4 Bingham problem: Proofs

This section is devoted to the proofs related to the Bingham problem.

4.1 Preliminaries

We require a local variation of Lemma 2.2. Recall that Cϕ (resp.,Cϕ,n andCn) denote generic constants whose value exclusively depends on ϕ(resp., on ϕ, nand on n).

Lemma 4.1 (local estimate) Given a bounded open set Ω ⊂ Rⁿ, u ∈ H¹(Ω)ⁿ, ϕ ∈ C_c^∞(Ω) withϕ≥0, and a positiveε≤2⁻⁴kϕ∇ϕk⁻²_L∞(Ω)ⁿ, there exists a unique solution u_ε∈H¹(Ω)ⁿ to

Z

Ω

u_ε·w+ε Z

Ω

ϕ⁴∇u_ε:∇w= Z

Ω

u·w for allw∈H¹(Ω)ⁿ. (4.1) Moreover, ϕ²∇²uε∈L²(Ω)^n×n×n, ϕ⁴∇³uε∈L²(Ω)^n×n×n×n, and

ku_εk_L2(Ω)ⁿ ≤ kuk_L2(Ω)ⁿ, (4.2) k∇u_εk_L2(Ω)^n×n ≤ 1

1−8εkϕ∇ϕk²_L∞(Ω)ⁿ

k∇uk_L2(Ω)^n×n, (4.3) k∇u_εk²_L2(Ω)^n×n+ ε

2kϕ²∆u_εk²_L2(Ω)ⁿ ≤ Z

Ω

∇u_ε:∇u+ 2⁵εkϕ∇ϕk²_L∞(Ω)ⁿk∇uk²_L2(Ω)^n×n, (4.4) kϕ²∇²u_εk_L2(Ω)^n×n×n ≤ kϕ²∆u_εk_L2(Ω)ⁿ+C_ϕ,nkuk_H1(Ω)ⁿ, (4.5) kDu_εk_L1(Ω)^n×n ≤ kDuk_L1(Ω)^n×n+Cϕ,nε kDuk_L1(Ω)^n×n+kϕ²∇u_εk_L2(Ω)^n×n

. (4.6)

Proof. Since the uniqueness is obvious, the proof focuses on the existence of a solution to (4.1).

Given anyγ >0, let u_εγ ∈H¹(Ω)ⁿ solve Z

Ω

uεγ·w+ε Z

Ω

(ϕ⁴+γ)∇u_εγ :∇w= Z

Ω

u·w for all w∈H¹(Ω)ⁿ. (4.7) The test function w=uεγ shows

ku_εγk_L2(Ω)ⁿ≤ kuk_L2(Ω)ⁿ. (4.8) Since

uεγ−εdiv((ϕ⁴+γ)∇u_εγ) =u in Ω (4.9) holds in the sense of distributions, we have div((ϕ⁴ +γ)∇u_εγ) = (u_εγ −u)/ε ∈ L²(Ω)ⁿ, so that (ϕ⁴ +γ)∇u_εγ ∈ H(div,Ω)ⁿ. The comparison of (4.7) with (4.9) proves ∇u_εγν = 0 in H^−1/2(∂Ω)ⁿ. Since

div((ϕ⁴+γ)∇u_εγ) = (ϕ⁴+γ)∆uεγ +∇u_εγ∇(ϕ⁴) (4.10) (the Laplacian acts componentwise), we have (ϕ⁴+γ)∆u_εγ ∈L²(Ω)ⁿ; whence ∆u_εγ ∈L²(Ω)ⁿ. This and the above homogeneous Neumann boundary conditions prove

Z

Ω

∇u_εγ :∇v=− Z

Ω

v·∆uεγ for allv ∈H¹(Ω)ⁿ. (4.11)

The test function v=uεγ −u in (4.11) and (4.9)-(4.10) verify Z

Ω

∇u_εγ : (∇u_εγ− ∇u) =−ε Z

Ω

∆u_εγ·

(ϕ⁴+γ)∆u_εγ+∇u_εγ∇(ϕ⁴)

. (4.12) The crucial point is the observation that

|∇(ϕ⁴)|= 4ϕ³|∇ϕ| ≤4kϕ∇ϕk_L∞(Ω)ⁿϕ². (4.13) This, a Cauchy–Schwarz inequality, and a weighted Young inequality result in

R

Ω∆uεγ ·(∇u_εγ∇(ϕ⁴))

