Partial regularity for BV^B local minimizers

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DEPARTMENT OF MATHEMATICS Master’s Degree in Mathematics

Partial regularity for BV

B

local minimizers

Candidate:

Federico Franceschini

Supervisor:

Prof. Luigi Ambrosio

Supervisor:

Prof. Jan Kristensen

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Introduction

Let us start describing the set-up we have in mind. The starting point is the following variational problem inf F [u] = ˆ Ω f (Bu(x)) dx : u ∈ W1,1(Rn, Rm), u ≡ g outside Ω , (P)

where the various ingredients are:

• Ω ⊂ Rnis a bounded open set with Ln(∂Ω) = 0;

• B is a partial differential operator of the first order with constant coefficients over Rn from

Rmto RN, for some N ∈ N, which is also elliptic;

• the lagrangian f : RN → R is regular enough, has linear growth and is strongly B-quasiconvex, a condition which we will introduce and that is almost minimal in order to ensure both lower semicontinuity and coercivity of F ;

• g ∈ W1,1(Rn, Rm) is fixed and should be thought as of the "boundary datum" of our problem. Furthermore we extend F to the whole L1(Rn, Rm) setting it equal to +∞. This is the natural generalization of the more classical problem

inf F [u] = ˆ Ω f (∇u(x)) dx : u ∈ W1,1(Rn, Rm), u ≡ g outside Ω ,

when we substitute the full gradient ∇ with B, which is just a linear combination of the derivatives. In this case the ∇-quasiconvexity is just Morrey’s standard quasiconvexity.

The direct method does not apply immediately to (P) because of the bad compactness properties of

W1,1, thus we need to relax the problem with respect to the natural notion of convergence, producing the "abstract" relaxed functional F . The definition in this case is

F [v] := inf

lim inf

k F [vk] : vk → v in L

1_(Rn₎_,

the key point being that F is L1-lower semicontinuous and finite on a larger domain than W1,1: in the classical case the space BV of maps of bounded variation shows up in this way, not surprisingly in this framework we are led to consider the space

BVB(Rn) :=nu ∈ L1(Rn, Rm) :Bu ∈ M(Rn, RN)o.

Using many nontrivial results (for example [19],[29],[5],[41]) we were able to prove an explicit local minimality condition for any u ∈ BVB minimizer of F (whose existence is much easier). This condition formally looks like

ˆ

Ω

f (Bu) ≤

ˆ

Ω

f (Bu + Bϕ) for every ϕ ∈ C_c∞(Ω, Rm), _(min) but notice that some clarification is needed since Bu is just a measure, we do not enter in details here. In the case B = ∇ Jan Kristensen and Franz Gmeineder recently showed (in [15]) that (min) entails an ε-regularity statement for u, the remarkable aspect being that only strong quasiconvexity was assumed on f . Adapting closely their argument and adding an assumption on f and B (the canceling condition 1.4.6), in this thesis we showed that the same result holds in the more general case of minimizers of F . Even if the argument is almost the same (see also [14] for the B = ∇ + ∇T case), the result is new and the adaptation involved some technical difficulties. In particular the new Weak

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Poincaré’s inequality established in Theorem 2.2.4 was crucial to make the argument work. With this picture in mind, let us briefly describe how the thesis is organized.

In the first Chapter we collect many technical tools and fix the notation. We start with the language of Generalized Young Measures, which is ratehr effective in the description of oscillation phenomena, it will allow us to understand the lower semicontinuity result Theorem 2.4.6. Then we recall what Sobolev-Slobodeckij spaces are and some technical properties concerning them, namely Gagliardo’s trace theorem and the Fubini property. Subsequently we fix the notation concerning partial differential operators and state a very deep result about solubility of overdetermined systems of PDEs closely related to "Ehrenpreis’ Fundamental Principle". Finally we state some elliptic regularity results which are somehow standard for the experts, even if an exact reference is not easy to find. Proofs are mostly omitted but careful references are given, some exceptions were made when very elegant arguments where available or no satisfactory references were found.

In the second Chapter we investigate the space BVB, whose existence was motivated before. We first investigate which properties stems from the ellipticity of B and then which further properties comes from the canceling assumption. We point out the Weak Poincaré’s inequality which is estab-lished by the means of the Open mapping Theorem in Fréchet spaces and the aforementioned Ehren-preis fundamental principle. Subsequently we define and motivate the notion of B-quasiconvexity, and show how it entails weak∗sequential lower semicontinuity of Bu 7→´ f (Bu). Incidentally, we shall make use of two recent results: the first is De Philippis and Rindler’s impressive generalization of Alberti’s "Rank one Theorem" (see [29]) and the second is Kircheim and Kristensen’s observation about convexity of homogeneous rank-one convex functions. Using the latter we will also finally show that in general BV $ BVB, fact that is known as "Ornstein non-inequality". In this chapter we provided full proofs only in the first part, in the second one we described more informally the methods of proof, in this way one can easily understand the line of reasoning and complete the rigorous (and heavier) arguments reading the original papers, which are very well written.

In the third and last chapter we attack problem (P). First we inspect the properties of the relaxed functional F and prove that recovery sequences converges in a stronger fashion in the interior of Ω. Unfortunately, because of the unclear boundary behavior, we will not be able to provide an explicit for-mula for F , differently from the B = ∇ case where this is possible (see [4]). Exploiting this analysis we will be able to prove that minimizers of F have the local minimality condition (min). In the last part we will prove the ε-regularity statement Theorem 3.4.3, the key trick here (devised in [15]) is a careful harmonic replacement, which is possible on suitable chosen balls. In this chapter full proofs are given. Let us conclude mentioning that the importance of the canceling hypothesis on B, at a closer look, seems to be null. We are nowadays working on a generalization of the result assuming only ellipticity.

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Notations

(i) We shall work in Rnwith the standard euclidean structure and identify Rn0= Rntrough this euclidean structure, furthermore we will always suppose n ≥ 2. The scalar product between two vectors x and y will be denoted with the dot: x · y, the action of a linear map A on a vector

x will be denoted with the lower dot A.x. We shall always identify linear maps with tensors in

the following way n

A : Rn→ RN, A linearo' RN ⊗ (Rn)0 ' RN⊗ Rn, (briefly: range ⊗ domain). On tensor products of spaces, such as RN ⊗ Rn, we will use the Euclidean structure induced by the factors, in this way the norm of a linear map A : Rn→ RN will be the Hilbert-Schmidt norm and not the smaller operator norm.

(ii) We shall systematically identify measures with dual vectors. In particular the space M(Ω, RN) of RN-valued, Borel measures on Ω with finite mass will be identified with

M(Ω, RN_{) ' C}

0(Ω, RN)∗, and similarly Mloc(Ω, RN) ' Cc(Ω, RN)∗

will denote the space of RN-valued Radon measures. The Lebesgue measure on Rn will be denoted by Lnbut sometimes we shall employ the lighter notation |E| = Ln(E) where

E ⊂ Rn is a Lebesgue-measurable set. We denote with Pk(Rn) the space of probability

measures on Rn that have finite kthmoment. We denote with Bn the open unit ball of Rn, that has measure ωn=Ln(Bn).

(iii) We denote with the angular bracket hµ, f i the duality between finite measures µ and a continuous functions f , generically both vector-valued.

(iv) If u : Rn→ R is a measurable function we denote its norm in some function space X either with kukX or with ku|X k, for the sake of readibility. We shall not identify maps that agree Ln

-almost everywhere.

