Properties of the energy functional

The aim of this section is to make Definition 3.2.2 rigorous by recalling the concept of convex functionals on measures, and to prove certain properties of the energy functional defined in Definition 3.2.2, which are needed for the proof of the main theorem. We start with some basic concepts concerning convex functions.

Definition 3.3.1. Let f :R →[0,∞] be a convex and lower-semicontinuous function withf(0) = 0.

We then define its convex conjugatef^∗:R→[0,∞] by f^∗(x) = sup

y∈R

(xy−f(y)), (3.3.1)

and its recession functionf_∞:R→[0,∞] by

f_∞(x) = lim

t→∞

f(tx)

t . (3.3.2)

Remark 3.3.2. Note thatf_∞ andf^∗ are convex. If f is symmetric, so aref_∞ andf^∗. Moreover, f_∞ is positively homogeneous.

For the notion of solution that we are aiming at, we need the concept of a convex function of a measure, which has been developed in [45].

Definition 3.3.3. Letψsatisfy (3.2.7) as well as Assumptions 3.2.1 (A3), (A5’). Define the set D^ψ={v∈ Cc⁰(O) :ψ^∗(v)∈L¹(O)}

and letµ∈ M(O). We then define the positive measureψ(µ)∈ M(O) by Z

O

η ψ(µ) :=_M(O)hψ(µ), ηiC₀⁰(O)

:= sup Z

O

vη dµ− Z

O

ψ^∗(v)η dx:v∈ D^ψ

(3.3.3)

forη∈ C0⁰(O), η≥0, and for generalη∈ C0⁰(O) we set

M(O)hψ(µ), ηiC₀⁰(O)=_M(O)hψ(µ), η∨0iC₀⁰(O)−M(O)hψ(µ),(−η)∨0iC⁰₀(O), according to Theorem 1.1 in [45].

Remark 3.3.4. As argued in Lemma 1.1 in [45], one can write forµ∈ M(O) Z

O

ψ(µ) =kψ(µ)kT V = sup Z

O

vdµ− Z

O

ψ^∗(v) dx:v∈ D^ψ

.

Remark 3.3.5. Letµ∈ M(O) with Lebesgue decompositionµ^a+µ^s, whereµ^ahas the densityh∈L¹(O) with respect to the Lebesgue measure. Then, by Theorem 1.1 in [45], we have

Z

O

η ψ(µ) = Z

O

η(x)ψ(h(x))dx+ Z

O

η ψ_∞(µ^s), (3.3.4)

where the recession functionψ∞is defined as in (3.3.2). In particular, this formulation shows the useful fact that

ψ(µ) =ψ(µ^a) +ψ(µ^s). (3.3.5)

Our next aim is to prove the lower-semicontinuity of the energy functional defined in Definition 3.2.2 and Definition 3.3.3. First, we show that the Radon measure ψ(µ) constructed in Definition 3.3.3 controls the norm of its original measureµin the following way.

Lemma 3.3.6. Let ψ satisfy (3.2.7) as well as Assumptions 3.2.1 (A3), (A5’). Letµ∈ M(O)and let y >0 such thatψ(y)>0 as demanded in Assumption 3.2.1 (A5’). Then

kψ(µ)kT V ≥ψ(y)

y kµkT V −ψ(y)|O|.

Proof. Forµ∈ M(O), denote byµ=µ^a+µ^sthe Lebesgue decomposition ofµwith respect to Lebesgue measure, and leth= ^dµ_dx^a be the Radon-Nikodym derivative ofµ^a. Asψ_∞(µ^s) is singular by Theorem 4.2 in [118], we can use the decomposition (3.3.4) to obtain

kψ(µ)kT V = Z

O

ψ(h) dx+kψ_∞(µ^s)kT V. (3.3.6) We now estimate the summands separately. For the absolutely continuous part we obtain using Assump-tion 3.2.1 (A5’)

Z

O

ψ(h) dx≥ ψ(y) y

Z

O|h|dx−ψ(y)|O|=ψ(y)

y kµ^akT V −ψ(y)|O|.

For the singular part, we note by Lemma 3.A.5 that for v ∈ C⁰c(O) being in D^ψ∞ is equivalent to

−ψ_∞(1)≤v≤ψ_∞(1), and for suchv,ψ^∗_∞(v)≡0. Thus, we get with Corollary 3.A.4 withk:= ^ψ(y)_y Z

O

ψ_∞(µ^s) = sup

v∈Dψ∞

Z

O

v dµ^s− Z

ψ_∞^∗ (v)dx

≥ sup

v∈C_c⁰(O)

−k≤v≤k

Z

O

v dµ^s=kkµ^skT V.