≤4kϕ∇ϕk_L^∞_(Ω)nkϕ²∆uεγk_L2(Ω)ⁿk∇u_εγk_L2(Ω)^n×n

≤8kϕ∇ϕk²_L∞(Ω)ⁿk∇u_εγk²_L2(Ω)n×n+¹₂kϕ²∆u_εγk²_L2(Ω)n. Its direct consequence

1 2

Z

Ω

(ϕ⁴+γ)|∆u_εγ|²−8kϕ∇ϕk²_L∞(Ω)ⁿk∇u_εγk²_L2(Ω)^n×n ≤ Z

Ω

∆u_εγ·

(ϕ⁴+γ)∆u_εγ+∇u_εγ∇(ϕ⁴) is substituted in (4.12) for the proof of

1−8εkϕ∇ϕk²_L∞(Ω)ⁿ

k∇u_εγk²_L2(Ω)^n×n+ ε 2

Z

Ω

(ϕ⁴+γ)|∆u_εγ|² ≤ Z

Ω

∇u_εγ :∇u. (4.14) Recall that ε≤2⁻⁴kϕ∇ϕk⁻²_L∞(Ω)ⁿ and deduce

k∇u_εγk_L2(Ω)^n×n≤ 1

1−8εkϕ∇ϕk²_L∞(Ω)ⁿ

k∇uk_L2(Ω)^n×n ≤2k∇uk_L2(Ω)^n×n. (4.15) The combination of (4.14)-(4.15) reveals that

k∇u_εγk²_L2(Ω)^n×n+ε 2

Z

Ω

(ϕ⁴+γ)|∆u_εγ|²≤ Z

Ω

∇u_εγ :∇u+2⁵εkϕ∇ϕk²_L∞(Ω)ⁿk∇uk²_L2(Ω)^n×n. (4.16) The bounds (4.15)-(4.16) onuεγ imply thatuεγ is bounded inH¹(Ω)ⁿandϕ²∆uεγ is bounded in L²(Ω)ⁿasγ →0. Hence, there exists a sequence γ_j →0 such that u_εγj * u_ε weakly inH¹(Ω)ⁿ andϕ²∆u_εγj * ϕ²∆u_εweakly inL²(Ω)ⁿ. This and the sequential weak lower semi-continuity of norms show that the weak limituε∈H¹(Ω)ⁿsolves (4.1) and, owing to (4.8) and (4.15)-(4.16), satisfies the estimates (4.2)-(4.4).

The further investigation of the unique solution uε∈H¹(Ω)ⁿ to (4.1) rewrites it as

u_ε−εdiv(ϕ⁴∇u_ε) =u in Ω (4.17)

in the sense of distributions. Since v_ε =ϕ²u_ε∈H¹(Ω)ⁿ has compact support, its extension by zero vε∈H¹(Rⁿ)ⁿ satisfies

∇v_ε=ϕ²∇u_ε+u_ε⊗ ∇(ϕ²)∈L²(Ω)^n×n (4.18) and, in the sense of distributions inRⁿ for any i, j= 1, . . . , n,

∂_ij²vε=ϕ²∂_ij²uε+∂iuε∂j(ϕ²) +∂i(ϕ²)∂juε+uε∂_ij²(ϕ²). (4.19)

Recall that ϕ²∆uε∈L²(Ω)ⁿ to deduce from (4.19) that ∆vε ∈L²(Ω)ⁿ satisfies

k∆v_εk_L2(Ω)ⁿ ≤ kϕ²∆u_εk_L2(Ω)ⁿ+C_ϕ,nku_εk_H1(Ω)ⁿ. (4.20) Since v_ε has compact support in Ω, v_ε ∈H¹(Rⁿ)ⁿ satisfies ∆v_ε ∈L²(Rⁿ)ⁿ. Invoking a Fourier transformation or the elliptic regularity of the Poisson problem on a large ball that includes Ω [15, Equ. (3,1,2,2)], we deduce that vε ∈H²(Rⁿ)ⁿ satisfies

k∇²vεk_L2(Ω)^n×n×n ≤ k∆v_εk_L2(Ω)ⁿ. (4.21) Recall (4.2)-(4.3) withε≤2⁻⁴kϕ∇ϕk⁻²_L∞(Ω)ⁿ to inferku_εk_H1(Ω)ⁿ ≤2kuk_H1(Ω)ⁿ. This and (4.19) show thatϕ²∇²uε∈L²(Ω)^n×n×n satisfies

kϕ²∇²u_εk_L2(Ω)^n×n×n ≤C_ϕ,nkuk_H1(Ω)ⁿ+k∇²v_εk_L2(Ω)^n×n×n ≤C_ϕ,nkuk_H1(Ω)ⁿ+kϕ²∆u_εk_L2(Ω)ⁿ

using (4.20)-(4.21) in the last step. This is (4.5).

