6 Approximation by Gamma-convergence

ε=0

= 1 + Z

B0

sin²α(x)

dist(x, K)dx>0 which proves that the second order minimality condition is fulfilled.

Remark 27. The above proposition actually proves that, in comparison with rectangles, only the square is a local minimizer in itsε-neighborhood. Indeed, if the edges have different length, a rectangle could not satisfy the first order condition (5.3) in its ε-neighborhood for translation of each edge. We could probably do the same for any regular polygon but the case of the rectangle is much simpler because of the orthogonality at the corners. Of course the previous proposition does not prove that regular polygons are minimizers, but that they are local minimizers for the considered class of variations.

6 Approximation by Gamma-convergence

In this last section we approximate the functional in (1.1) by a certain family of elliptic functionals, in view of numerical computations, as for eg. Ambrosio and Tortorelli [2] did for the Mumford-Shah functional. For this purpose we will use the result of Buttazzo and Santambrogio [7], which says that the average distance functional can be obtained as the limit whenp→+∞of the p-Compliance functional. We will also use the seminal work of Modica and Mortola [16] for the perimeter term.

In the sequel Ω will denote a bounded domain of R². For any p > 1, ϕ ∈ L^∞(Ω) and λ >0, we denote by uϕ,λ,p the unique solution u∈W^1,p(Ω) of the problem

−∆_pu+λϕu= 1,

The solutionu_ϕ,λ,p can be obtained for instance by minimizing the energy E_ϕ,λ,p(u) := 1

p Z

Ω

|∇u|^pdx− Z

Ω

u−λϕu²dx

among all functionsu∈W^1,p(Ω).

As a first step to approximate our minimization problem we begin with the following Γ-convergence result which is a little inspired from a Lemma given without proof in [7, Lemma 7].

Proposition 28. Let ϕ_ε : Ω →R⁺ be some nonnegative measurable bounded functions con-verging to some ϕ∈L^∞(Ω) when ε→ 0, and assume also that pε → +∞. Then the family of functionalsE_ϕε,λ,pε Γ-converges in L² to the functional

E_ϕ,λ(u) :=

( −R

Ωu+λR

Ωϕu² if u∈W^1,∞ and k∇uk_L^∞ ≤1,

+∞ otherwise. (6.1)

Proof. The limsup inequality is fairly simple. Let u∈W^1,∞ withk∇uk_∞≤1 be given. We have to find a recovery sequenceu_ε converging to u inL² and such that

lim sup

ε→0

E_ϕε,λ,pε(uε)≤E_ϕ,λ(u). (6.2) We simply take the identically constant sequence uε = u. Since |∇u| ≤ 1 we easily obtain

that 1 we conclude taking the limsup, actually proving (6.2) with an equality.

Now to prove the liminf inequality, we consider a sequence u_ε → u in L². We have to

Therefore, using the semicontinuity ofu7→R

|∇u|^q with respect to the weak convergence and H¨older inequality, we can compute for any fixed exponentq,

Z

The last inequality holds for any largeq. Now takingq →+∞ yields k∇uk_∞= lim

which proves both thatu ∈ W^1,∞(Ω) and k∇uk_∞ ≤ 1 (notice that u ∈ L^∞ by the Sobolev

We will also need the following proposition.

Proposition 29. Let K_λ ⊆ Ω ⊆ R² be a sequence of convex sets such that χ_Kλ converges toχK in L². Then K is also convex and denoting u_λ the minimizer of E_χKλ,λ, we have that u_λ(x) converges uniformly to x7→dist(x, K) when λ→+∞.

Proof. Firstly notice that since the sets K_λ are convex, the convergence of the characteristic functions inL² implies the convergence of K_λ for the Hausdorff distance and in the sense of compacts, thus it follows that K must be convex. To prove the proposition it is enough to prove that E_λ := E_χKλ,λ (as defined in (6.1)) Γ-converges in the uniform topology, to the

Indeed, the minimizer ofE∞is exactly x7→dist(x, K).

The limsup inequality is easy, taking u_λ := uχ_Ω\Kλ as a recovery sequence we have that u_λ converges to u inL^∞ and

E_λ(u) =− Z

Ω

u_λ thus taking the limsup we directly get

lim sup

λ

Eλ(uλ) =E∞(u).