Thus, we can continue (3.3.6) by kψ(µ)kT V ≥ ψ(y)

y kµ^akT V +kkµ^skT V −ψ(y)|O|=ψ(y)

y kµkT V −ψ(y)|O|, as required.

Proposition 3.3.7. In both settings of Definition 3.2.2, ϕ :H⁻¹ → [0,∞] is convex and lower-semi-continuous.

Proof. In the superlinear case, i. e. Definition 3.2.2 (i) applies, convexity and lower-semicontinuity ofϕ are proved on p. 68 in [7]. In the sublinear case, i. e. Definition 3.2.2 (ii) applies, convexity becomes clear by Remark 3.3.4. It remains to prove lower-semicontinuity in the sublinear case.

Step 1: As a preparatory step, we establish weak* lower-semicontinuity of the functional ˜ϕ:M(O)→ [0,∞),

˜

ϕ(µ) =kψ(µ)kT V, for which we have

˜

ϕ|M(O)∩H⁻¹ =ϕ.

Consider µ_n → µ weakly* for n → ∞. We can assume that ψ(µ_n) contains a subsequence which is bounded in TV norm (otherwise there is nothing to show). Then we select a subsequence (µ_nk)_k∈N such that kψ(µn_k)kT V →lim inf_n→∞kψ(µn)kT V fork→ ∞, from which we can choose a nonrelabeled subsequence (ψ(µn_k))_k∈N which converges weakly* to some ν ∈ M(O) (e. g. by Satz 6.5 in [1]). By Lemma 2.1 in [45], we get that

M(O)hψ(µ), ηiC₀⁰(O)≤^M(O)hν, ηiC⁰₀(O)

= lim

k→∞M(O)hψ(µ_nk), ηiC⁰₀(O)

≤ lim

k→∞kψ(µn_k)kT V kηkC₀⁰(O)

forη∈ Cc⁰(O),η≥0. Now, using thatψ(ρ) is a positive measure for anyρ∈ M(O) by (3.3.3), we obtain kψ(µ)kT V = sup

η∈C_c⁰(O) η∈[0,1]

M(O)hψ(µ), ηiC⁰₀(O)

≤ sup

η∈C_c⁰(O) η∈[0,1]

k→∞lim M(O)hψ(µn_k), ηiC⁰₀(O)

≤ sup

η∈C_c⁰(O) η∈[0,1]

lim

k→∞kψ(µ_nk)kT V = lim inf

n→∞ kψ(µ_n)kT V, as required.

Step 2: Assume now that (un)n∈N ⊂ H⁻¹, u∈ H⁻¹, and un → ufor n → ∞. Being the only non-trivial case, we can assume that (un)n∈N contains a subsequence (which we call again (un)) for which (ϕ(un))n∈Nis bounded. Thus, there are measures µn∈ M(O)∩H⁻¹ such that

u_n(η) = Z

O

η dµ_n for allη∈ Cc¹(O).

By definition of ϕ, ϕ(u_n) = kψ(µ_n)kT V, such that Lemma 3.3.6 implies that kµ_nkT V is bounded.

Thus, there is ˜µ∈ M(O) and an again nonrelabeled subsubsequence (µ_n)_n∈N such that µ_n *^∗ µ. For˜ η∈ Cc¹(O)⊆ Cc⁰(O) we have

Z

O

η d˜µ= lim

n→∞

Z

O

η dµn = lim

n→∞un(η) =u(η)≤ kukH⁻¹kηkH₀¹(O),

so ˜µ∈ M(O)∩H⁻¹ andu= ˜µ. Using the weak* lower-semicontinuity of ˜ϕfrom Step 1, we get ϕ(u) = ˜ϕ(˜µ)≤lim inf

n→∞ ϕ(µ˜ n) = lim inf

n→∞ ϕ(un). (3.3.7)

As this argument works for any bounded subsequence of (u_n)_n∈N, (3.3.7) is also true for the original sequence (u_n)_n∈N.

As one can see from the definition of the energy functionalϕin the second part of Definition 3.2.2, it has an explicit representation onH⁻¹\ M(O), where it is∞, and onL¹(O)∩H⁻¹, where it is an integral.

However, whenever we evaluateϕfor general measures inM(O)∩H⁻¹, e. g. in the uniqueness part of the proof of Theorem 3.2.6, we need an approximation reducing it to evaluations on L¹(O) functions.

This will be made precise in the following theorem, the proof of which will take the rest of this section.