(v) We shall use freely the language of, generally vector-valued, distributions and the basic theorems concerning them. When Ω ⊂ Rn is some open set D (Ω, RN) will denote the space of test functions and D0(Ω, RN) will denote its dual, the space of distributions. On the whole space we have the Schwartz functions S (Rn, RN) and its dual S0(Rn, RN), the space of tempered distributions. Given u ∈ S (Rn, RN) we define its Fourier Transform ˆu ∈S (Rξn, RN) by

the pointwise formula:

F u(ξ) = ˆu(ξ) = ˆ

Rn

u(x) e−ix·ξdx.

(vi) If A, B are sets then A ⊂ B means that every element of A is an element of B, so it’s always true that A ⊂ A. If f and g are quantities then we shall often use the notation f .a,b,cg, this

means that there is a positive constant C that depends only on the variables a, b, c such that

f ≤ C g. If we have f .a,b,cg and g .a,b,cthen we will write f ∼a,b,cg. In a slightly different

way, we will write f ≈a,b,c g if there is a positive constant C depending only on a, b, c such

that f = C g.

(vii) When f : Rn→ RN is a vector-valued function and k ≥ 1 we will denote with ∇kf : Rn→

RN ⊗k_Rn

the tensor field of its derivatives of order k. Analogous notation is employed when f is a distribution.

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Chapter No. 1

Preliminaries

In this first chapter we collect various analytical tools, it should not be completely skipped because some non standard notation is also developed.

We start introducing the language of Generalised Young Measures in the p = 1 set-up and we state the Reshetnyak continuity theorem in this framework. Then we briefly introduce the Sobolev-Slobodeckij spaces, recall Gagliardo’s trace theorem and state a Fubini-type property. Subsequently, we fix the notation for linear partial differential operators with constant coefficients and state a deep result about solubility of overdetermined systems. Finally, we give some Lp regularity results for systems of elliptic PDEs with constant coefficients.

1.1. Generalised Young Measures

The matter exposed in this section is a condensate of the results thoroughly proved in Section 3 of Kristensen and Rindler’s [20], see also Chapter 12 in the recent monograph [34]. We refer to Ambrosio, Fusco and Pallara [3] for more or less standard background in measure theory.

Throughout this section we fix an Ω ⊂ Rnopen and bounded, such that Ln(∂Ω) = 0, and some integer N ≥ 1. We start introducing the class E1(Ω, RN) of regular integrands:

Definition 1.1.1. A continuous function f : Ω × RN → R belongs to E1(Ω, RN) if:

lim

t→+∞

f (x0, tz0)

t =: f

∞

(x, z) exists in R, locally uniformly in x ∈ Ω, z ∈ RN. The function f∞(·, ·) is called the"strong recession function".

Let us clarify this definition giving a geometric interpretation which also gives quickly many properties of E1(Ω, RN). Recall that BN is the open unit ball in RN.

Proposition 1.1.2. Given a continuous function f : Ω × RN → R define T f : Ω × BN _{→ R trough}

the formula T f (x, ζ) := f x, ζ 1 − |ζ| (1 − |ζ|), x ∈ Ω, ζ ∈ BN.

Then f belongs to E₁(Ω, RN) if and only if T f extends by continuity to the closure of Ω × BN (i.e. is uniformly continuous).

The map T : E1(Ω, RN) → C

Ω × BNthat takes f to T f is thus a bijection and induces naturally a Banach space structure over E1. One can check that the induced norm is

kf k_E1 = sup

x∈Ω,z∈RN

|f (x, z)| 1 + |z| .

An easy consequence of this observation is that f∞: Ω ×RN → R must be continuous and positively one-homogeneous in its second argument.

Before going to explain what Young Measures are, let us stop to remark a key continuity property of these regular integrands.

Fix f ∈ E1and any RN-valued measure µ (which can be thought of an element of C(Ω, RN)∗). We take the (unique) decomposition of µ with respect to the Lebesgue measure µ = µac(x)Ln Ω + µs and further decompose the singular part µs in terms of its own total variation µs = dµ

s d|µs_|(x) |µs|. Then we defineˆ Ω f (x, µ) := ˆ Ω f (x, µac(x)) dx + ˆ Ω f∞ x, dµ s d|µs_|(x) d|µs|(x),

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and, more generally, for every Radon measure µ ∈ C0(Ω, RN)∗and Borel set A b Ω we define: ˆ A f (x, µ) := ˆ A f (x, µac(x)) dx + ˆ A f∞ x, dµ s d|µs_|(x) d|µs|(x), _(1.1.1)

we remark that the left-hand side of this equation is just a symbol, f (x, µ) not being a function.

Remark 1.1.3. If µs τ for some other measure τ then thanks to positive homogeneity of f∞(x, ·) we have (see Proposition 2.37 in [3])

ˆ Ω f∞ x, dµ s d|µs_|(x) d|µs|(x) = ˆ Ω f∞ x,dµ s dτ (x) dτ (x).

When µ is absolutely continuous with respect to Lnthis definition says nothing new, but why does our formula (1.1.1) give the "correct" extension to general measures? The answer is that (1.1.1) can be obtained as extension by continuity of the following functional

µ(x)Ln Ω 7−→

ˆ

Ω

f (x, µ(x)) dx

with respect to a suitable notion of convergence on the space of measures C(Ω, RN)∗. In order to explain it we need to define the area function E : RN → R+by

E(z) :=

q

1 + |z|2_{− 1 = hzi − 1,} _(1.1.2)

it is easily checked that E belongs to E1(Ω, RN) for every Ω and it has the nice property of being

strictly convex. Furthermore, simple computations shows that ˆ

Ω

E(µ) = |˜µ|(Ω) where ˜µ := (µ,Ln Ω) ∈ C(Ω, RN × R)∗.

Definition 1.1.4 (Area-strict convergence). Given µ and {µj}j∈Nelements of C0(Ω, RN)∗ we say

that {µj}j∈Nconverges "area-strictly" to µ and write

µj + µE as j → +∞, provided µ_j * µ in C∗ c(Ω, RN)∗and ˆ Ω E(µj) → ˆ Ω E(µ) as j → +∞.

As anticipated the main interest in this definition is that it ensures continuity of the functional

µ 7−→´_Ωf (x, µ) whenever f ∈ E1(Ω, RN):

Theorem 1.1.5 (Reshetnyak’s Continuity in the spirit of Kristensen and Rindler[21]). For every f ∈ E1(Ω, RN) we have lim j→+∞ ˆ Ω f (x, µj) = ˆ Ω f (x, µ) provided µj E + µ in C0(Ω, RN)∗.

Idea of the Proof. This Theorem follows directly from the classical Reshetnyak’s Continuity Theorem

(Theorem 2.39 in [3]) combined with a "perspective trick". This consists in adding one dimension to the all the measures, i.e. working with ˜µ := (µ,Ln Ω) instead of µ. With respect to these new variables we have

µj + µE if and only if µ˜j

|·|

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1.1. Generalised Young Measures

And also´_Ωf (x, µ) =´_ΩF (x, ˜µ) where F : Ω × Rn× R → R is the (continuous and positively 1 homogeneous in its second variable) map

F (x, z, t) :=

(

|t| f (x, z/|t|) _{if t 6= 0,}

f∞(x, z) _{if t = 0.}

The details of this argument can be found in the appendix of [21].

Remark. If we know something about the limit measure µ, we can relax the assumptions on f . In

fact, what is really needed in the proof is that the perspective integrand F has a ˜µ-neglegible set of discontinuity points (see Proposition 1.62, (b) in [3]).

As a simple application of this result one can see that the area-strict convergence implies the strict convergence.

