Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

Adolfo Arroyo-Rabasa; Guido De Philippis; Filip Rindler

doi:10.1515/acv-2017-0003

Published by De Gruyter January 11, 2018

Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

Adolfo Arroyo-Rabasa , Guido De Philippis and Filip Rindler

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2017-0003

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Abstract

We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints.

Keywords: Lower semicontinuity; functional on measures; generalized Young measure

MSC 2010: 49J45; 35J50; 28B05; 49Q20; 74B05

Communicated by Frank Duzaar

Funding source: Engineering and Physical Sciences Research Council

Award Identifier / Grant number: EP/L018934/1

Funding statement: Adolfo Arroyo-Rabasa is supported by a scholarship from the Hausdorff Center of Mathematics and the University of Bonn; the research conducted in this paper forms part of his Ph.D. thesis at the University of Bonn. Guido De Philippis is supported by the MIUR SIR-grant “Geometric Variational Problems” (RBSI14RVEZ). Filip Rindler acknowledges the support from an EPSRC Research Fellowship on “Singularities in Nonlinear PDEs” (EP/L018934/1).

A Proofs of the localization principles

In this appendix we prove Proposition 24 and Proposition 25.

Proof of Proposition 24.

In the following we adapt the main steps in proof of the localization principle at regular points which is contained in [29, Proposition 1]. The statement on the existence of an 𝒜-free and periodic generating sequence is proved in detail.

Let μj∈ℳ⁢(Ω;ℝN) be the sequence of asymptotically 𝒜-free measures which generates ν. In the following steps, for an open Ω′⊂ℝd, we will often identify a measure μ∈ℳ⁢(Ω′;ℝN) with its zero extension in ℳloc⁢(ℝd;ℝN), and similarly for a Young measure σ∈𝐘⁡(Ω′;ℝN) and its zero extension in 𝐘loc⁡(ℝd;ℝN).

Step 1. We start by showing that, for every r>0, there exists a subsequence of j’s (the choice of subsequence might depend on r) such that

r-d⁢T#(x0,r)⁢μj⁢→𝐘⁢σ(r) in 𝐘loc⁡(ℝd;ℝN).

Moreover, for ℒd-a.e. x0∈Ω, one can show that a uniform bound

supr>0〈〈𝟙K⊗|⋅|,σ(r)〉〉<∞ for every K⋐ℝd

holds; thus, by Lemma 5, there exists a sequence of positive numbers rm↓0 and a Young measure σ for which

σ(rm)⁢⇀*⁢σ in 𝐘loc⁡(ℝd;ℝN).

Step 2. For an arbitrary measure γ∈ℳ⁢(Ω;ℝN), the Radon–Nykodým differentiation theorem yields

r-dT#(x0,r)γ=d⁢γd⁢ℒd(x0+r⋅)ℒd+d⁢γd⁢|γs|(x0+r⋅)r-dT#(x0,r)|γs|.

Consider σ(r) as an element of 𝐘⁡(Q;ℝN). Fix φ⊗h∈C⁢(Q¯)×Lip⁡(ℝN). Using a simple change of variables, we get

〈〈φ⊗h,σ(r)〉〉=limj→∞⁡(∫Qφ⁢(y)⋅h⁢(d⁢μjd⁢ℒd⁢(x0+r⁢y))⁢dy+∫Q¯φ⁢(y)⋅h∞⁢(d⁢μjd⁢|μjs|⁢(x0+r⁢y))⁢d⁢(r-d⁢T#(x0,r)⁢|μjs|)⁢(y))

=r-d⁢limj→∞⁡(∫Qr⁢(x0)φ∘T(x0,r)⁢(x)⋅h⁢(d⁢μjd⁢ℒd⁢(x))⁢dx+∫Qr⁢(x0)¯φ∘T(x0,r)⁢(x)⋅h∞⁢(d⁢μjd⁢|μjs|⁢x)⁢d⁢|μjs|⁢(x))

(A.1)=r-d⁢〈〈(φ∘T(x0,r))⊗h,ν〉〉.