Theorem 3.3.8. Assume that ψ satisfies (3.2.7) as well as Assumptions 3.2.1 (A3), (A5’). Let ϕ be defined as in Definition 3.2.2 (ii) and u∈ M(O)∩H⁻¹. Then there exists a sequence(un)n∈N⊂L²(O) such that

u_n* u inH⁻¹, and (3.3.8)

ϕ(un)→ϕ(u) (3.3.9)

forn→ ∞.

Corollary 3.3.9. Since convex functions on a real Hilbert space are lower-semicontinuous if and only if they are weakly sequentially lower-semicontinuous (see e. g. Theorem 9.1 in [19]), Theorem 3.3.8 implies that ϕis the lower-semicontinuous hull ofϕ|^L2(O) inH⁻¹, which means that

ϕ= sup

( β:H⁻¹→[0,∞] :

β convex and lower-semicontinuous, β|^L2(O)≤ϕ|^L2(O) )

, (3.3.10)

wheresup denotes the pointwise supremum.

We will approach Theorem 3.3.8 by giving an explicit construction for the sequence (un)_n∈N, inspired by the construction in Lemma A6.7 in [1]. It will rely on applying the original functional to modified functions, which is why we first introduce several modifications to functions onO.

We next introduce further notation and recall some concepts relying on the regularity of the boundary.

Notations 3.3.10. Since the domainO is bounded and smooth, its boundary is locally the graph of a smooth function. More precisely, we recall from Section A6.2 in [1] that for each y ∈ ∂O there is a neighbourhood ˜U ⊂R^d, an orthonormal systeme1, . . . , ed ofR^d,r, h∈Rwithr > h >0, and a smooth bounded functiong:R^d−1→R, such that with the notation

x_,d:= (x₁, . . . , x_d−1), forx=

d

X

i=1

x_ie_i, we have

U˜ =

x∈R^d:|x_,d−y|< rand |x_d−g(x_,d)|< h , and forx∈U˜

x_d=g(x_,d) if and only ifx∈∂O, xd∈(g(x,d), g(x,d) +h) if and only ifx∈ O, and x_d∈(g(x_,d)−h, g(x_,d)) if and only ifx /∈ O. For technical reasons we set

U =

x∈U˜ :|x_,d−y|< r

2 and |x_d−g(x_,d)|< h 2

. (3.3.11)

The boundary∂Ois covered by those open setsU belonging to all possible reference pointsy. As∂Ois compact, we can choose a finite subcovering (U^j)^l_j=1, and for eachU^j, we denote the elements belonging to it by a superindex j, e. g.y^j, e^j_d, g^j, h^j,U˜^j. At last, we fix an open setU⁰ with U⁰ ⊂ O, such that O ⊂ ∪^lj=0U^j and we sete⁰_d:= 0.

Subordinate to the covering∪^lj=0U^j, let nowζ⁰, . . . , ζ^lbe a partition of unity onO, i. e. 0≤ζ^j≤1, ζ^j∈ Cc^∞(R^d),supp(ζ^j)⊆U^j for allj = 0, . . . , l, and

l

X

j=0

ζ^j = 1 onO.

For η : O →Rand µ ∈ M(O), we define ηext :R^d →R andµext ∈ M(R^d) as the extended function (resp. measure) by zero. Finally, we define forρ∈C_c^∞(R^d) with

supp(ρ)⊆B₁(0), Z

O

ρdx= 1, ρ(x) =ρ(−x) (3.3.12)

a Dirac sequence (ρδ)δ>0⊂C_c^∞(R^d) of mollifiers by ρ_δ(x) = 1

δ^dρx δ

. (3.3.13)

Forη∈L²(R^d), µ∈ M(R^d), we then define functionsρ_δ∗η,ρ_δ∗µ∈C^∞(R^d) by ρδ∗η(x) =

Z

R^d

ρδ(x−y)η(y) dy and ρδ∗µ(x) = Z

R^d

ρδ(x−y)dµ(y).

For brevity, we writeρδ∗η :=ρδ∗ηext forη∈L²(O).

The following construction allows to shift a function “away from the boundary”.

Definition 3.3.11. Letε >0 andη:O →R. Then we defineηε:O →Rby η_ε(x) =

l

X

j=0

ζ^j(x)η_ext(x−εe^j_d), (3.3.14) where we recall thate⁰_d is set to 0.

Remark 3.3.12. By this construction, we achieve thatη_ε= 0 on aw(ε)-neighbourhood of∂Owith w(ε) := min

dist(U⁰,O^c), min

j=1,...,l

min

ε 2, ε

2L^j,h^j 4 , h^j

4L^j

>0, (3.3.15) whereL^j denotes the Lipschitz constant ofg^j defined in Notations 3.3.10.