We now turn to the definition of Generalised Young Measures, here’s some of the general intuition behind this subject in the more familiar case p = 2. Instead of a sequence of measures {µj}, we

simply consider a bounded sequence {Vj} ⊂ L2(−1, 1), by general compactness result we may also

assume that Vj * V in L2. We know that there is no hope, in general, to enhance this convergence to strong L2, but why exactly? If one looks at standard examples there are just two pints where things can go bad:

(i) Vj oscillates, so Vj 9 V , say, in measure;

(ii) kVjkL2 _{9 kV k}_L2_{, this means, by Fatou, that there is a "loss of energy".}

Conversely if both the previous does not happen we can say that Vj → V strongly in L2. The aim of Generalised Young Measures is to quantify separatedly this two phenomena, in the following way. Suppose that the sequence {Vj} oscillates and look, from far away, at the sequence of their graphs.

The picture will start to blur "vertically" in some parts of (−1, 1) as the parameter j increase, because of the increasing oscillations, this blur having different intensities on the top of different x’s. If we fix an x and we look in the limit j → ∞, we can encode the information of the intensity of the blurred line that sits vertically on that x, in a probability measure νx on R. The family of probabilities (νx)x∈(−1,1)tells us a lot about the oscillation properties of {Vj}.

But this is half the work. What happens in the other case, when the family Vj loses energy? Again, this can be understood at the level of graphs, think Vjto be some properly rescaled centered Gaussians whose variance is shrinking, but L2norm is constant, but weakly converge to V = 0. As distribution theory suggests, it is intuitive that the "record of the loss mass" can be "saved" by placing a δ0in the origin, weighted with some number. In general, we may need a much wilder measure λ to remember the mass that escaped to ∞, and we may also need to keep track to the "direction" that it took when leaving (also in the 1-dimensional case, the mass can split and go 30% toward +∞ and 70% toward −∞). This additional information will be stored in the "concentration-angle measure".

One could reasonably suspect that these two phenomena actually splits: what if I carry mass to ∞ while oscillating madly? It’s hard to say. But if we look at {f (Vj)}jwhere f (z) behaves nicely at ∞ (for example if f is a regular integrand!) then we know what happens. This is because f tends to be radially constant (in a uniform fashion) when we are close to z = ∞, so "radial" oscillations of Vj, far form the origin z = 0, are not seen when we compose everything with f . This caveat is the only (heavy) restriction to our treatment.

With this picture in mind the following definitions and theorems, even if rather sophisticated to prove, should sound very plausible.

Definition 1.1.6 (Generalised Young Measures). The set of RN-valued Generalised Young Measures over Ω is denoted by Y (Ω, RN) and consists of all the triples

ν =(νx)x∈Ω, λν , (νx∞)x∈Ω

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(i) (Oscillation Measure) ν_xbelongs toP₁(RN) forLn-a.e. x ∈ Ω and for every φ ∈ C_c(RN)

the function Ω 3 x 7→ hνx, φi isLn-measurable. Furthermore we require

ˆ

Ω

hν_x, | · |i dx < +∞;

(ii) (Concentration measure) λνis a finite positive Borel measure on Ω;

(iii) (Concentration-Angle measure) ν_x∞ belongs to P(SN −1) for λν-a.e. x ∈ Ω and for every

φ ∈ C(SN −1) the function Ω 3 x 7→ hν_x∞, φi is λν-measurable.

We define the following duality pairing that lets us embed naturally Y (Ω, RN) ,→ E1(Ω, RN)∗:

hhν, f ii := ˆ Ω hν_x, f (x, ·)i dx + ˆ Ω

hν_x∞, f∞(x, ·)i dλ_ν(x) _{for every f ∈ E}₁(Ω, RN).

Playing with the identification between E1(Ω, RN) and C(Ω × B

N

) and with the Disintegration Theorem (Theorem 2.28 in [3]) one can prove the following useful formula:

Proposition 1.1.7 (Lemma 2 and Corollary 1 in [20]). The set (T−1)∗hY (Ω, RN)iconsist exactly of those positive measures τ ∈ CΩ × BN∗such that

ˆ Ω×BN φ(x) (1 − |z|) dτ (x, z) = ˆ Ω φ(x) dx for all φ ∈ C(Ω); here T : E1(Ω, RN) → C

Ω × BNis the previously defined isomorphism. In particular the set Y (Ω, RN) is weakly∗closed in E1(Ω, RN)∗.

This technical result allows us to extract information and in particular gives easily the following "Fundamental Theorem" of Young Measures:

Theorem 1.1.8 (Theorem 7 in [20]). Given a family of measures {µj}j∈Nin C(Ω, RN)∗such that

sup_j|µj|(Ω) < +∞, there exist a subsequence (not relabeled) and a Young measure ν ∈ Y (Ω, RN)

such that for every f ∈ E1(Ω, RN) we have

lim j ˆ Ω f (x, µj) = hhν, f ii = ˆ Ω hνx, f (x, ·)i dx + ˆ Ω hν_x∞, f∞(x, ·)i dλν(x).

Idea of the proof. Let us first see that C(Ω, RN)∗_{embeds isometrically in Y (Ω, R}N_{) via the} iden-tification ι : µ 7−→ ιµ = δ_µac_(x) x∈Ω, |µ s_|,_δ dµs d|µs|(x) x∈Ω ,

and one easily checks that hhιµ, f ii =

ˆ

Ω

f (x, µ) for every integrand f ∈ E1(Ω, RN).

Then one applies Banach Alaoglu in E1(Ω, RN)∗ and use the weak∗closedness of Y (Ω, RN). Let us remark with this language Theorem 1.1.5 reads as:

µj + µ in ΩE ⇐⇒ µjgenerates the Young measure δµac_(x) x∈Ω, |µ s_|,_δ dµs d|µs|(x) x∈Ω ,

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1.1. Generalised Young Measures

Definition 1.1.9 (Barycenter of a Young Measure). Given ν ∈ Y (Ω, RN) we define the barycenter [ν] ∈ C(Ω, RN₎∗_{as the measure}

[ν] := [νx]Ln Ω + [νx∞]λν,

where for every x ∈ Ω we set

[νx] := ˆ RN y dνx(y), [νx∞] := ˆ SN −1 y dν_x∞(y).

It is easy to check that if {µ_j} generates ν then µj

∗

* [ν] in C(Ω, Y )∗.

We derive a simple lower semicontinuity result for convex regular integrands

Proposition 1.1.10. Let f ∈ E1(Ω, RN) such that f (x, ·) is a convex function for every x ∈ Ω, then

we have

f (x, z + z0) ≤ f (x, z) + f∞(x, z0) for every x ∈ Ω, and z, z0 ∈ RN. (1.1.3)

Furthermore if µj ∗ * µ in C(Ω, Rn)∗then lim inf j ˆ Ω f (x, µj) ≥ ˆ Ω f (x, µ). (1.1.4)

Proof. The first inequality stems from the monotonicity of incremental ratios, just send t → +∞ in

the following

f (x, z + tz0) − f (x, z)

t ≥ f (x, z + z

0

) − f (x, z) provided t ≥ 1.

Concerning the lower semicontinuity statement we extract a subsequence such that µj → ν ∈

Y (Ω, RN) and then exploit Jensen and 1.1.3: lim j ˆ Ω f (x, µj) = ˆ Ω hf (x, ·), ν_xi + hf∞(x, ·), ν_x∞iλac_ν (x) dx + ˆ Ω hf∞(x, ·), ν_x∞i dλs_ν ≥ ˆ Ω f (x, [νx]) + f∞(x, [νx∞]λacν (x)) dx + ˆ Ω f∞(x, [ν_x∞]) dλs_ν(x) ≥ ˆ Ω f (x, [νx] + [νx∞]λacν (x)) dx + + ˆ Ω f∞(x, [ν_x∞]) dλs_ν(x) = ˆ Ω f (x, µac(x)) dx + ˆ Ω f∞(x, µs) = ˆ Ω f (x, µ).

