dirichlet forms of diffusion type which describe the variational behavior of possibly highly non-homogeneous and non-isotropic bodies. Structural Harnack inequality and Saint-Venant type energy decays are proved for their local solutions. Estimates for the Green functions are also derived.
From the point of view of partial differential equations the theory elaborated in the paper finds its main motivation in the study of structural properties of second order degenerate elliptic operators. New important results, such as Harnack’s inequality, estimates for the Green function and bounds for energy decay, are obtained for weighted uniformly elliptic operators in divergence form with measurable coefficients, as well as for uniformly subelliptic (selfadjoint) second order operators with bounded measurable coefficients.

MSC:
31C25 Dirichlet forms
35J70 Degenerate elliptic equations
31B35 Connections of harmonic functions with differential equations in higher dimensions
74E05 Inhomogeneity in solid mechanics
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