Proof. The numberw(ε) is obviously strictly positive by the construction of the covering (U^j)^l_j=0. To show the support property, let j ∈ {0,1, . . . , l} and U_ε^j := U^j ∩((U^j ∩ O) + εe^j_d). By definition, ηext(x−εe^j_d) = 0 ifx∈U^j\U_ε^j. By the definition ofζ^j, we furthermore conclude thatζ^j(x)ηext(x−εe^j_d) = 0 forx /∈U_ε^j. Consequently,

η_ε:x7→

l

X

j=0

ζ^j(x)η_ext(x−εe^j_d) is supported on

U_ε:=

l

[

j=0

U_ε^j,

such that it remains to show that dist(U_ε,O^c)≥w(ε), or equivalently, that dist(U_ε^j,O^c)≥w(ε) for all j∈ {0, . . . , l}.

For j = 0, this is trivial by construction ofU_ε⁰ =U⁰ and w(ε). For j = 1, . . . , l, using the coordinate system (x^j_,d, x^j_d) we can rewrite

U_ε^j ={x∈U^j:x^j_d> g^j(x^j_,d) +ε}. Hence, we can compute for anyx∈U_ε^j, i. e.x=

x^j_,d, g^j(x^j_,d) +ε⁰

for someε⁰ ∈(ε,^h₂^j), andy∈∂O∩U˜^j kx−yk²=kx_,d−y_,dk²+|g(x_,d) +ε⁰−g(y_,d)|²

≥ kx,d−y,dk²+ (ε⁰− |g(x,d)−g(y,d)|)²,

wherek·kdenotes the Euclidean norm both inR^d and inR^d−1. LettingL^j be the Lipschitz constant of g^j, we can then argue that eitherkx_,d−y_,dk> _2L^ε_j or

|g(x_,d)−g(y_,d)| ≤L^j ε 2L^j = ε

2, such that dist(U_ε^j, ∂O ∩U˜^j) is at least min_ε

2,_2L^ε_j . By similar arguments, we can obtain from the construction ofU^j in (3.3.11) (note thatr^j> h^j by construction) that

dist(U_ε^j,( ˜U^j)^c)≥min h^j

4 , h^j 4L^j

,

such that we conclude

dist(U_ε^j, ∂O) = min{dist(U_ε^j, ∂O ∩U˜^j),dist(U_ε^j, ∂O ∩( ˜U^j)^c)}

≥min{dist(U_ε^j, ∂O ∩U˜^j),dist(U_ε^j,( ˜U^j)^c)}

≥min ε

2, ε 2L^j,h^j

4 , h^j 4L^j

≥w(ε).

This allows to define the following approximating objects foru∈ M(O)∩H⁻¹.

Definition 3.3.13. Letε >0 and 0< δ≤ ^w(ε)2 . Foru∈H⁻¹, we define forη∈H₀¹(O) uε(η) =_H⁻¹hu, ηεiH₀¹(O)

and uε,δ(η) =_H−1hu, ρδ∗ηεiH¹₀(O). (3.3.16) These functionals are inH⁻¹ by Lemma 3.3.14 and Lemma 3.3.15 below. Foru∈ M(O), we define for η∈ C0⁰(O)

˜

uε(η) =_M(O)hu, ηεiC₀⁰(O)

and u˜ε,δ(η) =_M(O)hu, ρδ∗ηεiC₀⁰(O). (3.3.17) These functionals are in M(O) by Lemma 3.3.16 below. Ifu ∈ M(O)∩H⁻¹, the uniqueness of the linear continuation allows to conclude that

uε, uε,δ∈ M ∩H⁻¹, as well as uε= ˜uεand uε,δ= ˜uε,δ.

Lemma 3.3.14. Let ε >0 andη∈H₀¹(O). Then the mapH₀¹(O)3η7→ηε∈H₀¹(O) is linear, and kη_εkH₀¹(O)≤CkηkH₀¹(O),

where C only depends on the localizing functions (ζ^j)^l_j=0, the number of covering sets l, the Poincar´e constant of the domainO and the spatial dimension d.

Proof. The proof of the linearity claim is straightforward and therefore skipped. In order to prove boundedness, letV^j =U^j∩ OandU_ε^j :=U^j∩((U^j∩ O) +εe^j_d) as before. We first note

kη_εkH₀¹(O)=

l

X

j=0

ζ^jη_ε^j _H1

0(O)

≤

l

X

j=0

ζ^jη_ε^j _H1

0(O), (3.3.18)

where we have written

η^j_ε∈H¹(R^d), η^j_ε(x) =ηext(x−εe^j_d).