Remark. This result holds true also for those f ∈ C(Ω × RN) such that f (x, ·) is convex for every

x ∈ Ω and lim sup|z|→+∞

f (x,z)

1+|z| < +∞.

Consider the following situation: we have a sequence {µj} ⊂ C(Ω, RN)∗ that generates ν ∈

Y1(Ω, RN_{), what can we say about the measures {µ}j ω} where ω ⊂ Ω is some open set? Do they

generate a Young measure? If so, can it be expressed in terms of ν? The answer is contained in the following lemma, which says that things can go wrong only on the boundary:

Lemma 1.1.11 (Restriction of Young measures). Suppose that {µj} ⊂ C(Ω, RN)∗ generates a

1-Young measureν = (νx, λν, νx∞) ∈ Y1(Ω, Y ). Fix an open bounded subset ω b Ω such that

λν(∂ω) =Ln(∂ω) = 0.

Then the sequence ˜µj := µj ω ∈ C(ω, RN)∗generates the 1-Young measure ˜ν = (˜νx, λν˜, ˜νx∞) ∈

Y1_{(ω, R}n_{) where}

νx = ˜νx forLn-a.e. x ∈ ω, λν ω = λν ω = λν˜, νx∞= ˜ν

∞

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We conclude repeating briefly the intuitive introduction, with two concrete examples of appli-cation of the Fundamental Theorem. Consider a sequence of vector fields {Vj} which is bounded

in L1(Ω, RN) and hence converges (extracting a subsequences) weakly∗ to some measure µ ∈

C(Ω, RN)∗_{. By Vitali’s theorem we know that in order to have V}_j _{→ µ in total variation (and then}

µ Ln) we must require two necessary conditions

(i) No oscillations: the sequence {Vj} converge in measure to some vector field V ,

(ii) No concentration: the family {Vj} is equiintegrable.

The classical example where (i) fails is Vj(x) = sin(2πj(x · e)) with N = 1 and e some unit vector.

In this case the Young measure generated is

₁ √ 1 − z2L 1 _{(−1, 1)} x∈Ω , 0 , n/a ! .

At a level of graphs oscillations (in the limit) generates on the top of every point x the same diffused measure√_1−z1 2L

1 _{(−1, 1). On the other side this sequence is bounded and hence equi-integrable,}

this is the reason why the concentration measure λ is trivial. The classical example where (ii) fails is Vj(x) = (−1)j j

n

|A|1A(jx) where A is some neighborhood of

0 ∈ Rn_{and N = 1. These approximations of the identity with alternate signs are clearly converging} to 0 in measure and weakly∗. This implies that no oscillation occurs hence ν∞is trivial. But they concentrate (in the L1norm) on the origin. While they concentrate they spread mass equally towards +∞ and −∞. In the end one can verify that

(δ0)x∈Ω, δ0, 1 2(δ++ δ−)x∈Ω .

What we have seen in these examples can be generalized: whenever we have an equiintegrable sequence we can say that it will generate only Young Measures which have no concentration. On the other hand, whenever we have a sequence that is converging in measure, it will generate a trivial oscillation measure (a delta).

Conversely one can detect concentration and oscillation of a given sequence by the inspection of the Young Measure that it generates. We shall not need anything sophisticated in this direction so we do not go further into this topic.

1.2. Sobolev-Slobodeckij spaces and Gagliardo’s trace theorem

In this section we collect two important facts about Sobolev-Slobodeckij spaces Ws,p. The first is a very important result: Gagliardo’s Trace Theorem which, by the way, motivates the definition of these spaces. Then we will prove that these spaces have the so called "Fubini property" on balls. An self contained account of the general properties of Sobolev-Slobodeckij spaces can be found in the recent monograph [22] by G. Leoni, for a quicker introduction see also [26].

We will deal with real valued functions but the results are clearly true also for vector-valued maps. In this section Ω ⊂ Rnwill be an open bounded set with Lipschitz boundary.

For s ∈ (0, 1), p ∈ [1, ∞) and a measurable function u : Ω → R define

[u]_Ws,p_(Ω):= ˆ Ω ˆ Ω |u(x) − u(y)|p |x − y|n+sp dx dy 1/p .

This quantity is called the Gagliardo seminorm of u in Ω. The Sobolev-Slobodeckij space of smoothness 0 < s < 1 and integrability 1 ≤ p < ∞ on the domain Ω is defined by

Ws,p(Ω) :=nu ∈ Lp(Ω) : [u]_Ws,p_(Ω)< +∞ o

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1.2. Sobolev-Slobodeckĳ spaces and Gagliardo’s trace theorem

It is well-known that Ws,p(Ω) is a separable Banach space (reflexive if 1 < p < ∞) under the norm kuk_Ws,p_(Ω):= kuk_Lp_(Ω)+ [u]_Ws,p_(Ω).

We will see that by Poincaré’s inequality every measurable function u such that [u]Ws,p_(Ω) < +∞ must lie in Lp(Ω), at least if Ω has finite measure.

An important remark is in order now, to avoid any confusion with other definitions.

Remark. Suppose for simplicity Ω = Rn. The space Ws,p(Rn), as the name suggest, can be thought as of a "fractional Sobolev space", nevertheless this terminology is potentially dangerous since it is often employed for the Bessel-potential spaces Hs,p, whose norm is defined as:

kuk_Hs,p_(Rn₎ := F −1 (1 + |ξ|2)s/2u(ξ)ˆ _Lp_(Rn₎= (1 − ∆) s/2_u _Lp_(Rn₎.

The main caveat here is that in general Ws,p(Rn) 6= Hs,p(Rn), more precisely when 0 < s < 1 we have that:

Ws,p(Rn) = Hs,p(Rn) _{if and only if} p = 2.

For the reader familiar with the theory of Besov spaces Bp,qs and Triebel-Lizorkin spaces Fp,qs we can

state the following equivalences for every 0 < s < 1 and p > 1:

Ws,p∼ B_p,ps = F_p,ps _and Hs,p∼ F_p,2s ,

see Chapter 17 in [22] for the first and section 6.2.2 in [16] for the second.

The importance of the Sobolev-Slobodeckij spaces stems from the fact that they provide the sharp characterization of the "boundary values" of functions in W1,p(Ω), at least for p > 1. Before stating the result we shall give a meaning to the space Ws,p(∂Ω). The definition is not surprising, even if it hides a subtlety: measurability is required with respect to the surface measure, so regularity assumptions on ∂Ω are forced. For 1 < p < +∞ and 0 < s < 1 set:

Ws,p(∂Ω) := u ∈ Lp(∂Ω) : ˆ ∂Ω ˆ ∂Ω |u(x) − u(y)|p |x − y|n−1+sp dσ(x) dσ(y) < +∞ ,

here the measure σ on the boundary ∂Ω is the standard Hausdorff (n − 1)-dimensional measure, this is well defined because Ω has Lipschitz boundary.

We are now ready to state Gagliardo’s Trace Theorem [12]:

Theorem 1.2.1 (Traces). Let Ω ⊂ Rnbe open, bounded with Lipschitz boundary and 1 < p ≤ +∞. Then there exist a unique linear bounded operator, called the "Trace Operator", such that

tr_∂Ω : W1,p(Ω) → W1−1/p,p(∂Ω),

that extends the pointwise restriction u 7→ u|_∂Ω, defined for u ∈ C∞(Ω). Furthermore, this operator

is surjective and admits a linear and bounded right inverse.

There exist a unique linear bounded operator, also called the "Trace Operator", such that

tr∂Ω : BV (Ω) → L1(∂Ω),

that extends the pointwise restriction u 7→ u|_∂Ωdefined for u ∈ C∞(Ω). Furthermore, the restriction

of this operator to W1,1(Ω) is surjective, but it never admits any linear and bounded right inverse.

