Divergence-measure fields: generalizations of Gauss-Green formula - - PowerPoint PPT Presentation

Divergence-measure fields: generalizations

• f 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; RN)

• E

divφ dx = −

• ∂E

φ · νE dHN−1, where νE is the interior unit normal to ∂E.

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 3 / 27

BV theory

u : Ω ⊂ RN → R is a function of bounded variation in Ω, u ∈ BV (Ω), if u ∈ L1(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 χE is a (locally) BV function in Ω. By the polar decomposition of Radon measures, DχE = νE||DχE||, for some Borel function νE with norm 1 ||DχE||-a.e. Relevant subsets of the topological boundary of E:

the reduced boundary, (De Giorgi) ∂∗E := {x ∈ Ω : ∃ limr→0

DχE (B(x,r)) ||DχE ||(B(x,r)) = νE(x) ∈ SN−1}, on which the unit

vector νE is well defined and called measure theoretic interior unit normal, since we have the blow-up property (E − x)/r → {(y − x) · νE ≥ 0} in measure as r → 0 for any x ∈ ∂∗E; the measure theoretic boundary, (Federer) ∂mE := Ω \ (E 0 ∪ E 1), where E d := {x ∈ RN : limr→0

|E∩B(x,r)| |B(x,r)|

= d}, which satisfies ∂mE ⊃ ∂∗E and HN−1(∂mE \ ∂∗E) = 0. Hence, we can integrate on ∂mE or ∂∗E with respect to HN−1 indifferently.

||DχE|| = HN−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

• Ω

χEdivφ dx = −

• Ω

φ · dDχE = −

• Ω

φ · νEd||DχE|| 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; RN)

• E

divφ dx = −

• ∂∗E

φ · νE dHN−1. 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 ≪ HN−1; precise representative: any BV function u admits a representative u∗ well defined HN−1-a.e. which satisfies u∗(x) = limε→0(u ⋆ ρε)(x) HN−1-a.e. for any mollification of u. In particular, if E is a set of finite perimeter, χ∗

E = χE 1 + 1

2χ∂∗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; RN) ∩ L∞(Ω; RN) and E ⊂⊂ Ω be a set of finite perimeter, then

• E 1 d div(u) = divu(E 1) = −
• ∂∗E

uνE · νE dHN−1,

• E 1∪∂∗E

d div(u) = divu(E 1 ∪ ∂∗E) = −

• ∂∗E

u−νE · νE dHN−1, where E 1 is the measure theoretic interior of E and u±νE are respectively the interior and the exterior trace; that is, the approximate limits of u in HN−1-a.e. x ∈ ∂∗E restricted to Π±νE (x) := {y ∈ RN : (y − x) · (±νE(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; RN) 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; RN) ∩ L∞(Ω; RN).

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 8 / 27

Divergence-measure fields: definition

Definition

A vector field F ∈ Lp(Ω; RN), 1 ≤ p ≤ ∞ is said to be a divergence-measure field, and we write F ∈ DMp(Ω; RN), if divF is a finite Radon measure on Ω. A vector field F is a locally divergence-measure field, and we write F ∈ DMp

loc(Ω; RN), if F ∈ DMp(W ; RN) 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; RN)

BV (Ω; RN) ∩ Lp(Ω; RN) ⊂ DMp(Ω; RN). Indeed if F = (F1, ..., FN) ∈ Lp(Ω; RN) with Fj ∈ BV (Ω) for j = 1, ...N, then it is clear that DiFj are finite Radon measure for each i, j and so divF = N

j=1 DjFj is

also a finite Radon measure. The condition divF = µ, with µ Radon measure, allows for cancellations; hence, for N ≥ 2, the inclusion is strict. For example, F(x, y) = (sin

• 1

x − y

• , sin
• 1

x − y

• )

satisfies F ∈ DM∞(R2; R2) \ BVloc(R2; R2).

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 13 / 27

Absolute continuity and Leibniz rule

If N ≥ 2 and F ∈ DMp

loc(Ω; RN) for N N−1 ≤ p ≤ ∞, then we have

||divF|| ≪ HN−q, where q :=

p p−1 is the conjugate exponent of p.

This result is sharp: if 1 ≤ p <

N N−1, then for any arbitrary signed Radon

measure µ with compact support inside Ω there exists F ∈ DMp

loc(Ω; RN)

such that divF = µ. On the other hand, if

N N−1 ≤ p ≤ ∞, then for any

s > N − q there exists a field F ∈ DMp

loc(Ω; RN) such that ||divF|| is not

Hs absolutely continuous. Therefore, if F ∈ DM∞(Ω; RN), then ||divF|| ≪ HN−1. If g ∈ BV (Ω) ∩ L∞(Ω) with compact support in Ω and F ∈ DM∞(Ω; RN), we have gF ∈ DM∞(Ω; RN) and div(gF) = g∗divF + F · Dg, where g∗ is the precise representative of g and F · Dg is the weak-star limit

• f F · ∇(g ∗ ρδ) as δ → 0, which satisfies ||F · Dg|| ≪ ||Dg||. Hence, it is in

particular possible to use this formula in the case g = χE with E ⊂⊂ Ω of finite perimeter.