We now analyze the summands separately, where we make use of the fact that for all j ∈ {1, . . . , l}, ζ^j∈ Cc^∞(U^j) andζ^jη^j_εis supported onV^j. In the following, (∂i)^d_i=1represent the weak partial derivatives of first order. We then compute fori∈ {1, . . . , d}

∂_i(ζ^jη^j_ε) _L2

(O)=

∂_i(ζ^jη_ε^j)

_L2(Vj)≤

(∂_iζ^j)η^j_ε

_L2(Vj)+ ζ^j∂_iη_ε^j

_L2(Vj)

≤C η_ε^j

_L2(Vj)+ Z

V^j

∂i(ηext(x−εe^j_d))

2

dx ¹₂

≤CkηkL²(O)+ Z

U_ε^j

(∂iη)(x−εe^j_d)

2

dx ¹₂

≤CkηkL²(O)+k∂_iηkL²(O).

This yields

ζ^jη_ε^j

2 H₀¹(O)=

d

X

i=1

∂i(ζ^jη_ε^j)

2 L²(O)≤

d

X

i=1

CkηkL²(O)+k∂iηkL²(O)

2

≤Ckηk²H₀¹(O)+ 2

d

X

i=1

k∂iηk²L²(O)≤Ckηk²H₀¹(O),

where C may depend on d,O (through the Poincare constant) andζ^j. Thus, we can continue (3.3.18) by

kηεkH₀¹(O)≤

l

X

j=0

ζ^jη_ε^j _H1

0(O)≤(l+ 1)CkηkH₀¹(O), as required.

Concerning the mollification step, we note that by Remark 3.3.12, ρ_δ ∗η_ε(x) = 0 if dist(x, ∂O)≤ ^w(ε)2

and 0< δ≤ ^w(ε)2 , so that in this case we can restrict ρδ∗ηε toO to get aCc¹(O) function. By a slight abuse of notation, we then write

(ρδ∗ηε)|^O =ρδ∗ηε∈ Cc¹(O)⊆H₀¹(O)∩ C0⁰(O). (3.3.19) Also for this step, we have to ensure linearity, which is clear, and an estimate on theH₀¹(O) norm, which is done in the following lemma.

Lemma 3.3.15. Let ε >0 and0< δ≤ ^w(ε)2 . Then the mapH₀¹(O)3η7→ηε,δ∈H₀¹(O) is linear, and kρδ∗ηεkH₀¹(O)≤CkηkH¹₀(O) for allη∈H₀¹(O),

whereC is the constant from Lemma 3.3.14.

Proof. The proof of linearity is straightforward. In order to show boundedness, for anyg∈L²(O) such thatρδ∗g= 0 onO^c we can compute

kρ_δ∗gk²L²(O)= Z

R^d

Z

R^d

ρ_δ(x−y)g_ext(y) dy 2

dx

≤ Z

R^d

Z

R^d

ρδ(x−y) (gext(y))²dydx

= Z

R^d

Z

R^d

ρδ(x−y)dx(gext(y))²dy

=kgextk²L²(R^d)=kgk²L²(O),

(3.3.20)

where in the second step we could apply Jensen’s inequality sinceρδ(x−y) dy is a probability measure for eachx∈R^d. By Remark 3.3.12 for alli∈ {1, . . . , d},ρ_δ∗(∂_iη_ε) vanishes outside ofOif 0< δ≤ ^w(ε)2 . Henceg in (3.3.20) can be replaced by each partial derivative∂_iη_ε which yields

kρδ∗ηεk²H₀¹(O)=

d

X

i=1

k∂i(ρδ∗ηε)k²L²(O)=

d

X

i=1

kρδ∗∂i(ηε)k²L²(O)

≤

d

X

i=1

k∂_iη_εk²L²(O)=kη_εk²H₀¹(O)≤Ckηk²H₀¹(O),

where the second equality can be found e. g. in Section 2.23 in [1] and the last inequality is the statement of Lemma 3.3.14.

Lemma 3.3.16. Let ε >0,0< δ≤ ^w(ε)2 andη∈ Cc⁰(O). Then, the map Cc⁰(O)3η7→(ηε, ρδ∗ηε)∈(Cc⁰(O))² is linear. Furthermore, we have

kρ_δ∗η_εk∞≤ kη_εk∞≤ kηk∞, (3.3.21) wherek·k∞ denotes the supremum norm.

Proof. The proof of the linearity claim is straightforward. In order to show boundedness, we first note that for 0 < δ ≤ ^w(ε)2 , ρδ ∗ηε ∈ C⁰c(O) by construction and Remark 3.3.12. To obtain (3.3.21), we estimate for arbitraryx∈ O

|ηε(x)| ≤

l

X

j=0

ζ^j(x)

ηext(x−εe^j_d) ≤

l

X

j=0

ζ^j(x)kηk∞=kηk∞, which yields the second relation. The first one can be seen by

|ρδ∗ηε(x)| ≤ Z

R^d

ρδ(x−y)kηεk∞ dx=kηεk∞, which concludes the proof.