Since div(ϕ⁴∇u_ε) = ϕ⁴∆uε+∇u_ε∇(ϕ⁴), (4.17) shows ∇(ϕ⁴∆uε) ∈ L²(Rⁿ)^n×n with compact support in Ω. This and the above arguments for the proof of (4.20)-(4.21) eventually lead to ϕ⁴∇³u_ε∈L²(Ω)^n×n×n×n; further details on this are omitted here.

It remains to prove (4.6). Define jδ as in (2.4) but let the argumentξ be a (symmetric) matrix rather than a vector. Recall the above properties ofj_δ hold in the example at hand as well. For instance, the convexity of j_δ implies that

j_δ(Duε)−j_δ(Du)≤j⁰_δ(Duε) : (Duε−Du). (4.22) Moreover,jδ∈C² is convex (its Hessian is positive semi-definite) andjδas well as its derivative j_δ⁰ are globally Lipschitz continuous with|j_δ⁰(F)| ≤1 and|F| |j_δ⁰⁰(F)| ≤1 for anyF ∈R^n×n. We remark that for a symmetric matrix S and an arbitrary matrixA, one hasS:A=S :A^t=S : (A+A^t)/2. Consequently,

j_δ(Du_ε)−j_δ(Du)≤j_δ⁰(Du_ε) :∇(u_ε−u). (4.23) Recall (4.17) to compute, for anyi, j = 1, . . . , n, the derivatives in the sense of distributions as follows:

ε⁻¹[∇(u_ε−u)]_ij = [∇(div(ϕ⁴∇u_ε))]_ij =

n

X

k=1

∂_jk² (ϕ⁴∂_kuⁱ_ε)

=

n

X

k=1

∂_k ϕ⁴∂_jk² uⁱ_ε+∂_j(ϕ⁴)∂_kuⁱ_ε

=

n

X

k=1

∂_k ϕ⁴∂_jk² uⁱ_ε+ 2∂_j(ϕ⁴)(Du_ε)_ki−∂_j(ϕ⁴)∂_iu^k_ε

=

n

X

k=1

∂_k

ϕ⁴∂_jk² uⁱ_ε+ 2∂_j(ϕ⁴)(Du_ε)_ki−δ_ik∂_j(ϕ⁴) tr(Du_ε)

−∂_j(ϕ⁴) ∂_iu^k_ε−δ_iktr(Du_ε) with the Kronecker deltaδik. Since Pn

k=1∂k(∂iu^k_ε−δiktr(Duε)) = 0, this is recast as

∇(u_ε−u) =ε

n

X

k=1

∂k

ϕ⁴∂k∇u_ε+ 2(Duε)^k⊗ ∇(ϕ⁴)−tr(Duε)e^k⊗ ∇(ϕ⁴)

−ε (∇u_ε)^t−tr(∇u_ε) Id

∇²(ϕ⁴).

(4.24)

The integral over Ω of the combination of (4.23)-(4.24) reads

An integration by parts in the first term on the right-hand side of (4.25) shows that

n term on the right-hand side of (4.25) shows the identity below, which is followed by a Cauchy–

Schwarz inequality with respect to the symmetric and positive semi-definite Hessian ofj_δ. This proves a.e. in Ω, the last term is controlled by ^C

ϕ,n2

The combination of (4.25)-(4.28) involves the cancellation of the upper bound (4.26) with the first term on the right-hand side of (4.27). This proves

Z and a Cauchy–Schwarz inequality over the support ofϕ concludes the proof of (4.6).

