Divergence-measure fields: generalizations of Gauss-Green formula Giovanni E. Comi (SNS) work in collaboration with K. Payne Technische Universitaet Dresden (TUD) Mathematik AG Analysis & Stochastik May 26, 2016 G. E. Comi (SNS) Divergence-measure fields May 26, 2016 1 / 27
Plan 1 Survey of preexisting theory 2 Definition of divergence-measure fields: motivations and first properties 3 Main result and consequences G. E. Comi (SNS) Divergence-measure fields May 26, 2016 2 / 27
Classical Gauss-Green formula Theorem Let E ⊂⊂ Ω be an open regular set; that is, int (¯ E ) = E and ∂ E is a C 1 ( N − 1) -manifold. Then ∀ φ ∈ C 1 c (Ω; R N ) � � φ · ν E d H N − 1 , div φ dx = − E ∂ E where ν E is the interior unit normal to ∂ E. G. E. Comi (SNS) Divergence-measure fields May 26, 2016 3 / 27
BV theory u : Ω ⊂ R N → R is a function of bounded variation in Ω, u ∈ BV (Ω), if u ∈ L 1 (Ω) and the distributional gradient Du is a finite Radon measure; that is, a vector valued Borel measure with finite total variation on Ω. A set E of (locally) finite perimeter in Ω is a set whose characteristic function χ E is a (locally) BV function in Ω. By the polar decomposition of Radon measures, D χ E = ν E || D χ E || , for some Borel function ν E with norm 1 || D χ E || -a.e. Relevant subsets of the topological boundary of E : the reduced boundary , (De Giorgi) D χ E ( B ( x , r )) ∂ ∗ E := { x ∈ Ω : ∃ lim r → 0 || D χ E || ( B ( x , r )) = ν E ( x ) ∈ S N − 1 } , on which the unit vector ν E is well defined and called measure theoretic interior unit normal , since we have the blow-up property ( E − x ) / r → { ( y − x ) · ν E ≥ 0 } in measure as r → 0 for any x ∈ ∂ ∗ E ; the measure theoretic boundary , (Federer) ∂ m E := Ω \ ( E 0 ∪ E 1 ), where E d := { x ∈ R N : lim r → 0 | E ∩ B ( x , r ) | = d } , which satisfies ∂ m E ⊃ ∂ ∗ E and | B ( x , r ) | H N − 1 ( ∂ m E \ ∂ ∗ E ) = 0. Hence, we can integrate on ∂ m E or ∂ ∗ E with respect to H N − 1 indifferently. || D χ E || = H N − 1 � ∂ ∗ E (De Giorgi’s theorem). G. E. Comi (SNS) Divergence-measure fields May 26, 2016 4 / 27
Gauss-Green formula for sets of finite perimeter We just need to apply the definition of distributional derivative � � � χ E div φ dx = − φ · dD χ E = − φ · ν E d || D χ E || Ω Ω Ω and then De Giorgi’s theorem. Theorem ( De Giorgi and Federer) Let E ⊂ Ω be a set of locally finite perimeter. Then ∀ φ ∈ C 1 c (Ω; R N ) � � φ · ν E d H N − 1 . div φ dx = − E ∂ ∗ E Aim: to weaken the regularity hypotheses on the vector fields. Strategy: to characterize the divergence in a weak sense (as a Radon measure) and the trace as an approximate limit or the density of a Radon measure. G. E. Comi (SNS) Divergence-measure fields May 26, 2016 5 / 27
Fine properties of BV functions Important properties of BV functions: if u ∈ BV (Ω), then Du ≪ H N − 1 ; precise representative: any BV function u admits a representative u ∗ well defined H N − 1 -a.e. which satisfies u ∗ ( x ) = lim ε → 0 ( u ⋆ ρ ε )( x ) H N − 1 -a.e. for any mollification of u . In particular, if E is a set of finite perimeter, E = χ E 1 + 1 χ ∗ 2 χ ∂ ∗ E ; if u ∈ BV (Ω) and supp ( u ) ⊂⊂ Ω, then Du (Ω) = 0; Leibniz rule: if u , v ∈ BV (Ω) ∩ L ∞ (Ω), then uv ∈ BV (Ω) ∩ L ∞ (Ω) and D ( uv ) = u ∗ Dv + v ∗ Du . G. E. Comi (SNS) Divergence-measure fields May 26, 2016 6 / 27
Gauss-Green formula for BV vector fields Theorem ( Vol’pert) Let u ∈ BV (Ω; R N ) ∩ L ∞ (Ω; R N ) and E ⊂⊂ Ω be a set of finite perimeter, then � � E 1 d div ( u ) = div u ( E 1 ) = − u ν E · ν E d H N − 1 , ∂ ∗ E � � d div ( u ) = div u ( E 1 ∪ ∂ ∗ E ) = − u − ν E · ν E d H N − 1 , E 1 ∪ ∂ ∗ E ∂ ∗ E where E 1 is the measure theoretic interior of E and u ± ν E are respectively the interior and the exterior trace; that is, the approximate limits of u in H N − 1 -a.e. x ∈ ∂ ∗ E restricted to Π ± ν E ( x ) := { y ∈ R N : ( y − x ) · ( ± ν E ( x )) ≥ 0 } . G. E. Comi (SNS) Divergence-measure fields May 26, 2016 7 / 27