We next analyze howϕas given in Definition 3.2.2 (ii) acts on the approximating measures from Definition 3.3.13. First, we state that ifµis absolutely continuous with respect to the Lebesgue measure, so isµ_ε, which we show by giving its density.

Lemma 3.3.17. Let ε >0,h∈L¹(O)andµ:=hdx∈ M(O). Then µε has the density O 3x7→

l

X

j=0

ζ^j(x+εe^j_d)h_ext(x+εe^j_d) with respect to the Lebesgue measure.

Proof. Forη∈ Cc⁰(O), we compute Z

O

η dµε= Z

O





l

X

j=0

ζ^j(x)ηext(x−εe^j_d)



µ(dx)

= Z

R^d





l

X

j=0

ζ^j(x)η_ext(x−εe^j_d)



h_ext(x) dx

=

l

X

j=0

Z

R^d

ζ^j(x+εe^j_d)ηext(x)hext(x+εe^j_d) dx

= Z

O

η(x)

l

X

j=0

ζ^j(x+εe^j_d)h_ext(x+εe^j_d) dx,

as required. The switching of integration domains is possible as the integrands are supported onO by Remark 3.3.12 or by assumption, respectively.

A more direct construction ofµε,δis given by the following lemma.

Lemma 3.3.18. Let ε >0,0< δ≤ ^w(ε)2 andµ∈ M(O). Then, the measure

˜

µε,δ:= ((ρδ∗µext)|^Odx)_ε∈ M(O) (3.3.22) coincides with µε,δ.

Proof. We apply ˜µε,δto η∈ Cc⁰(O) and obtain Z

O

ηd˜µε,δ= Z

O

ηd((ρδ∗µext)|^Odx)ε

= Z

O

ηε(ρδ∗µext)|^Odx

= Z

R^d

(ηε)ext(ρδ∗µext) dx

= Z

R^d

ρδ∗(ηε)extdµext

= Z

O

ρδ∗ηεdµ,

where for the last step, we used Remark 3.3.12 to extend the integration domain. We conclude by noticing that the last term is precisely the definition of_M(O)hµε,δ, ηiC_c⁰(O).

In the rest of this section, we will argue that the sequence µ1

n,¹₂w(_n¹)

n∈N

is an approximation ofµ∈ M(O)∩H⁻¹ in the sense of Theorem 3.3.8. First we address the regularity ofµ_ε,δ, whereε >0 and 0< δ≤ ^w(ε)2 .

Lemma 3.3.19. Letε >0,0< δ≤ ^w(ε)2 andµ∈ M(O). Then, the measureµε,δ has a bounded density with respect to Lebesgue measure.

Proof. The fact thatµ_ε,δhas a density with respect to Lebesgue measure follows from its characterization in Lemma 3.3.18 and Lemma 3.3.17. This density is bounded in space since

l

X

j=0

ζ^j(x+εe^j_d)(ρδ∗µext)(x+εe^j_d)

≤(l+ 1) sup

x∈R^d

|ρδ(x)| kµkT V .

The first part of the following proposition allows to deduce property (3.3.8), while the second part is needed for the further proof of (3.3.9).

Proposition 3.3.20. Let ρbe as in (3.3.13), and for eachε >0, let0< δ_ε≤ ^w(ε)2 . 1. Forη∈H₀¹(O), we have

ρ_δε∗η_ε→η forε&0 in H₀¹(O). (3.3.23) 2. Forη∈ Cc⁰(O), we have

ρδ_ε∗ηε→η forε&0 inCc⁰(O). (3.3.24) Proof. Throughout this proof, we will writeδinstead ofδε, always assuming that 0< δ ≤^w(ε)2 .

Proof of part 1: It is enough to show that for alli∈ {1, . . . , d}

k∂i(ρδ∗ηε)−∂iηkL²(O)→0 forε→0. (3.3.25) By the density ofC0^∞(O) in H₀¹(O), for anyβ >0 we can chooseϕ∈ C0^∞(O) such that

maxn

kϕ−ηkL²(O),k∂iϕ−∂iηkL²(O)

o≤ β

6(l+ 1) ˜C, (3.3.26) where

C˜ := max

j=1,...,lmax (sup

R^d

∂iζ^j ),1

.