4.2 Proof of Theorem 1.6

The test functionsv= 0 and v= 2u in (1.31) provide u_ε to the solutionu∈H to (1.31). Recall thatu_εsatisfies (4.17) andϕhas compact support, so thatuεis equal tououtside the support ofϕ; whenceuε∈H (also in case of Dirichlet boundary conditions). Thus, if u∈H₀¹(Ω)ⁿ, we have also u_ε ∈H₀¹(Ω)ⁿ. The test functionv =u_ε∈H in (1.31) plus elementary algebra proves

αku_ε−uk²_L2(Ω)ⁿ+µ|u_ε−u|²_H1(Ω)ⁿ+ from below by (4.6), and (4.17) rewrites the integralR

Ωf·(u−u_ε) =−εR

Ωf·div(ϕ⁴∇u_ε). These arguments and a division byε >0 lead in (4.32) to the estimate

µ Recall the convention on the generic constants Cnϕ and estimates like (4.13) or the estimate k∇u_εk_L2(Ω)^n×n ≤2k∇uk_L2(Ω)^n×n from (4.3) (for smallε) to rewrite this as

µ Cnϕ

kϕ²∆uεk²_L2(Ω)ⁿ≤ kDuk_L1(Ω)^n×n+k∇uk_L2(Ω)^n×n+µk∇uk²_L2(Ω)^n×n

+kfk_L2(Ω)ⁿkϕ²∆u_εk_L2(Ω)ⁿ+kfk_L2(Ω)ⁿk∇uk_L2(Ω)^n×n. (4.34)

Except the term kfk_L2(Ω)ⁿkϕ²∆uεk_L2(Ω)ⁿ, that shall be absorbed, the a priori bounds (4.30)-(4.31) control the remaining terms on the right-hand side of (4.34). This leads to

µ Cnϕ

kϕ²∆uεk²_L2(Ω)ⁿ ≤(1 α + 1

µ)kfk²_L2(Ω)ⁿ+ 1

√αµkfk_L2(Ω)ⁿ. (4.35) The combination with (4.5) bounds kϕ²∇²uεk_L2(Ω)^n×n×n independently ofε >0. Moreover, the first two terms in (4.32) have been neglected before, but lead to the control of

kϕ²∇²u_εk_L2(Ω)^n×n×n+ku_εk_H1(Ω)ⁿ+α

εku−u_εk²_L2(Ω)ⁿ+µ

ε|u−u_ε|²_H1(Ω)ⁿ

by a global upper bound independently ofε >0 asε→0. Hence, there exists a sequenceε→0 (not labelled explicitly) such that u_ε,∇u_ε, ϕ²∇²u_ε are bounded and convergence component-wise weakly in L²(Ω) to the weak limit u,∇u, ϕ²∇²u. In particular, ϕ²∇²u ∈ L²(Ω)^n×n×n.

This and (4.35) lead to the assertion (1.32).

4.3 Proof of Theorem 1.7

The test functionsv= 0 and v= 2u in (1.37) provide α

Z

Ω

|u|²+ψ(u) = Z

Ω

f ·u. (4.36)

This leads to the explicit bounds on u inL²(Ω)ⁿ, namely kuk_L2(Ω)ⁿ ≤ kfk_L2(Ω)ⁿ

α and ψ(u)≤

kfk²_L2(Ω)n

α . (4.37)

Given any µ >0, let u_µ ∈H solve (1.31) with eitherH =H₀¹(Ω)ⁿ (Dirichlet) or H =H¹(Ω)ⁿ (Neumann). Theorem 1.6 proves that uµ ∈ H∩H_loc² (Ω)ⁿ satisfies (1.32) (with uµ replacing u therein) for any ϕ∈C_c^∞(Ω) with ϕ≥0. Apply Lemma 4.1 as in the proof of Theorem 1.6 and define u_ε as an approximation to u_µ (replace u in the lemma by u_µ) and recall the estimate (4.33) (with uµ replacingu). Since f ∈L²(Ω)ⁿ∩H_loc¹ (Ω)ⁿ, the first term in the right-hand side of (4.33) reads

− Z

Ω

f·div(ϕ⁴∇u_ε) = Z

Ω

ϕ⁴∇f :∇u_ε ≤ α

2kϕ²∇u_εk²_L2(Ω)^n×n+ 1

2αkϕ²∇fk²_L2(Ω)^n×n. (4.38) Rewrite the a priori bounds (4.30)-(4.31) as

kDu_µk_L1(Ω)^n×n+µk∇u_µk²_L2(Ω)^n×n ≤ kfk²_L2(Ω)ⁿ

α . (4.39)

Use (4.38)-(4.39) in (4.33) and follow the arguments in the proof of Theorem 1.6 to deduce α

2kϕ²∇u_εk²_L2(Ω)^n×n ≤ 1

2αkϕ²∇fk²_L2(Ω)^n×n+C_nϕ kϕ²∇u_εk_L2(Ω)ⁿ +

kfk²_L2(Ω)n α

!