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 14 / 27

Main result

Theorem

Let F ∈ DM∞(Ω; RN). If E ⊂⊂ Ω is a set of finite perimeter, then there exist interior and exterior normal traces of F on ∂∗E; that is, there exist (Fi · νE), (Fe · νE) ∈ L∞(∂∗E; HN−1) such that divF(E 1) = −2χEF · DχE(Ω) = −

• ∂∗E

Fi · νE dHN−1, divF(E 1 ∪ ∂∗E) = −2χΩ\EF · DχE(Ω) = −

• ∂∗E

Fe · νE dHN−1, where χEF · DχE and χΩ\EF · DχE are the weak-star limits, respectively, of the sequences χEF · ∇(χE ∗ ρδ) and χΩ\EF · ∇(χE ∗ ρδ) as δ → 0, up to a

• subsequence. Moreover,

||Fi · νE||L∞(∂∗E;HN−1) ≤ ||F||L∞(E;RN), ||Fe · νE||L∞(∂∗E;HN−1) ≤ ||F||L∞(Ω\E;RN).

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 15 / 27

References

The statement of this theorem is analogous to the result shown in the papers [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). [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).

[S] M. ˇ SILHAV´ Y, Divergence measure fields and Cauchy’s stress theorem Rend.

• Sem. Mat. Univ. Padova 113 (2005), 15-45.

However, the method of proof used in the thesis follows Vol’pert’s idea ([VH]), which consists in using Leibniz rule in order to obtain identities between Radon measures, and then in evaluating them over Ω, without exploiting the approximation theory through sets with smooth boundary exposed in [CTZ]. [VH] A.I. VOL’PERT, S.I. HUDJAEV, Analysis in Classes of Discontinuous Functions and Equation of Mathematical Physics, Martinus Nijhoff Publishers, Dordrecht, 1985.

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 16 / 27

Sketch of the proof

Key facts necessary to the proof:

1 if F ∈ DM∞(Ω; RN) has compact support in Ω, then

divF(Ω) = 0;

2 div(χ2

EF) = div(χEF).

Indeed, since E ⊂⊂ Ω, then χEF has compact support in Ω and so div(χEF)(Ω) = 0. The next steps consist in exploiting Leibniz rule in the second identity.

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 17 / 27

A few calculations

We deduce 1 2χ∂∗EdivF + F · DχE − 2χEF · DχE = 0. (1) If we subtract (1) from div(χEF) = χE 1divF + 1 2χ∂∗EdivF + F · DχE, we get div(χEF) = χE 1divF + 2χEF · DχE. If we add (1) instead, we get div(χEF) = χEdivF + 2F · DχE − 2χEF · DχE = χEdivF + χΩ\EF · DχE. Evaluating these formulas over Ω we obtain the desired result, since the absolute continuity χEF · DχE, χΩ\EF · DχE ≪ ||DχE|| = HN−1∂∗E implies the existence of the interior and exterior normal traces as summable functions, by the Radon-Nikodym theorem.

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 18 / 27

L∞ bounds on the traces

To show that (Fi · νE), (Fe · νE) ∈ L∞(∂∗E, Hn−1), we apply Besicovitch differentiation theorem and we use the fact that χEF · ∇(χE ⋆ ρδ)

∗

⇀ χEF · DχE and χΩ\E∇(χE ⋆ ρδ)

∗

⇀ 1

2DχE. Hence, we can find a sequence rj → 0 such that

(Fi · νE)(x) = lim

rj→0

• 2χEF · DχE(B(x, rj))

||DχE||(B(x, rj))

• = lim

rj→0

• lim

δk→0 2

• B(x,rj)

χEF · ∇χδk dx lim

δk→0

• B(x,rj)

|∇χδk| dx

• ≤ 2||F||L∞(E;Rn) lim

rj→0

   1 − lim

δk→0 |

• B(x,rj)

χΩ\E∇χδk dx| lim

δk→0

• B(x,rj)

|∇χδk| dx     = 2||F||L∞(E;RN) lim

rj→0

• 1 − 1

2 |DχE(B(x, rj))| ||DχE||(B(x, rj))

• = ||F||L∞(E;RN),

by the properties of reduced boundary. (Fe · νE) is treated similarly.

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 19 / 27

Remarks

1 The Gauss-Green formula can be applied also to F ∈ DM∞

loc(Ω; RN): indeed,

since we consider sets of finite perimeter E relatively compact in Ω, then we can always find an open set W such that E ⊂⊂ W ⊂⊂ Ω for which F ∈ DM∞(W ; RN) and the theorem holds.

2 It can be shown that the product rule for divergence measure fields is

consistent with the one for BV fields; hence this result is consistent with the Vol’pert’s theorem.