Asϕ_ext, ζ^j ∈ Cb¹(O) for eachj∈ {1, . . . , l}, we can chooseε₀>0 small enough, such that for allx∈R^d andy, z∈B_ε0(x)

∂iζ^j(y)ϕext(z)−∂iζ^j(x)ϕext(x)

≤ β 6(l+ 1)|O|¹²

(3.3.27) and

ζ^j(y)∂iϕext(z)−ζ^j(x)∂iϕext(x)

≤ β

6(l+ 1)|O|¹². (3.3.28) We approach (3.3.25) by splitting the term under consideration into the more convenient pieces

k∂_i(ρ_δ∗η_ε)−∂_iηkL²(O)=kρ_δ∗∂_iη_ε−∂_iη_extkL²(R^d)

=kρδ∗∂i(ηε−ϕε) +ρδ∗∂iϕε−∂iϕext+∂iϕext−∂iηextkL²(R^d)

≤ kρ_δ∗∂_i(η_ε−ϕ_ε)kL²(R^d)+kρ_δ∗∂_iϕ_ε−∂_iϕ_extkL²(R^d)+k∂_iϕ_ext−∂_iη_extkL²(R^d)

= (I) + (II) + (III).

We estimate the summands separately. For the first one we get with the convolution estimate (e. g.

Section 2.13 in [1])

(I)≤ k∂i(ηε−ϕε)kL²(R^d)

where we used (3.3.26) in the last step. For the second term, we recall that (ζ^j)^l_j=0is a partition of unity on the support ofϕ. Thus, we can compute

(II)≤ withε0as for (3.3.27). Noting thatρδ integrates to 1 for anyδ >0 and using Jensen’s inequality in the second step, we obtain

(V)²_j = of Remark 3.3.12 to see that the integrand of the outer integral is supported onO. Thus, we can restrict the integration domain to obtain

(3.3.30) =

While we have used (3.3.28) in the second step, the estimate for (IV)_j uses (3.3.27) instead and gets the same result. We conclude

(II) =

Finally the estimate

(III)≤β

3 (3.3.32)

is obvious by property (3.3.26). Collecting (3.3.29), (3.3.31), and (3.3.32), we obtain k∂_i(ρ_δ∗η_ε−η)kL²(O)≤β

only by choosingεsmall enough and adapting 0< δ≤^w(ε)2 , which proves (3.3.23).

Proof of part 2: Sinceη is now assumed to be continuous and to have compact support, it is uniformly continuous. For arbitraryβ >0, we can thus fixε₀>0 such that for allx, y∈R^d

We now turn to prove Property (3.3.9). Recall the definition of a convex function of a measure from Definition 3.3.3. We need some more lemmas on measures obtained by this technique, the first of which can be found in Equation (2.11) in [45].

Lemma 3.3.21. Let ψsatisfy (3.2.7)as well as Assumptions 3.2.1 (A3),(A5’). Letµ∈ M(R^d)and let (ρδ)δ>0 be a family of mollifying kernels as specified in (3.3.12)and (3.3.13). Then

Z

Remark 3.3.22. Given the assumptions on ψ, the theory of Definition 3.3.3 indeed also applies to finite measures onR^d (cf. p. 202 in [118]).

Lemma 3.3.23. Let ψ satisfy (3.2.7) as well as conditions Assumptions 3.2.1 (A3),(A5’). For µ ∈ M(O)we have which allows us to write

Z

We note that forvsatisfying the conditions of D¹,vext satisfies the conditions ofD², while the involved integrals agree due to the definition ofµext andψ^∗(0) = 0. This yields “≥”.

Conversely, forvsatisfying the conditions ofD² we can define ˜v=v|^O. ˜v satisfies the conditions of D¹. Furthermore, we have

Z

O

˜ vdµ=

Z

R^d

vdµext and Z

O

ψ^∗(˜v) dx≤ Z

R^d

ψ^∗(v) dx due toψ^∗≥0.

Thus, we have found an element inD¹ being larger than or equal to Z

R^d

vdµext− Z

R^d

ψ^∗(v) dx, which yields “≤”, completing the proof.

The key tool to prove the approximation property (3.3.9) is the following proposition.

Proposition 3.3.24. Let ε > 0, 0 < δ ≤ ^w(ε)2 and µ ∈ M(O). Let ψ satisfy (3.2.7) as well as Assumptions 3.2.1 (A3),(A5’). Then,

kψ(µ_ε,δ)kT V ≤ kψ(µ)kT V. (3.3.35) Proof. Recall Notations 3.3.10 and letV^j=U^j∩O. Let (ξα)α>0⊂C_c⁰(R^d) be a sequence of non-negative cut-off functions compactly supported inO, which converge to 1 pointwise inOforα→0, and each of which is monotonically increasing on eachV^j ine^j_d direction.