. (4.40)

This andCnϕkϕ²∇u_εk_L2(Ω)^n×n ≤C_nϕ² /α+^α₄kϕ²∇u_εk²_L2(Ω)^n×n provide theε-independent bound ε^−1/2ku_µ−uεk_L2(Ω)+kϕ²∇u_εk_L2(Ω)^n×n ≤ Cnϕ

α 1 +kfk_L2(Ω)ⁿ+kϕ²∇fk_L2(Ω)^n×n

. (4.41) The first term in (4.41) stems from the first term in (4.32) and was simply suppressed in the above formulas. Given any fixed µ > 0, there exists a sequence ε→ 0 (not labelled explicitly) such that uε → uµ inL²(Ω)ⁿ and ϕ²∇u_ε ∈L²(Ω)^n×n is bounded by the µ-independent upper bound in (4.41). Recall ku_µk_L2(Ω)ⁿ ≤ kfk_L2(Ω)ⁿ/α from (4.31). Hence, there exists a sequence µ → 0 (not labelled explicitly) such that u_µ * u^∗ ∈L²(Ω)ⁿ converges weakly in L²(Ω)ⁿ with ku^∗k_L2(Ω)ⁿ ≤ kfk_L2(Ω)ⁿ/α andϕ²∇u^∗ ∈L²(Ω)^n×n bounded by the upper bound in (4.41).

It remains to prove that u^∗ =u solves (1.37). Recall thatu_µ∈H solves (1.31) and deduce the elementary boundµR

Ω∇u_µ: (∇v− ∇u_µ)≤ ^µ₄ R

Ω|∇v|² for any test functionv∈H. Recall that ψ is defined either by (1.35) (Dirichlet) or by (1.33) (Neumann), thatψ(v) =||Dv||_L1(Ω)^n×n for v∈H, and that α|| • ||_L2(Ω)ⁿ+ψ(•) is sequentially weakly lower semi-continuous. This and the above weak convergence lead in the limit as µ→0 in (1.31) to

Z

Ω

f ·(v−u^∗)≤α Z

Ω

u^∗·(v−u^∗) +ψ(v)−ψ(u^∗) forv∈H. (4.42) Lemma 4.2 below shows thatu^∗ =u and the uniqueness of the weak limit also implies uµ* u

inL²(Ω)ⁿ not only forsomesequence µ→0 but for any.

Lemma 4.2 Any function u^∗ ∈L²(Ω)with (4.42) is equal to the solution u to (1.37).

Proof. In the Dirichlet case, suppose v ∈ L²(Ω)ⁿ satisfies ψ(v) < ∞ with ψ defined by (1.35). A modification of Lemma A.1 for BD (rather than BV) functions shows that v can be approximated by a sequence vm ∈ H₀¹(Ω)ⁿ such that vm → v in L² and ψ(vm) → ψ(v) as m→ ∞.

In the Neumann case, suppose v ∈ L²(Ω)ⁿ satisfies ψ(v) < ∞ with ψ defined by (1.33). The existence of a sequencevm∈H¹(Ω)ⁿsuch thatvm→vinL²(Ω)ⁿandψ(vm)→ψ(v) asm→ ∞ follows from standard regularization [11].

Take the admissible test function v = v_m ∈ H in (4.42) for m and then pass to the limit as m → ∞ to deduce that (1.37) holds for the test function v ∈ L²(Ω)ⁿ. The uniqueness of the solutions to (1.37) concludes the proof. Note that the assumption ∂Ω Lipschitz applies only

here in each of the two cases.

Remark 4.3 The above passage to the limit as µ→0 and uµ* u in L²(Ω)ⁿ also shows

µ→0limµ Z

Ω

|∇u_µ|² = 0 and lim

µ→0ψ(uµ) =ψ(u).

(The proof compares the limit of (4.30) (written in terms ofuµ replacingu therein) with (4.36) and utilizes the sequential weak lower semi-continuity of ψ and ψ(u_µ) =kDu_µk_L1(Ω)^n×n.)