Now for the liminf inequality, assume that uλ is a sequence that converges uniformly to u∈ L^∞. Then since K_λ converges to K is Hausdorff distance, taking the limit inλR

Kλu²_λ, which is uniformly bounded in λ, it is easy to see that u = 0 a.e. on K. Moreover by the uniform convergence we obtain that u is Lipschitz and k∇uk_∞ ≤ 1. Therefore taking the liminf inE_λ(u_λ) we get the desired inequality that finishes the proof of the Proposition.

Remark 30. We could change the convexity assumption in Proposition 29 into some different one like for instance uniformly Lipschitz domains, and still get the Γ-convergence result.

Next we define the following functionals F_ε,λ(ϕ) := (1−ε)

Z

Ω

u_ϕ,λ,1

ε dx+λ₁ Z

Ω

ϕ dx+λ₂ Z

Ω

ε|∇ϕ|²+W(ϕ) ε dx

. (6.5)

Where W(t) := t²(1−t)² is the traditional double well potential involved in Modica and Mortola’s functional. Then we finally denote the limiting functional in ε, defined on L^p0(Ω) for p₀>2

F_0,λ(ϕ) :=

( R

Ωu_K,λ+λ₁|K|+λ₂H¹(∂K) ifϕ∈BV(Ω) and ϕ=χ_K for someK

+∞ Otherwise.

whereu_K,λ is defined as the minimizer for E_χK,λ (defined in (6.1)). Finally we define F_0,∞(ϕ) :=

( R

Ωd(x, K∪∂Ω)dx+λ1|K|+λ2H¹(∂K) ifϕ=χK,K convex

+∞ Otherwise.

Theorem 31. Let λ₂6= 0. Then we have the following diagram F_ε,λ−→^Γ

ε→0F_0,λ −→^Γ

λ→+∞F_0,∞

for the L² topology, the second convergence holding under the extra assumption that F_0,λ is +∞ if K is not convex.

Remark 32. It is not clear wether one could let λ depend on εand pass to the limit only a single time instead of letting first ε → 0 and then λ → +∞, separately. Indeed, such a “uniform” Γ-convergence result would need the convergence of solutions u_{ϕε,λε,p(ε)} to the solution uϕ,∞,∞. This convergence holds provided the rate of convergence of ϕ_ε to ϕ is fast enough compared to the convergence of λε to +∞. Otherwise the Γ-limit, that can be characterized using capacitary measures, could be different from the one we would expect.

It is therefore difficult to find a proper dependance of λ with respect to ε to make this convergence true for any ϕε → ϕ, that have a uniform bound of F_ε(ϕε), say (this would be needed in the proof of the Γ-liminf). This is somehow not a real problem in view of numerical computations. Indeed, to approximate the average distance functional F_0,∞ one should first fix a very largeλand then minimizeF_ε,λfor someεsmall. Our convergence result guaranties that for εsmall enough, we will approximateF0,∞, at least as much as F_0,λ does.

Remark 33. For the second convergence we need to restrict the class to convex sets only.

This is not convenient for numerical computations because it is difficult to implement any topological or convexity constraint with the computer.

Proof of Theorem 31 (first convergence). We prove first the liminf inequality. Let ϕ_ε be a sequence of functions converging toϕinL². We want to prove that

F₀(ϕ)≤lim inf

ε→0 F_ε(ϕε).

Without loss of generality we may assume that the liminf is finite and is a real limit. Since λ₂ 6= 0 we know that R

Ω W(ϕε)

ε is bounded by a uniform constant, thus passing to the limit we deduce that R

It is also clear that

Z

Ω

ϕ_ε→ |K|.

Therefore to prove the liminf inequality it is enough to prove that lim inf direct consequence of Lemma 28, which precisely says that u_ϕε,λ,pε converges to u_K,λ inL², yielding an equality in (6.6).

Let us now prove the limsup inequality. Let ϕ=χK be fixed. By Modica and Mortola’s result and standard approximation by polygonal domains we know that there exists a sequence ϕ_ε that converges to ϕinL² and such that

Then it remains to prove the convergence of the p-compliance term, and this again directly comes from Proposition 28 as in the proof of the Γ-liminf.

Proof of Theorem 31 (second convergence). We prove first the liminf inequality. Let ϕ_λ = χK_λ be a sequence of characteristic functions converging to ϕ = χK in L², with Kλ and K convex. Then by convexity of K_λ it is well known that the volume and perimeter are continuous with respect to this convergence. It is therefore enough to prove that

F(K)≤lim inf

λ→+∞

Z

Ω

u_Kλ,λ and this follows from Proposition 29.