For a proof of this result see to the original paper [12] (in italian) or Chapter 18 in [22]. A quick proof of the p = 1 case can also be found in [24].

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Now that we have motivated why these spaces are interesting, we could develop a theory of Ws,p spaces along the same lines followed for the classical spaces W1,p. That is to prove the Sobolev embedding on the whole space, Poincaré inequalities of the first kind, extension theorems for regular domains Ω and the Poincaré inequalities of second kind. This is indeed possible and as one could guess the results are quite similar, but the proofs are rather different with respect to the non-fractional case: in Ws,pwe cannot handle pointwise the distributional derivative. There are two possible roads at this point: the first is a "PDEs-style" approach that goes through by-hand elementary estimates (see [26]). The second is somewhat more "Harmonic Analysis-style" approach, that embeds the whole discussion in the broader context of Besov spaces and interpolation theory (see [22] or [37]). An advantage of the first is that it is less sophisticated and works directly in domains. An advantage of the second is that is more systematic and allows one to use Fourier analysis techniques when working on Rn. Luckily, we shall only need the following basic embedding theorem which is a union of Theorem 6.5 and Theorem 7.1 in [26].

Theorem 1.2.2. Let 0 < s < 1 and p ≥ 1 and suppose sp < n. Then for every measurable function u with compact support we have

kuk_Lp∗_(Rn₎.n,s,p[u]Ws,p_(Rn₎ where p∗ = p∗(n, s, p) :=

np n − sp.

It follows that Ws,p(Rn) ,→ Lq(Rn) for every p ≤ q ≤ p∗. Furthermore, the embedding Ws,p(Rn) ,→ Lq_loc(Rn) is compact whenever 1 ≤ q < p∗, also in the limiting case sp = n, p∗ = +∞.

We also have the following Poincaré inequality which we state for the boundary space Ws,p(∂Ω), but an identical proof works for Ws,p(Ω).

Theorem 1.2.3 (Fractional Poincaré). For any bounded, Lipschitz, domain Ω ⊂ Rn, 0 < s < 1, p ≥ 1 the following inequality holds:

ku − (u)_∂Ωk_Lp_(∂Ω) ≤ ( diam(∂Ω)n−1+sp σ(∂Ω) ˆ ∂Ω ˆ ∂Ω |u(x) − u(y)|p |x − y|n−1+sp dσ(x)dσ(y) )1/p ,

for every measurable function u : ∂Ω → R. Proof. We use integral Minkowski inequality:

ku − (u)_∂Ωk_Lp_(∂Ω) = ∂Ωy

u(x) − u(y) dσ(y)

_Lp_(∂Ω x) ≤ 1 σ(∂Ω) ˆ ∂Ωy

ku(x) − u(y)k_Lp_(∂Ω_x₎dσ(y)

now raising this inequality to the p ≥ 1 power and using Jensen we get

ku − (u)∂Ωkp_Lp_(∂Ω) ≤ 1 σ(∂Ω) ˆ ∂Ω ˆ ∂Ω

|u(x) − u(y)|pdσ(x)dσ(y)

≤ 1 σ(∂Ω) ˆ ∂Ω ˆ ∂Ω

|u(x) − u(y)|pdiam(∂Ω)

n−1+sp

|x − y|n−1+sp dσ(x)dσ(y),

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1.3. Good restrictions and the quantitative Fubini property

1.3. Good restrictions and the quantitative Fubini property

We now turn our attention to a finer restriction property of Sobolev functions, in order to introduce it we start with a simple example. Consider a function u ∈ W1,2(Rn) and some ball BR(x0), by

Fubini theorem we can write ˆ R 0 ˆ ∂Br(x0) |∇_τu|2dσ ! dr ≤ ˆ BR(x0) |∇u|2_dx,

where ∇τu(x) := ∇u(x) − x(∇u · x)|x|−2 is the tangential gradient and we used the pointwise inequality |∇τu| ≤ |∇u|. Formally, this inequality is telling us that v := u|_∂

rB(x0) has finite

W1,2(∂Br(x0)) norm for almost every r. By the mean value theorem we also have the quantitative

estimate R k∇vk2_L2_(∂B r∗(x0)) ≤ k∇uk 2 L2_(B R(x0))

holds true for some r∗ between 0 and R. Thus we found a quantitatively good restriction of u on a well-chosen sphere. By the trace theorem we already knew that the restriction of u to any sphere is no better than W1/2,2(∂Br(x0)), but with some care we found a sphere such that the restriction of

u is actually W1,2_(∂B

r∗(x₀)) with quantitative bounds. A generalization of this argument will be

useful later. Let us now make rigorous statements, first we make a simple observation that permits us to restrict any L1locfunction to a lot of spheres ∂Br(x0).

Recall that if u ∈ Lploc(Rn, RN) (p < ∞) we can define the singular set Su by the identity

Rn\ Su :=

(

x ∈ Rn: exists a unique zx∈ RN such that lim

r→0+

Br(x)

|u(y) − zx|pdy = 0

)

,

and we have Ln(Su) = 0. We may then define the precise representative ˜u : Rn\ Su → RN as the map that sends x 7−→ zx. By the Fubini theorem we immediately have the following

Proposition 1.3.1 (Lebesgue spheres). Given u ∈ Lploc(Rn) and x0 ∈ Rn then forL1-a.e. t > 0

we have

Hn−1_(∂B

t(x0) ∩ Su) = 0, u|˜_∂B

t(x0)is well-defined and lies in L

p_(∂B

t(x0), Hn−1).

For these t’s we will say that ∂Bt(x0) is a "Lebesgue Sphere" for u. Furthermore, the map t 7→

´ ∂Bt(x0)|˜u|∂Bt(x0)| dσ isL 1_{-measurable and} kuk_Lp_(B R(x0))= ˆ R 0 ˆ ∂Bt(x0) u|˜_∂B_t_(x₀₎ p dσ ! dt.

This proposition allows us to state the following elementary but fundamental inequality, whose proof can be found in [14], Proposition 8.25.

Lemma 1.3.2 (Fubini inequality for spheres). Given 0 < s < 1, 1 ≤ p < ∞ then for every u ∈ L1_loc(Rn), x0∈ Rnand R > 0 we have

ˆ R 0 [˜u|_∂B t(x0)| W s,p_(∂B t(x0))]pdt .n,s,p[u | Ws,p(BR(x0))]p,

in particular the constant does not depend on the radius R.

For more regular maps we can strengthen this definition and make it quantitative, here’s the definition:

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Definition 1.3.3 (Good Spheres). Given 0 < s < 1, 1 ≤ p < ∞, u ∈ Wlocs,p(Ω) and a ball

B(x, r) b Ω we say that “∂Br(x) is a good sphere for u" if the following two conditions holds:

(i) ∂Br(x0) is a Lebesgue Sphere for u,

(ii) there exist a smooth and positive mollifier % such that

lim inf

ε→0+ ku ∗ %ε| W

s,p_{(∂B(x))k < +∞,}

where as usual %ε(x) := ε−n%(x/ε).

There is a plenty of good spheres, which enjoy nice properties, the following proof is a consequence of Fatou’s lemma and the Fubini estimate 1.3.2.

Proposition 1.3.4 (Good restrictions). Given 0 < s < 1, 1 ≤ p < ∞ pick any u ∈ Wlocs,p(Rn) and

x0 ∈ Rn. Denoting with ˜u the precise representative of u there holds:

(i) forL1-a.e. t > 0 the sphere ∂Bt(x0) is a "good sphere" for u;

(ii) limεu ∗ %ε(x) = ˜u(x) for every x /∈ Su;

(iii) if ∂Bt(x0) is a "good sphere" for u, then ˜u|_∂B

(iv) if ∂Bt(x0) is a "good sphere" for u and Ws,p(∂B(x0)) ,→ X compactly, where X is some

Banach space, then

(u ∗ %_ε_)|

∂Bt(x0) → ˜u|∂Bt(x0) in X .