3 The L∞ estimates are sharp. 4 Exploiting Leibniz rule, we can easily prove that, for any φ ∈ C 1

c (Ω),

• E 1 φ ddivF = −
• ∂∗E

φ(Fi · νE) dHN−1 −

• E

F · ∇φ dx (2)

• E 1∪∂∗E

φ ddivF = −

• ∂∗E

φ(Fe · νE) dHN−1 −

• E

F · ∇φ dx. (3)

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 20 / 27

Jump component of the divergence

We have the following representation formula for the jump component of the divergence of F; that is, for any set of finite perimeter E ⊂⊂ Ω we have χ∂∗EdivF = (Fi · νE − Fe · νE) HN−1∂∗E in the sense of Radon measures on Ω. Hence, we obtain also ||divF||(∂∗E) =

• ∂∗E

|Fi · νE − Fe · νE| dHN−1 and, for any Borel set B ⊂ ∂∗E, divF(B) =

• B

(Fi · νE − Fe · νE) dHN−1. If F is continuous, interior and exterior normal traces coincide on ∂∗E as functions in L∞(∂∗E; HN−1), and they admit a representative which is the classical scalar product F · νE. Therefore, the divergence of continuous vector fields does not have jump component (||divF||(∂∗E) = 0).

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 21 / 27

Gluing and extension theorems

As a consequence of the Gauss-Green formula we obtain gluing and extension theorems: let F1 ∈ DM∞(Ω; RN), F2 ∈ DM∞(RN \ W ; RN) and W ⊂⊂ E ⊂⊂ Ω for some set of finite perimeter E, then we can glue F1 and F2 along the boundary of E; that is, if we set F(x) =

• F1(x) se x ∈ E

F2(x) se x ∈ RN \ E we have F ∈ DM∞(RN; RN); if F ∈ DM∞(Ω; RN) and HN−1(∂Ω) < ∞, then we can extend F to 0

• utside of Ω, in such a way that the extension ˆ

F is in DM∞(RN; RN).

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 22 / 27

The Gauss-Green formula on particular unbounded sets of finite perimeter

Theorem

Let W be a bounded open set, F ∈ DM∞(RN \ W ; RN) and E ⊃⊃ W be a bounded set of finite perimeter. Then

• E 0 φ ddivF = −
• ∂∗E

φ(Fi · νRN\E) dHN−1 −

• RN\E

F · ∇φ dx (4) for any φ ∈ C 1

c (RN \ W ).

This result can be deduced by applying (2) to (B(0, R) \ E)1 = B(0, R) ∩ E 0 for R large enough so that supp(φ) ⊂⊂ B(0, R) and by observing that RN \ (E 1 ∪ ∂mE) = E 0.

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 23 / 27

Relations between normal traces of complementary sets

The formula (4) can be used to prove the gluing theorem, since it gives a Gauss-Green formula for F2 on the set RN \ E. Another interesting consequence is the following relation between the exterior normal trace on ∂∗E and the normal trace interior with respect to the complement.

Proposition

If F ∈ DM∞(RN; RN) and E is a bounded set of finite perimeter, then (Fe · νE) = −(Fi · νRN\E) HN−1-a.e. on ∂∗E, where these functions are respectively the exterior normal trace of F on ∂∗E and the interior normal trace on ∂∗E taken with opposite orientation.

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 24 / 27

Applications to hyperbolic systems of conservation laws

A general hyperbolic system of conservation laws is the following Cauchy problem ut + divxf (u) = 0 in Rd+1

+

:= (0, +∞) × Rd, u = g su {0} × Rd, where u : Rd+1

+

→ U ⊂ Rm, f ∈ C 1(U; Rm×d), divxf (u) is (at least formally) the divergence with respect to x of the matrix f and g is the initial datum. In order to select a unique weak solution we impose that it is satisfied the Lax entropy inequality ∂tη(u) + divxq(u) ≤ 0 in the sense of distributions for any entropy pair (η, q). Given a weak entropy solution u(t, x) ∈ L∞(Rd+1

+

; Rm), the field (η(u), q(u)) is in DM∞

loc(Rd+1 +

; Rd+1) and there exists a positive Radon measure µη such that div(t,x)(η(u), q(u)) = −µη.

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 25 / 27

Traces on hyperplanes

Let u be a weak entropy solution satisfying the following vanishing mean property

• n the half balls B+((τ, y), r) := B((τ, y), r) ∩ {(t, x) ∈ Rd+1 : t > τ}:

lim

r→0

1 r d+1

• B+((τ,y),r)

|q(u(t, x)) − q(u)r(τ, y)| dt dx = 0 for Hd-a.e. (τ, y) ∈ ∂{t > τ}, where q(u)r(τ, y) is the average of q(u) over the half ball. Then, for any τ > 0, η(u) has an essentially bounded trace η(u)tr Hd-a.e. on the hyperplane {(t, x) ∈ Rd+1 : t = τ}; that is, lim

r→0

1 ωdr d+1

• C+((τ,y),r)

η(u(t, x)) dt dx = η(u)tr(τ, y), where C +((τ, y), r) is the cylinder {(t, x) ∈ Rd+1 : 0 < t − τ < r, |x − y| < r}. Therefore, if we choose η(u) = uj, j = 1, ..., m, we obtain the trace of each component of u.

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 26 / 27

Thank you for your attention!

• G. E. Comi (SNS)

Divergence-measure fields May 26, 2016 27 / 27