The limsup also directly follows from Proposition 29 in an even weaker form, taking as a recovery sequence for a convex setK the identically constant sequenceχ_K.

References

[1] Luigi Ambrosio and Carlo Mantegazza. Curvature and distance function from a manifold.

J. Geom. Anal., 8(5):723–748, 1998. Dedicated to the memory of Fred Almgren.

[2] Luigi Ambrosio and Vincenzo Maria Tortorelli. Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math., 43(8):999–

1036, 1990.

[3] Dorin Bucur, Ilaria Fragal`a, and Jimmy Lamboley. Optimal convex shapes for concave functionals. personal communication.

[4] Giuseppe Buttazzo, Edoardo Mainini, and Eugene Stepanov. Stationary configura-tions for the average distance functional and related problems. Control Cybernet., 38(4A):1107–1130, 2009.

[5] Giuseppe Buttazzo, Edouard Oudet, and Eugene Stepanov. Optimal transportation problems with free Dirichlet regions. InVariational methods for discontinuous structures, volume 51 of Progr. Nonlinear Differential Equations Appl., pages 41–65. Birkh¨auser, Basel, 2002.

[6] Giuseppe Buttazzo, Aldo Pratelli, Sergio Solimini, and Eugene Stepanov. Optimal ur-ban networks via mass transportation, volume 1961 of Lecture Notes in Mathematics.

Springer-Verlag, Berlin, 2009.

[7] Giuseppe Buttazzo and Filippo Santambrogio. Asymptotical compliance optimization for connected networks. Netw. Heterog. Media, 2(4):761–777 (electronic), 2007.

[8] Giuseppe Buttazzo and Eugene Stepanov. Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem. Ann. Sc. Norm. Super. Pisa Cl.

Sci. (5), 2(4):631–678, 2003.

[9] Michel C. Delfour and Jean-Paul Zol´esio. Shape analysis via distance functions: local theory. InBoundaries, interfaces, and transitions (Banff, AB, 1995), volume 13 ofCRM Proc. Lecture Notes, pages 91–123. Amer. Math. Soc., Providence, RI, 1998.

[10] Herbert Federer. Curvature measures. Trans. Am. Math. Soc., 93:418–491, 1959.

[11] Herbert Federer. Geometric measure theory. Die Grundlehren der mathematischen Wis-senschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.

[12] Antoine Henrot and Michel Pierre. Variation et optimisation de formes, volume 48 of Math´ematiques & Applications (Berlin) [Mathematics & Applications]. Springer, Berlin, 2005. Une analyse g´eom´etrique. [A geometric analysis].

[13] Antoine Lemenant. About the regularity of average distance minimizers inR².To Appear in Journal of Convex Analysis.

[14] Antoine Lemenant. A presentation of the average distance minimizing problem.Preprint Cvgmt.

[15] Carlo Mantegazza and Andrea Mennucci. Hamilton-jacobi equations and distance func-tions on riemannian manifolds. Appl. Math. Opt., 47(1):1–25, 2003.

[16] Luciano Modica and Stefano Mortola. Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5), 14(3):526–529, 1977.

[17] Emanuele Paolini and Eugene Stepanov. Qualitative properties of maximum distance minimizers and average distance minimizers in Rⁿ. J. Math. Sci. (N. Y.), 122(3):3290–

3309, 2004. Problems in mathematical analysis.

[18] Filippo Santambrogio and Paolo Tilli. Blow-up of optimal sets in the irrigation problem.

J. Geom. Anal., 15(2):343–362, 2005.

[19] Leon Simon. Lectures on geometric measure theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University. Australian National University Centre for Mathematical Analysis, Canberra, 1983.

[20] Eugene Stepanov. Partial geometric regularity of some optimal connected transporta-tion networks. J. Math. Sci. (N. Y.), 132(4):522–552, 2006. Problems in mathematical analysis. No. 31.

[21] Paolo Tilli. Some explicit examples of minimizers for the irrigation problem. J. Conv.

Anal., 17(2):583–595, 2010.

Antoine Lemenant Edoardo Mainini lemenant@ann.jussieu.fr edoardo.mainini@unipv.it

Universit´e Paris Diderot - Paris 7 Dipartimento di Matematica ‘F. Casorati’, U.F.R de Math´ematiques Universi`a degli Studi di Pavia,

Site Chevaleret via Ferrata 1, 27100 Pavia (Italy) Case 7012

75205 Paris Cedex 13 (France)