Summary of the preexisting theory Classical Gauss-Green formula: fields φ ∈ C 1 c (Ω; R N ) and open regular sets as integration domains. BV theory: new characterization of sets based on the properties of the distributional gradient of their characteristic function and Leibniz rule in the sense of Radon measures. De Giorgi and Federer: extension to sets of finite perimeter. Vol’pert: extension to vector fields in BV (Ω; R N ) ∩ L ∞ (Ω; R N ). G. E. Comi (SNS) Divergence-measure fields May 26, 2016 8 / 27
Divergence-measure fields: definition Definition A vector field F ∈ L p (Ω; R N ) , 1 ≤ p ≤ ∞ is said to be a divergence-measure field , and we write F ∈ DM p (Ω; R N ), if div F is a finite Radon measure on Ω. A vector field F is a locally divergence-measure field , and we write F ∈ DM p loc (Ω; R N ), if F ∈ DM p ( W ; R N ) for any open set W ⊂⊂ Ω. G. E. Comi (SNS) Divergence-measure fields May 26, 2016 9 / 27
Brief history These new function spaces were introduced in the early 2000s by many authors for different purposes. 1 Chen and Frid were interested in the applications to the theory of systems of conservation laws with the Lax entropy condition and achieved a Gauss-Green formula for divergence-measure fields on open bounded sets with Lipschitz deformable boundary. Later, Chen, Torres and Ziemer extended this result to the sets of finite perimeter in the case p = ∞ . 2 Degiovanni, Marzocchi, Musesti, ˇ Silhav´ y and Schuricht wanted to prove the existence of a normal trace under weak regularity hypotheses, in order to achieve a representation formula for Cauchy fluxes, contact interactions and forces in the context of continuum mechanics. 3 Ambrosio, Crippa and Maniglia studied a class of these vector fields induced by functions of bounded deformation, with the aim of extending DiPerna-Lions theory of the transport equation. G. E. Comi (SNS) Divergence-measure fields May 26, 2016 10 / 27
Relevant bibliography [ACM] L. AMBROSIO, G. CRIPPA, S. MANIGLIA, Traces and fine properties of a BD class of vector fields and applications , Ann. Fac. Sci. Toulouse Math. XIV 527-561 (2005). [CF1] G.-Q. CHEN, H. FRID, Divergence-measure fields and hyperbolic conservation laws , Arch. Ration. Mach. Anal. 147 no. 2, 89-118 (1999). [CF2] G.-Q. CHEN, H. FRID, Extended divergence-measure fields and the Euler equations for gas dynamics , Comm. Math. Phys. 236 no. 2, 251-280 (2003). [CT1] G.-Q. CHEN, M. TORRES, Divergence-measure fields, sets of finite perimeter, and conservation laws , Arch. Ration. Mach. Anal. 175 no. 2, 245-267 (2005). [CT2] G.-Q. CHEN, M. TORRES, On the structure of solutions of nonlinear hyperbolic systems of conservation laws , Comm. on Pure and Applied Mathematics, Vol. X, no. 4, 1011-1036 (2011). G. E. Comi (SNS) Divergence-measure fields May 26, 2016 11 / 27
Relevant bibliography [CTZ] G.-Q. CHEN, M. TORRES, W.P. ZIEMER, Gauss-Green Theorem for Weakly Differentiable Vector Fields, Sets of Finite Perimeters, and Balance Laws , Comm. on Pure and Applied Mathematics, Vol. LXII, 0242-0304 (2009). [DMM] M. DEGIOVANNI, A. MARZOCCHI, A. MUSESTI, Cauchy fluxes associated with tensor fields having divergence measure , Arch. Ration. Mech Anal. 147 (1999), no. 3, 197-223. [Sc] F. SCHURICHT, A new mathematical foundation for contact interactions in continuum physics Arch. Rat. Math. Anal. 184 (2007) 495-551. [Si] M. ˇ SILHAV´ Y, Divergence measure fields and Cauchy’s stress theorem Rend. Sem. Mat. Univ. Padova 113 (2005), 15-45. G. E. Comi (SNS) Divergence-measure fields May 26, 2016 12 / 27
Comparison with BV (Ω; R N ) BV (Ω; R N ) ∩ L p (Ω; R N ) ⊂ DM p (Ω; R N ). Indeed if F = ( F 1 , ..., F N ) ∈ L p (Ω; R N ) with F j ∈ BV (Ω) for j = 1 , ... N , then it is clear that D i F j are finite Radon measure for each i , j and so div F = � N j =1 D j F j is also a finite Radon measure. The condition div F = µ , with µ Radon measure, allows for cancellations; hence, for N ≥ 2, the inclusion is strict. For example, � 1 � � 1 � F ( x , y ) = (sin , sin ) x − y x − y satisfies F ∈ DM ∞ ( R 2 ; R 2 ) \ BV loc ( R 2 ; R 2 ) . G. E. Comi (SNS) Divergence-measure fields May 26, 2016 13 / 27
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