Leth∈L¹(O) and µ=hdx. In the following argument, we will needξ_α(x)≥ξ_α(x−εe^j_d) forx∈V^j, wherex−εe^j_d is not a priori inO. However, sinceξ_α= 0 outside ofO, it is clear that the statement is valid even ifx−εe^j_d∈ O/ . By the convexity ofψ, the construction of (ζ^j)^l_j=0 and Lemma 3.3.17, we then estimate

Z

O

ξ_αψ(µ_ε) = Z

O

ξ_α(x)ψ





l

X

j=0

ζ^j(x+εe^j_d)h_ext(x+εe^j_d)



dx

≤ Z

O

ξα(x)

l

X

j=0

ζ^j(x+εe^j_d)ψ(hext(x+εe^j_d)) dx

= Z

R^d

ξα(x)

l

X

j=0

ζ^j(x+εe^j_d)ψ(hext(x+εe^j_d)) dx

= Z

R^d

ψ(hext(x))

l

X

j=0

ξα(x−εe^j_d)ζ^j(x)dx. (3.3.36)

We note that Pl

j=0ξ_α(x−εe^j_d)ζ^j(x) is supported on O by Remark 3.3.12. Furthermore, by the con-struction ofξ_α, we have

ξ_α(x−εe^j_d)≤ξ_α(x)

for allx∈V^j, so this holds especially forx∈ O for whichζ^j(x)>0. Thus, we can continue (3.3.36) =

l

X

j=0

Z

O

ξα(x−εe^j_d)ζ^j(x)ψ(h(x)) dx

≤ Z

O l

X

j=0

ζ^j(x)ξ_α(x)ψ(h(x)) dx

= Z

O

ξα(x)ψ(h(x)) dx= Z

O

ξαψ(µ).

(3.3.37)

For a positive Radon measure µ, we have µ(O) = sup{µ(K) : K ⊆ Ocompact}. Since any such K is included in

K_α:={x∈ O: dist(x,O^c)≥α}

for αsmall enough, we can as well writeµ(O) = lim_α→0µ(Kα). Then, noting that ξα ≥1K_α, we can argue by definition of the Radon measure of compact sets that

µ(O)≥ Z

O

ξαdµ≥µ(Kα)^α→0−→µ(O), thusµ(O) = lim_α→0R

Oξ_αdµ.

Hence, we conclude by (3.3.37) forµ=hdx,h∈L¹(O), that Z

O

ψ(µ_ε) = lim

α→0

Z

O

ξ_αψ(µ_ε)≤ lim

α→0

Z

O

ξ_αψ(µ) = Z

O

ψ(µ). (3.3.38)

Using (3.3.38), Lemma 3.3.21 and Lemma 3.3.23, we then obtain for 0< δ ≤^w(ε)2

Z

O

ψ(µε,δ) = Z

O

ψ(((ρδ∗µext)|^Odx)ε)

≤ Z

O

ψ((ρδ∗µext)|^O) dx

= Z

R^d

ψ(ρδ∗µext)1_Odx

≤ Z

R^d

ψ(ρδ∗µext) dx≤ Z

R^d

ψ(µext) = Z

O

ψ(µ), which finishes the proof.

Corollary 3.3.25. Together with Remark 3.3.4, Proposition 3.3.24 immediately implies lim sup

ε&0

Z

O

ψ(µε,δ_ε)≤ Z

O

ψ(µ),

whereµ∈ M(O)and0< δε≤ ^w(ε)2 for eachε >0.

Proof of Theorem 3.3.8. Foruas in Theorem 3.3.8, we show that the sequence (un)_n∈N:=

u1 n,¹₂w(¹_n)

n∈N

, wherewwas defined in Remark 3.3.12, meets all requirements.

By construction,u_n ∈ M(O)∩H⁻¹ for alln∈N, and by Lemma 3.3.19, the density of u_n is bounded and thus in L²(O). Property (3.3.8) is proved in the first part of Proposition 3.3.20. For Property (3.3.9), note that Corollary 3.3.25 especially shows that (ψ(u_n))_n∈N is uniformly bounded in the TV norm, which means that it contains a subsequence that converges weakly* toψ(u) by Proposition 3.3.20, Corollary 3.3.25 and Lemma 2.1 in [45]. Since this argument can be carried out for any subsequence, we get weak* convergence for the whole sequence and, also by Lemma 2.1 in [45],

ψ

u1 n,¹₂w(_n¹)

_{T V} =

Z

O

ψ u1

n,¹₂w(¹_n)

→ Z

O

ψ(u) =kψ(u)kT V as n→ ∞. This yields (3.3.9) and thereby concludes the proof.

Im Dokument Stochastic partial differential equations arising in self-organized criticality (Seite 75-87)