For example we can take X = Lq(∂B_t(x₀)) for 1 ≤ q < np/(n − sp);

(v) for any 0 < r < R there is a Borel set G ⊂ (r, R) such thatL1(G) > 0 and h ˜ u|_∂B t(x0)| W s,p_(∂B t(x0)) ip .n,s,p 1 (R − r)[u | W s,p_(B R(x=)]p for every t ∈ G.

1.4. Linear partial differential operators with constant coefficients

In this section we introduce some vocabulary to describe the algebraic structure of differential operators with constant coefficients. Then we state a very deep result about the solubility of systems of PDEs with constant coefficients in convex domains.

From now on X, Y and Z will be finite dimensional real Hilbert spaces one should really think

X, Y, Z to be just Rd for some d = dX, dY, dZ, but the abstract notation seems lighter. We will always identify X, Y, Z with their duals so that a linear map a : X → Y will be tough as an element of Y ⊗ X instead of Y ⊗ X0.

We define the differential operator

B : C∞_(Rn_{, X) → C}∞_(Rn_{, Y ),} _{Bu(x) :=} X

|α|≤k

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1.4. Linear partial differential operators with constant coefficients

where α is an multi-index in Nn, k ≥ 1 is some integer and Bα ∈ Y ⊗ X are R-linear maps from

X to Y , whose coefficients are fixed real numbers. In this setup we will say that

"B is a differential operator over Rnbetween X and Y of order k”.

The principal part of B is given by

Bk_{u :=} X

|α|=k

Bα∂αu,

and an operator will be said homogeneous if it coincides with its principal part. An equivalent way of thinking an homogeneous diferential operator B is as a fixed linear operator β : X ⊗ (kRn) → Y , defined implicitly trough the relation

β[∇ku(x)] =Bu(x) for every u ∈ D(Rn, X), x ∈ Rn.

Trough the Fourier transform we can associate to every operator an (Y ⊗ X)-valued polynomial in the dual variables ξ1, . . . , ξn:

c

B =B[ξ] :=c X

|α|≤k

ξαBα.

This linear operator-valued polynomial is called the symbol of B, and similarly we have the notion of principal symbol.

Finally we can define the formal adjoint of B to be the operator over Rnfrom Y to X given by Bv(x) := X

|α|≤k

(Bα)†∂αv(x),

where (Bα)† : Y → X is the adjoint of Bα, X, Y being Hilbert spaces. Concretely the symbol c

B†_{[ξ] is obtained taking the adjoint of the matrix}

c

B[ξ]. One should use some care when working with real or complex Hilbert spaces as the adjoint is not the transpose.

As a general rule many qualitative properties of differential operators are determined by their principal part, hence we will restrict from now on to this special class of operators.

Definition 1.4.1 (Wave cone/Singular cone). Let B be an homogeneous differential operator over Rn_{between X and Y of degree k ≥ 1. Then its wave cone is the subset of X defined by:}

Λ_B:= [

|ξ|=1

kerB[ξ]c

.

Notice that it is indeed an R-cone.

The idea is that ΛB contains the directions along which a vector field can oscillate without being detected by B. Namely, if u0 ∈ kerB[ξc ₀] and h : R → R then u(x) := u₀h(x · ξ₀) satisfies Bu = 0 in the sense of distributions. Hence there is a complete loss of information, h being any function.

We now give some definitions that rule out many operators.

Definition 1.4.2 (Constant rank). Let B be an homogeneous differential operator over Rnbetween X and Y of degree k ≥ 1. We will say thatB has costant rank if

dim kerB[ξ] = r for every |ξ| = 1,c

for some fixed 0 ≤ r ≤ dim X.

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Definition 1.4.3 (Ellipticity). Let B be an homogeneous differential operator over Rn between X and Y of degree k ≥ 1. We will say thatB is elliptic if Λ_B= {0}. This happens if and only if there

exist δ > 0 such that

B[ξ] uc _Y ≥ δ |ξ| k_|u| X for every ξ ∈ Rn, u ∈ X.

Remark 1.4.4. Ellipticity is also easily seen to be equivalent to

detBct[ξ] ◦B[ξ]c

6= 0 _{for every ξ ∈ S}n−1.

For the sake of completeness we give the strongest notion of ellipticity, eben if we shall not need it. The idea is to allow ξ to assume complex values. To do so we first complexify our Hilbert spaces

X, Y and then extend the linear maps Bαforcing C-linearity, so that:

Bα: XC:= X + iX → Y + iY =: YC, Bα: u + iv 7→ Bα(u) + iBα(v)

At this point the polynomial B[ξ] can be evaluated on every ξ ∈ Cc n _{producing a C-linear map} between the complex vector spaces XCand YC. Furthermore XCand YChave a natural structure of Hilbert spaces, obtained promoting the real scalar product of X and Y to an Hermitian one. To stress the difference we may denote it asBcC[·]. The definition of wave cone still makes sense:

ΛC_B:= [

|ξ1|2+...+|ξn|2=1

kerBcC[ξ]

.

Let us remark that even if at a formal level this construction may seem sophisticated, at a practical level it is extremely natural: just promote every ξj to a complex number in you formulas. One just needs to be careful when doing scalar products as the new Hermitian structure involves complex conjugation, so for example the transpose of a matrix has no interesting properties and needs to be replaced by the adjoint.

Definition 1.4.5 (C-ellipticity). Let B be an homogeneous differential operator over Rn_{between X}

and Y of degree k ≥ 1. We will say thatB is elliptic if ΛC_B= {0}. This happens if and only if there

exist δ > 0 such that

Bc C_{[ξ] u} _YC ≥ δ |ξ| k

Cn|u|_XC for every ξ ∈ Cn, u ∈ XC.

Note that in the case n ≥ 2 the C−ellipticity forces dim X < dim Y , just write in down things in coordinate and look at the determinant of the square matrix B[ξ]: if n ≥ 2 an homogeneousc polynomial cannot vanish only in the origin.

We shall now present another condition of somewhat different nature, that will play crucial role in establishing certain limiting inequalities.

Definition 1.4.6 (Canceling). Let B be an homogeneous differential operator over Rn between X and Y of degree k ≥ 1. ThenB it is said to be canceling if

VB:= \

ξ∈Rn_\{0}

ranB[ξ] = {0}.c

Remark. This condition is equivalent to the non existence of a fundamental solution for B. Notice

that the cone VBis the wave cone of the formal adjoint of B, and that if B is real elliptic then it can be canceling only in the case dim Y > dim X. Finally it can be shown by purely algebraic means that if n ≥ 2 and B is C-elliptic, then B is canceling.

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1.4. Linear partial differential operators with constant coefficients

Example (Gradient). In this case X = R, Y = RnandBu = (∂₁u, . . . , ∂nu). It is easy to see that

this operator is C-elliptic and canceling (n ≥ 2). One computesB[ξ]s = sξ and so ranc B[ξ] = Rξ,c kerB[ξ] = 0.c

Example (Symmetric gradient). In this case X = Rn, Y = Rn×nandBu = B[∇u] = ∇u+(∇u)†, where B is proportional to the orthogonal projection of Rn×n → Sym_n. We first find the wave cone, hence try to solve

0 =B[ξ]v = B[v ⊗ ξ] ⇔ vc _iξ_j+ v_jξ_i = 0.

If we multiply the last equation by v_iξjand sum over i, j we find

|v|2_|ξ|2_{= −|(v|ξ)|}2 _{⇔ ξ = 0 or v = 0,}

hence ΛC_B = {0}. We show thatB is also canceling if n ≥ 2. Pick a matrix y ∈ T

|ξ|=1ranB[ξ]c

and so:

for every ξ ∈ Sn−1exist v(ξ) ∈ Rnsuch that vi(ξ)ξj+ vj(ξ)ξi = yij.