4.4 Proof of Theorem 1.8

In the first part of the proof of the characterization, suppose u solves (1.37). Theorem 1.7 asserts that Du ∈L²_loc(Ω)^n×n∩L¹(Ω)^n×n and that u is the weak limit in L²(Ω)ⁿ, uµ * u as µ → 0, of the solution u_µ ∈ H to the viscous problem (1.31). For any µ > 0, the solution u_µ ∈H∩H_loc¹ (Ω)ⁿ satisfies the bound (1.38) and is itself the weak limit in H as δ →0 of the solution u_µδ ∈H to the regularized problem

Z

is symmetric and satisfies|σ_µδ| ≤1 a.e. The pointwise boundedness leads to a sequence δ →0 (not labelled explicitly) such that σµδ *^∗ σµ weak star in L^∞(Ω)^n×n with |σ_µ| ≤ 1 a.e. The passage to the limit in (4.44) leads to

Z The same arguments apply in the passage to the limit asµ→0 and the weak convergenceuµ* u inL²(Ω)ⁿtowards the solution u to (1.37) and some weak star limitσ ofσ_µ inL^∞(Ω)^n×nwith no information on the boundary for test functions in H=H₀¹(Ω)ⁿ from (4.47).

The traces in BD are considered in [2] and the normal traces in H(div,Ω) are well known [17].

Forσ ∈H(div,Ω)ⁿ∩L^∞(Ω)^n×n_sym, the normal traceσν ∈H^−1/2(∂Ω) belongs toL^∞(∂Ω)ⁿ, as we now show. Since this result can be of independent interest, we state it as a separate lemma.

Lemma 4.4 (traces in BD) If σ ∈ H(div,Ω)ⁿ∩L^∞(Ω)^n×n_sym satisfies |σ| ≤ 1 a.e. in Ω, then the normal trace σν ∈H^−1/2(∂Ω)ⁿ satisfies σν ∈L^∞(∂Ω)ⁿ with|σν| ≤1 a.e. on the boundary

∂Ωand the integration by parts formula Z

Proof of the lemma. The first step of the proof recalls the tracevb ∈L¹(∂Ω)ⁿ of a function v∈L²(Ω)ⁿ withDv ∈L¹(Ω)^n×n_sym [2] that satisfies the integration by parts formula

Z

Ω

(v·div Φ + Φ :Dv) = Z

∂Ω

v_b·Φν for all Φ∈C¹(Rⁿ)^n×n_sym. (4.48) Since vb is unique, we simplify the notation and writev rather thanvb in the Lebesgue integral over the boundary ∂Ω.

Thesecond stepdesigns a sequence of smooth approximations Φ_j ∈C_c^∞(Rⁿ)^n×n_sym with (i)|Φ_j| ≤1 in Ω for allj∈N, (ii) its strong convergence Φj →σinH(div,Ω)ⁿ, (iii) its weak star convergence Φj *^∗ σ inL^∞(Ω)^n×n_sym, and (iv) Φj|_∂Ω *^∗ σν (weak star) in L^∞(∂Ω)^n×n_sym, asj→ ∞.

It is a standard argument in the approximation of Sobolev functions by mollification in the Lipschitz domain Ω [12, p. 128-129] to consider a suitable partition of unityPN

i=0ζi= 1. Then, forε >0 sufficiently small, one sets Φε:= (ζ0σ)∗ηε+PN

i=1σi∗ηε, where the functionηεis the mollification kernel, σ_i(x) := (ζ_iσ)(x+ετ_i) for all x ∈ Ω∩ω_i, and the translation vector τ_i is such that B(x, ε) +ετ_i ∈ Ω for all x ∈ Ω∩ω_i and all i∈ {1, . . . , N} with ζ_i supported in ω_i. Since ζi is scalar-valued, we have div(ζiσ) =σ∇ζ_i+ζi(divσ). The arguments in [12] then show that Φε ∈ C_c^∞(Rⁿ)^n×n with strong convergence to σ in L²(Ω)^n×n and strong convergence of divergence div Φ_ε →divσinL²(Ω)ⁿ, i.e., strong convergence inH(div,Ω)ⁿ. Moreover, Φ_ε takes symmetric values and max|Φ_ε| ≤1 +Cεfor some constant C, that depends only on the smooth and fixed functionsζ0, . . . , ζ_N in the partition of unity. The pointwise boundedness implies the weak star convergence of a subsequence Φ_εj.