Step 3. We now let r=rm in (A.1) and quantify its values as m→∞. This will allow us to characterize σ in terms of ν. Let {gl:=φl⊗hl}l⊂C(Q¯)×Lip(ℝN) be the dense subset of 𝐄⁡(Q;ℝN) provided by Lemma 4 and further assume that x0 verifies the following properties: x0 is a Lebesgue point of the functions

(A.2)x↦〈hl,νx〉+〈hl∞,νx∞〉⁢d⁢λνd⁢ℒd⁢(x) for all l∈ℕ,

and x0 is a regular point of the measure λν, that is,

(A.3)d⁢λνsd⁢ℒd⁢(x0)=limr↓0⁡λνs⁢(Qr⁢(x0))rd=0.

Consider σ as an element of 𝐘⁡(Q;ℝN). Setting r=rm in (A.1) and letting m→∞, we get

〈〈gl,σ〉〉=limm→∞⁡rm-d⁢〈〈φl∘T(x0,rm)⊗hl,ν〉〉

=limm→∞(⨍Qrm⁢(x0)φl(x-x0rm)[〈hl,νx〉+〈hl∞,νx∞〉d⁢λνd⁢ℒd(x)]dx

+1rd∫Qrm⁢(x0)¯φl(x-x0rm)〈hl∞,νx∞〉dλνs(x))

=∫Q〈gl⁢(y,⋅),νx0〉⁢dy+∫Q〈gl∞⁢(y,⋅),νx0∞〉⁢d⁢λνd⁢ℒd⁢(x0)⁢dy.

Here, we have used (A.2) and the dominated convergence theorem to pass to the limit in the first summand, and with the help of (A.3), we used that

∫Qr⁢(x0)¯φl⁢(x-x0r)⁢〈hl∞,νx∞〉⁢dλνs⁢(x)≤∥φ∥∞⋅Lip⁡(hl)⋅λνs⁢(Qr⁢(x0))=o⁢(rd)

to neglect the second summand in the limiting process. Since the set {gl} separates 𝐘⁡(Q;ℝN), Lemma 4 tells us that σy=νx0,σy∞=νx0∞,λσ=d⁢λνd⁢ℒd⁢(x0)⁢ℒd for ℒda-e. y∈Q, and that λσs is the zero measure in ℳ⁢(Q¯), as desired.

Step 4. We use a diagonalization principle (where j is the fast index with respect to m) to find a subsequence (μj⁢(m)) such that

γm:=rm-dT#(x0,rm)μj⁢(m)→𝐘σ in 𝐘loc⁡(ℝd;ℝN).

Step 5. Up to this point, the localization principle presented in [29, Proposition 1] has been adapted to Young measures without imposing any differential constraint. Here we additionally require σ to be an 𝒜k-free Young measure; this is achieved by showing that (γm) is asymptotically 𝒜k-free (on bounded subsets of ℝd). To this end, let us note that

𝒜⁢μj=θj

with ∥θj∥W-k,q→0 as j→∞. By scaling we can write

𝒜k⁢γm=rm-d⁢𝒜k⁢(T#x0,rm⁢μj⁢(m))=-∑h=0k-1𝒜h⁢(rk-h⁢rm-d⁢T#(x0,r)⁢μj⁢(m))+rmk-d⁢T#(x0,rm)⁢θj⁢(m).

Since ∥rmk-d⁢T#(x0,rm)⁢θj∥W-k,q≤C⁢(m)⁢∥θj∥W-k,q and for every open U⋐ℝd there exists a positive constant CU such that

supm∈ℕ⁡rm-d⁢T#(x0,rm)⁢|μj|⁢(U)≤CU,

arguing as in Proposition 16 we can choose a further subsequence j⁢(m)→∞ such that

𝒜k⁢γm=𝒜k⁢(rm-d⁢T#(x0,r)⁢μj⁢(m))→0 in Wloc-k,q⁢(ℝd)

and this shows that σ is an 𝒜k-free Young measure.