Setting i = j and ξi = 0 we discover yii= 0, hence vi(ξ) = 0, hence y = 0.

Example (Cauchy-Riemann). Here n = 2, X = Y = R2andBu = (∂₁u1− ∂2u2, ∂1u2+ ∂2u1).

A direct computation yields the symbol

c

B[ξ] = ξ1 −ξ2

ξ2 ξ1

!

,

henceB is real elliptic and so cannot be canceling (dim X = dim Y ). Setting ξ1 = iξ2 one sees

directly thatB is not C-elliptic.

Example (Divergence). Here X = Rn, Y = R andBu = div u. Since dim Y = 1 we have no hope to be canceling (cf. Malgrange-Ehrenpreis Theorem). Since dim X < dim Y we will not be elliptic. The symbol of the operator is the 1 × n matrixB[ξ] = ξc †. Notice thatBc†[ξ] = ξ = ˆ∇.

Example. We consider the operator over R2from X = R2to Y = R3, given by

Bu =    ∂1u1 ∂2u2 ∂1u2+ ∂2u1   , B[ξ] =c    ξ1 0 0 ξ2 ξ2 ξ1   .

It’s easy to see that the matrixB[ξ] is injective for every ξ ∈ Cc 2\ {0}. Suppose we have v ∈ VB.

Then in particular v ∈ ranB[ec ₁] = Re₁+ Re₃and v ∈ ranB[ec ₂] = Re₂+ Re₃, hence v = v₃e₃.

There must also exist a vector u(ξ) such that v =B[ξ]u(ξ). Writing explicitly this equation we havec    0 0 v3   =    ξ1u1(ξ) ξ2u2(ξ) u1(ξ) ξ2+ u2(ξ) ξ1    for every ξ ∈ S n−1_.

This tells us v3= 0 just pluggin in any ξ such that ξ1 6= 0 and ξ2 6= 0. SoB is canceling.

Example. We can consider a couple of differential operators (B0,B1) on the same X as a whole.

DefineB = B0⊕B1from X to Y := Y0⊕ Y1setting Bα = Bα0 ⊕ B1α. The wave cone becomes

Λ_B₀⊕B1 = [ |ξ|=1 kerBc₀[ξ] ∩ kerBc₁[ξ] ⊂ Λ_B₀ ∩ Λ_B₁,

so if one is elliptic, the sum clearly stays elliptic. The dual relation becomes V_B₀⊕_B1 = \ |ξ|=1 ranBc₀[ξ] ∪ ranBc₁[ξ] ⊃ V_B₀∪ V_B₁.

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We now turn our attention to a very natural problem. Consider an opens set Ω ⊂ Rn and a k-homogeneous differential operator B over Rn from X to Y . Given some smooth function

f ∈ C∞(Ω, Y ) can we solve the linear system

Bu = f for some u ∈ C∞(Ω, X)? (?)

Let us consider the familiar example of the De Rham cohomology where Y =Vk+1Rn, X = VkRn and B = d is the exterior differential. The previous system amounts exactly to ask that the differential form f is exact. Then two necessary conditions must be imposed:

(i) local compatibility on f , that is df = 0;

(ii) topological triviality of Ω, that is Hk+1(Ω) = 0.

As is known these two conditions are also sufficient to ensure solubility of (?). Furthermore the first condition can be checked in principle by brute force computation, while the second is satisfied for (say) all contractible domains Ω. The remarkable fact is that these two conditions can be generalized to every operator B to ensure solubility of (?). The easy piece is the generalization of the local compatibility condition (i). To this end consider the family MBof all (constant complex coefficients) operators α : C∞(Rn, YC) → C∞(Rn, C) such that α ◦ B ≡ 0, clearly this tells us that to solve Bu = f we necessarily have αf = 0. Then MBhas a natural structure of C[ξ1, . . . , ξn]-module via

the obvious action:

p · α := p(∂/∂x1, . . . , ∂/∂xn) ◦ α, for all p ∈ C[ξ], α ∈ MB.

Since MB is a submodule of the Noetherian module C[ξ1, . . . , ξn], it is finitely generated by some

operators {α1, . . . , αN}. This means that, even for general B’s, just a finite number of compatibility

conditions on f should be checked, namely:

α(f ) = 0 for every α ∈ MB ⇐⇒ αj(f ) = 0 for every j = 1, . . . , N.

We shall not investigate which is the right generalization of the topological condition (ii), but we will just say that it is fulfilled if whenever Ω is a convex set. Explicitly, we are claiming that this new conditions

(i)0 for every α ∈ MBwe have α(f ) = 0;

(ii)0 Ω ⊂ Rnis open and convex,

implies that (?) has a solution, that is to say: there exists u ∈ C∞(Ω, X) such that Bu(x) = f (x) for every x ∈ Ω. Needless to say, this result is highly nontrivial. The proof relies on the so called "Ehrenpreis fundamental principle" stated by L. Ehrenpreis in 1960 and proved (in the correct form) for the first time by V. Palamodov in 1965 (see the survey [38] and the references therein). The proof uses heavily the cohomology of coherent analytic sheaves (Cartan’s Theorem B), which was being developed in those years. The argument was later simplified using Hörmander’s L2estimates for the ∂ operator, a simplified proof, still sufficient for our claim, can be found in Chapter 7 of the monograph [17]. However, for further purpose, we need the full power of this theory whose proofs allow one to replace C∞(Ω, X) with other function spaces, such as the space of distributions D0(Ω, X). The following statement is a particular case of Theorem 1, Chapter 7 in Palamodov’s monography [28].

Theorem 1.4.7. Let Ω ⊂ Rn be an open convex set and let B be a k-order differential operator over Rnfrom X to Y with constant coefficients. Denote with M_Bits C[ξ]-module of compatibility conditions. If f ∈D0(Ω, Y ) satisfies

α(f ) = 0 inD0(Ω) for every α ∈ M_B, then there exists u ∈D0(Ω, X) such thatBu = f in D0(Ω, Y ).

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1.5. Some regularity results for elliptic systems of PDEs

1.5. Some regularity results for elliptic systems of PDEs

In this section we will use the terminology and notation introduced in the previous section. We recall two preliminary facts, the first regards homogeneous distributions and their Fourier transform, the second is a formula for the singular supports of convolutions. We refer to Hörmander’s monography [18] for both of them:

Theorem 1.5.1 (Theorems 7.1.16 and 7.1.18 in [18]). Let T ∈ D0(Rn) be an homogeneous

distri-bution of degree h, then T ∈ S0(Rn) and ˆT is homogeneous of degree −n − h. Furthermore if T ∈ C∞(Rn\ {0}) so does ˆT .

Theorem 1.5.2 (Theorem 4.2.5 in [18]). If u, v are distributions in Rn, one of which has compact support, then

sing spt{u ∗ v} ⊂ sing spt u + sing spt v.

We shall need also the following well-known endpoint weak estimate for the Sobolev-Hardy-Littlewood inequality, we enclose a quick proof.

Lemma 1.5.3. If n ≥ 2, then for every function f ∈S (Rn) there holds: 1 | · |n−1 ∗ f L n n−1,∞_(Rn₎ . nkf kL1_(Rn₎,

where the Lorentz quasinorm k · k

L n n−1,∞_(Rn₎is defined by kf k Ln−1n ,∞_(Rn₎:= sup λ>0 λ |{x ∈ Rn: |f (x)| > λ}|n−1n .