The third stepsubstitutes Φ = Φεj for the sequence Φεj →σ inH(div,Ω)ⁿ from step two for a fixed v∈L²(Ω)ⁿ withDv ∈L¹(Ω)^n×n_sym in (4.48). The weak star convergence in L^∞(Ω)^n×n_sym and inL^∞(∂Ω)^n×n_sym leads to the asserted integration by parts formula in the passage to the limit as j→ ∞.

The fourth step concludes the proof: Given any σ := σ(x) ∈ R^n×nsym and ν = ν(x) ∈ Rⁿ with

|σ| ≤ 1 = |ν| (at almost every point x ∈ ∂Ω), the linear functional ϕ7→ ϕ·σν = σ : (ϕν) has a unique Riesz representation z ∈Rⁿ in the Hilbert space (Rⁿ,((•,•))) with respect to the scalar product

((a, b)) := (aν) : (bν) fora, b∈Rⁿ. (4.49) Then ((z, ϕ)) = σ : (ϕν) for all ϕ ∈ Rⁿ implies (zν)ν = σν and |zν|² = ((z, z)) = σ : (zν)≤ |σ| |zν| ≤ |zν|, whence |zν| ≤ 1. This pointwise definition ofσ(x)7→ z(x) at a.e. x∈∂Ω specifies a measurable function z∈L^∞(∂Ω)ⁿ. Let us resume the proof of Theorem 1.8. Lemma 4.4 applies to the solutionv=uand its trace ub ∈L¹(∂Ω). Substitute divσ =αu−f from (4.46) in the resulting identity and compare with (4.36). This proves

ψ(u) = Z

Ω

σ:Du−β Z

∂Ω

(zν) : (u_bν) (4.50) for β = 1 in the Dirichlet case and β = 0 in the Neumann case. (The integration by parts formula (4.47) shows that σν = 0 in the sense of traces in H^−1/2(∂Ω)ⁿ if H = H¹(Ω)ⁿ, but traces are involved in the Dirichlet case displayed forβ= 1.) The appendix reports on the traces of BV and BD functions and the weak definition of ψ for the Dirichlet (β = 1) and Neumann (β = 0) case. Lemma A.2 clarifies foru∈L¹(Ω)ⁿwith Du∈L¹(Ω)^n×nthat

ψ(u) = Z

Ω

|Du|+β Z

∂Ω

|u_bν|. (4.51)

The equality of (4.50)-(4.51) implies that σ :Du=|Du|a.e. in Ω and, if β = 1, that (zν) : (u_bν) =|u_bν|a.e. on∂Ω. Since|σ| ≤1 a.e. in Ω and|zν| ≤1 a.e. in∂Ω, the preceding identities imply that equality holds in the Cauchy–Schwarz inequality (pointwise a.e. in Ω).

Consequently, σ and zν are (pointwise a.e. in Ω) a non-negative multiple of Du and ubν, respectively. This leads to the final formulas and concludes the proof of (1.40).

The second part of the proof establishes the converse implication: Suppose u∈L²(Ω)ⁿsatisfies Du∈L¹(Ω)^n×nand (1.40). Lemma 4.4 applies tov=uand the arising integrals therein satisfy (from (1.40)) that

σ:Du=|Du| a.e. in Ω and σb: (ubν) =−|u_bν| a.e. on∂Ω (the second identity holds for β= 1 only). It follows (for β= 0 and β = 1 as above) that

− Z

Ω

u·divσ= Z

Ω

|Du|+β Z

∂Ω

|u_bν|.

The combination of this with divσ=αu−f ∈L²(Ω)ⁿand (4.51) proves thatu satisfies (4.36).

Given any v ∈H, an integration by parts with divσ = αu−f ∈L²(Ω)ⁿ does not involve any boundary terms (σν = 0 if β = 0 and H =H¹(Ω)ⁿ, while v ∈ H₀¹(Ω)ⁿ for β = 1). This and

|σ| ≤1 a.e and ψ(v) =kDvk_L1(Ω)^n×n forv∈H verify Z

Ω

(f −αu)·v= Z

Ω

σ :Dv≤ψ(v). (4.52)

The comparison of (4.36) and (4.52) proves thatu^∗ :=usatisfies (4.42). Thus Lemma 4.2 shows thatu solves (1.37) and concludes the proof of Theorem 1.8.

A Appendix

Dans le document ${\textit H}^1$ regularity of the inviscid total variation and Bingham minimisers for ${\textit H}^1$ data (Page 22-31)