A divergence measure vectorfield is an ℝn valued measure on an open subset U of ℝn whose weak divergence in U is a (signed) measure. The paper uses the product rule for the product of the divergence measure by a function from W 1,∞(U) 3 established in Šilhavý [Šilhavý, M., submitted, 2007] to prove the divergence theorem for the divergence measure vectorfields on bounded open sets U. It is shown that the surface integral of the normal component of the vectorfield occurring in the classical divergence theorem has to be replaced by a continuous linear functional on the space of Lipschitz functions on the boundary1 the volume integral contains the duality pairing occurring in the product rule. The boundary of U is arbitrary, it can be even fractal in the sense that the normal to ∂U cannot be defined.
Chen, G.-Q. and Frid, H.On the theory of divergence-measure fields and its applications. Boletim da Sociedade Brasileiro de Mathemática , 321-33 (2001).
2.
Chen, G.-Q. and Frid, H.Extended divergence-measure fields and the Euler equations for gas dynamics. Communications in Mathematical Physics, 236251-280 (2003).
Šilhavý, M.Normal traces of divergence measure vectorfields on fractal boundaries. (Preprint, Dipartimento di Matematica, University of Pisa, October 2005.)
5.
Degiovanni, M. , Marzocchi, A. and Musesti, A.Virtual powers on diffused subbodies and normal traces of tensor-valued measures. (2007). (In press.)
6.
Whitney, H.Geometric Integration Theory, Princeton University Press , Princeton, 1957.
7.
Anzellotti, G.Pairings between measures and bounded functions and compensated compactness. Annali di Matematica Pura ed Applicata, 135293-318 (1983).
8.
Témam, R.Problémes Mathématiques en Plasticité, Gauthier-Villars , Paris, 1983.
9.
Kohn, R. and Témam, R.Dual spaces of stresses and strains, with applications to Hencky plasticity . Applied Mathematics and Optimization, 101-35 (1983).
Chen, G.-Q. and Torres, M.Divergence measure fields, sets of finite perimeter, and conservation laws . Archive for Rational Mechanics and Analysis, 175245-267 (2005).
12.
Šilhavý, M.Divergence measure fields and Cauchy's stress theorem. Rendiconti del Seminario Matematico della Universitá di Padova, 11315-45 (2006).
13.
Šilhavý, M.Cauchy's stress theorem and tensor fields with divergences in Lp . Archive for Rational Mechanics and Analysis, 116223-255 (1991).
14.
Degiovanni, M. , Marzocchi, A. and Musesti, A.Cauchy fluxes associated with tensor fields having divergence measure. Archive for Rational Mechanics and Analysis, 147197-223 (1999).
15.
Marzocchi, A. and Musesti, A.Decomposition and integral representation of Cauchy interactions associated with measures. Continuum Mechanics and Thermodynamics, 13149-169 (2001).
16.
Marzocchi, A. and Musesti, A.On the measure theoretic foundation of the second law of thermodynamics. Mathematical Models and Methods in Applied Sciences, 12721-736 (2002).
17.
Marzocchi, A. and Musesti, A.Balanced powers in continuum mechanics. Meccanica, 38369-389 (2003).
18.
Marzocchi, A. and Musesti, A.The Cauchy stress theorem for bodies with finite perimeter. Rendiconti del Seminario Matematico della Universitá di Padova, 1091-11 (2003).
19.
Marzocchi, A. and Musesti, A.Balance laws and weak boundary conditions in continuum mechanics. Journal of Elasticity, 38239-248 (2004).
20.
Schuricht, F.A new mathematical foundation for contact interactions in continuum physics. Archive for Rational Mechanics and Analysis, (2006). Electronic version DOI 10.1007/s00205-006-0032-6
21.
Lucchesi, M., Šilhavý , M. and Zani, N.A new class of equilibrated stress fields for no-tension bodies. Journal of Mechanics of Materials and Structures, 1503-539 (2006).
22.
Podio-Guidugli, P.Examples of concentrated contact interactions in simple bodies. Journal of Elasticity, 75167- 186 (2004).
23.
Podio-Guidugli, P.On concentrated contact interactions, in Proceedings of the SISSA Meeting "Variational Problems in Materials Science", Trieste, September, 2004Birkhauser, Basel, 2005.
24.
Šilhavý, M.Geometric integration theory and Cauchy's stress theorem. (Multiplied notes from the Ravello Summer School in Mathematical Physics, 2004.)
25.
Šilhavý, M.Divergence measure vectorfields: their structure and the divergence theorem. (2007). (To appear.)
26.
Federer, H.Geometric Measure Theory, Springer, New York, 1969.
27.
Ambrosio, L., Fusco, N. and Pallara, D.Functions of Bounded Variation and Free Discontinuity Problems, Clarendon, Oxford, 2000.