Step 6. So far we have shown that [σ]=A0⁢ℒd with

A0:=〈id,νx0〉+〈id,νx0∞〉⁢d⁢λνd⁢ℒd⁢(x0)∈ℝN,

and that σ is generated by a sequence (μj)⊂ℳ⁢(Q;ℝN) satisfying 𝒜k⁢μj→0. Note that without loss of generality we may assume that the measures μj are of the form uj⁢ℒd, where uj∈L1⁢(Q;ℝN). Indeed, since

γR:=T#(x0,R)μj→μj area strictly in ℳloc⁢(ℝd;ℝN),∥𝒜k(γR-μj)∥Wloc-k,q⁢(ℝd)→0 as R↑1,

and

γR∗ρε→γR area strictly in ℳloc⁢(ℝd;ℝN),∥𝒜k⁢(γR-γR∗ρε)∥Wloc-k,q⁢(ℝd)→0 as ε↓0,

we might use a diagonalization argument (relying on the weak*-metrizability of bounded subsets of 𝐄(Q;ℝN)* and Remarks 2 and 6), where ε appears as the faster index with respect to R, to find a sequence with elements uj:=γR⁢(j)∗ρε⁢(R⁢(j)) such that

uj⁢ℒd⁢→𝐘⁢σ∈𝐘loc⁡(ℝd;ℝN) and 𝒜k⁢uj→0 in Wloc-k,q⁢(ℝd).

Using (2.3), we get

|uj|⁢ℒd⁢⇀*⁢|[σ]|=|A0|⁢ℒd in ℳloc⁢(ℝd).

Hence, |uj|⁢ℒd⁢⇀*⁢Λ in ℳ⁢(Q¯) with Λ⁢(∂⁡Q)=0. We are now in a position to apply Lemma 15 to the sequences (uj) and (vj:=A0) to find a sequence zj∈Cper∞⁢(Q;ℝN)∩ker⁡𝒜k with ∫Qzj⁢dy=0 and such that (up to taking a subsequence)

(A.4)(A0+zj)⁢ℒd⁢→𝐘⁢σ in 𝐘⁡(Q;ℝN).

Since the properties of x0 that were involved in Steps 1–3 are valid at ℒd-a.e. x0∈Ω, the sought localization principle at regular points is proved. ∎

Proof of Proposition 25.

The proof of the localization principle at singular points resembles the one for regular points, with a few exceptions.

Step 1. In comparison to Step 1 from the regular localization principle, we here choose

cr(x0):=|λνs|(Qr(x0))-1>0

and we define σr as

cr⁢(x0)⁢T#(x0,r)⁢μj⁢→𝐘⁢σ(r) in 𝐘loc⁡(ℝd;ℝN).

Moreover, by [27, Lemma 2.4 and Theorem 2.5] and (A.7) below, at λνs-a.e. x0∈Ω, it is possible to show that

supr>0〈〈𝟙K⊗|⋅|,σ(r)〉〉=supr>0|λνs|⁢(x0+r⁢K)+ℒd⁢(x0+r⁢K)|λνs|⁢(Qr⁢(x0))<∞ for every K⋐ℝd.

By the compactness of 𝐘loc⁡(ℝd;ℝN), see Lemma 5, there exists a sequence of positive numbers rm↓0 and a Young measure σ for which

σ(rm)⁢⇀*⁢σ in 𝐘loc⁡(ℝd;ℝN).