Proof. It is an exercise to prove that the translation-invariant norm:

kf k0 Ln−1n ,∞_(Rn₎:= sup |E|−1/n ˆ E |f | : E ⊂ Rn_{, 0 < |E| < +∞}_,

is equivalent to the Lorentz seminorm. Furthermore is easy to check that k| · |1−n_k

Ln−1n ,∞_(Rn₎= ω

1 1−n

n < +∞,

so by Minkowski integral inequality applied to the proper norm | · |0

L n n−1,∞: 1 | · |n−1 ∗ f Ln−1n ,∞_(Rn₎ ∼_n ˆ Rn f (y) | · −y|n−1dy 0 Ln−1n ,∞_(Rn₎ ≤ ˆ Rn f (y) | · −y|n−1 0 Ln−1n ,∞_(Rn₎ dy (integral Minkowski) = ˆ Rn |f (y)| | · | 1−n 0 L n n−1,∞_(Rn₎ dy (translation invariance) ∼nk| · |1−nk L n n−1,∞_(Rn₎ ˆ Rn |f (y)| dy ≈nkf kL1_(Rn₎.

We are now ready to state and prove our first regularity result which is essentially taken from [32] and [7].

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Theorem 1.5.4. Let Ω ⊂ Rn be an open set and let B be a k-order, elliptic, homogeneous, differential operator over Rn from X to Y . Suppose that u ∈D0(Ω, X) is some distribution and

thatBu = µ ∈ Cc(Ω, Y )∗is a locally finite Borel measure. Then

∂αu ∈ L

n n−1,∞

loc (Ω, X) for every α ∈ N

n_{, |α| = k − 1.}

In particular u ∈ W_lock−1,p(Ω, X) for every 1 ≤ p < n/(n − 1).

Proof. We first prove a preliminary representation formula result for v ∈ S0(Rn_{, X). We fix some} multi index α ∈ Nnof length |α| = k − 1 ≥ 0 and take the Fourier transform of ∂αv. This is an analytic function and thus can be evaluated and manipulated pointwise. In particular for any ξ 6= 0 we exploit ellipticity in the following way:

d ∂α_{v(ξ) = (−i)}|α|_ξα_v(ξ)_ˆ = (−i)|α|ξαBc†(ξ)B(ξ))c −1 c B†_(ξ)d_Bv(ξ) = m(ξ)Bv(ξ),d

where m : Rn\0 → X ⊗Y defined by m(ξ) = (−i)|α|ξα c B†_(ξ) c B(ξ))−1 c B† (ξ) is clearly smooth and (−1)-homogeneous. Furthermore, m is also locally integrable so can be extended trivially to a (−1) homogeneous distribution that we will call again m. Invoking Theorem 1.5.1 we can say that :

F−1

m(x) ∈ C∞(Rn\ {0}, X ⊗ Y ) and is (1 − n)-homogeneous,

we define for x 6= 0 the function K(x) := F−1m(x) and, since by homogeneity K ∈ L1loc(Rn, X ⊗

Y ), we can extend it to trivially to a tempered distribution K ∈ S0(Rn_{, X ⊗ Y ). By the previous} computation we know that in S0(Rn, X ⊗ Y ) there holds:

sptK −F−1m= {0}, _so K −F−1m =X

β

cβ∂βδ0 where the sum is finite.

But by the same Theorem 1.5.1 we know that K − F−1m is (1 − n)-homogeneous while a sum of derivatives of δ0 is at least (−n)-homogeneous, so it must be cβ = 0 for every β and we conclude

that K = F−1m as tempered distributions.

By the previous computation we have that the following equality holds in S0(Rn, X): d

∂α_{v = m.}_{Bv + "a finite number of derivatives of δ}_d

0”,

taking the anti-Fourier transform we finally get:

∂αv = K ∗Bv + "a polynomial" in S0(Rn, X). (1.5.1) We are now ready to do the proof. Fix some ω b Ω and a plateau function % ∈ D (Ω, [0, 1]) which is identically 1 in an open set ω0 with ω b ω0. Then we use formula 1.5.1 with v := %u, (extend trivially to Rn) and the Leibniz rule:

∂α(%u) = K ∗ (B(%u)) + "a polynomial” = K ∗ (%µ) + k X j=1 K ∗ bj ∇j%, ∇k−ju+ "a polynomial”,

in the sense of tempered distributions. Here the {bj} are bilinear maps that depend on B, whose

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1.5. Some regularity results for elliptic systems of PDEs

We start with the first, for which we use the previous lemma after a regularization procedure. No-tice that by smoothness and homogeneity the kernel K obeys the bound |K(x)| .B |x|1−n so by Lemma 1.5.3 we have

kK ∗ ((%µ) ∗ φε)k L

n

n−1,∞_(Rn₎.n,Bk(%µ) ∗ φεkL1(Rn) ≤ |µ|(spt % + εB)

where {φε} si a family of standard mollifiers and ε is sufficiently small, hence we can conclude by

Fatou sending ε → 0+. Concerning the second term we have for every j = 1, . . . , k

sing spt K = {0} and sing spt bj

∇j%, ∇k−ju⊂ Rn\ ω0,

because ∇j% ≡ 0 in ω0. Using Theorem 1.5.2 we infer sing spt K ∗ bj ∇j%, ∇k−ju⊂ {0} + Rn\ ω0 = Rn\ ω0,

this tells us that the second term is a smooth function on ω0so it belongs to Ln/(n−1),∞(ω, X). The third term is polynomial so it also belongs to Ln/(n−1),∞(ω, X).

Thus we proved that for every ω b Ω and every multi-index α ∈ Nn, |α| = k − 1 the distributions

∂αu, restricted to ω, are in fact Ln/(n−1),∞(ω, X) functions. By classical Sobolev embeddings and inclusions between local Lorentz spaces we deduce u ∈ Wlock−1,p(Ω, X).

The second result is concerns a boundary value problem and is adapted from [15], even if is a sum of uncountably contributions. Ee introduce some language first: take as before an elliptic differential operator B over Rn from X to Y , we will assume for simplicity that B is homogeneous and has order k = 1, so

Bu =

n

X

`=1

∂`u for suitable linear maps B`: X → Y, ` = 1, . . . , n.

Let Q : Y × Y → R be a symmetric bilinear form, recall that any symmetric Q gives rise naturally to a unique linear mapQ : Y → Y trough the scalar product on Y , that is requiring:ˆ

Q[y, y0] = ( ˆQ.y) · y0 = y · ( ˆQ.y0_{) for every y, y}0 ∈ Y, for the ease of notation we shall writeQ = Q.ˆ

We will say that th bilinear map Q satisfies the B-Legendre-Hadamard ellipticity condition of order if there are two positive constants Λ, λ such that:

(

λ|x|2|y|2 _{≤ Q[x ⊗}

Bξ, x ⊗Bξ] for every x ∈ X, ξ ∈ Rn,

Q[y, y] ≤ Λ|y|2

for every y ∈ Y. (B-LH)

Here we used the bilinear map (x, ξ) 7→ x ⊗Bξ which has the following implicit definition:

x ⊗_Bξ =B[ξ].x =c

n

X

`=1

ξ`B`.x is a vector of Y, .

Notice that this operation is linked with the Leibniz rule B(ψu) = ψBu + ∇ψ ⊗Bu whenever

ψ ∈D(Rn) and u ∈ D0(Rn_{, X), We also remark that the ellipticity of B implies: |ξ ⊗}_Bx|Y ∼B

|ξ|2

Rn|x|2_X_.

We relate this notion of ellipticity with the standard Legendre-Hadamard ellipticity using some algebra. The bilinear map ⊗B: : X × Rn→ Y can be represented by a unique linear map β on the

tensor product β : X ⊗ Rn→ Y in such a way that

Partial regularity for BV^B local minimizers