Step 2. The calculations of the second step, for the constant cr⁢(x0), is

〈〈φ⊗h,σ(r)〉〉=limj→∞(∫Qφ(y)⋅h(cr(x0)rdd⁢μjd⁢ℒd(x0+ry))dy

+∫Q¯φ(y)⋅h∞(cr(x0)rdd⁢μjd⁢|μjs|(x0+ry))d(r-dT#(x0,r)|μjs|)(y))

=r-dlimj→∞(∫Qr⁢(x0)φ∘T(x0,r)(x)⋅h(cr(x0)rdd⁢μjd⁢ℒd(x))dx

+∫Qr⁢(x0)¯φ∘T(x0,r)(x)⋅h∞(cr(x0)rdd⁢μjd⁢|μjs|(x))d|μjs|(x))

(A.5)=r-d〈〈φ∘T(x0,r)⊗h(cr(x0)rd⋅),ν〉〉.

Step 3. The assumptions of the third step are substituted by assuming that x0 is a λνs-Lebesgue point of the functions

(A.6)x↦〈|⋅|,νx∞〉,x↦〈hl∞,νx∞〉 for all l∈ℕ.

We further require that

(A.7)limr↓0⁡rdλνs⁢(Qr⁢(x0))=limr↓0⁡cr⁢(x0)⁢rd=0

and that

(A.8)limr↓0cr(x0)∫Qr⁢(x0)[〈|⋅|,νx〉+〈|⋅|,νx∞〉d⁢λνd⁢ℒ(x)]dx=0.

Hence, defining S:={x0∈Ω:(A.6), (A.7) and (A.8) hold}, we have λνs⁢(Ω∖S)=0. Fix x0∈S. Setting r=rm in (A) and letting m→∞ gives

〈〈𝟙Q⊗|⋅|,σ〉〉=limm→∞〈〈𝟙Q⊗|⋅|,σ(rm)〉〉

=limm→∞cm(x0)(∫Qrm⁢(x0)[〈|⋅|,νx〉+〈|⋅|,νx∞〉d⁢λνd⁢ℒ(x)]dx+∫Qrm⁢(x0)¯〈|⋅|,νx∞〉dλνs(x))

=〈|⋅|,νx0∞〉limm→∞(∫Q¯d(cm(x0)T#(x0,rm)λνs)(y))

=∫Q¯〈|⋅|,νx0∞〉dγ(y)

for some γ∈Tan⁡(λνs,x0), where we have used that x0∈S. Moreover,

γ(Q¯)=〈〈𝟙Q¯⊗|⋅|,σ〉〉≥limm→∞|λνs|⁢(Qrm⁢(x0))|λνs|⁢(Qrm⁢(x0))=1,

which implies γ≠0. Testing with gl=φl⊗hl, we obtain by (A.6) and a similar argument to the one above, that

〈〈gl,σ〉〉=∫Q¯φl⁢(y)⁢〈hl∞,νx0∞〉⁢dγ⁢(y).

From the above equations we deduce that σy=δ0 for ℒd-a.e. y∈Q, σy∞=νx0∞ and λσ=γ∈Tan⁡(λνs,x0)∖{0}.

Step 4. The arguments of Step 4 remain unchanged except that this time one gets

γm:=cmT#(x0,rm)μj⁢(m)→𝐘σ in 𝐘⁡(Q,ℝN).

Step 5. This is similar to the corresponding step in the proof of the regular localization principle.

Step 6. Differently from the case at regular points, we want to additionally show λσ⁢(Q)=1 and λσ⁢(∂⁡Q)=0. There exists 0<ε<1 such that λσ⁢(∂⁡Qε)=0. Up to taking r′=ε⁢r (and thus as rm′=rm⁢ε) in the arguments of Steps 1–4 above, we may assume without loss of generality that λσ⁢(∂⁡Q)=0 and λσ⁢(Q)=1. This proves the localization principle at singular points. ∎

Acknowledgements

The authors would like to thank the anonymous referee for her/his careful reading of the manuscript which led to a substantial improvement of the presentation.

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Received: 2017-01-27

Revised: 2017-10-04

Accepted: 2017-12-20

Published Online: 2018-01-11

Published in Print: 2020-07-01

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Frontmatter

Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

The Cauchy problem for the Finsler heat equation

Regularity for scalar integrals without structure conditions

The Nitsche phenomenon for weighted Dirichlet energy