For every $f \in L^N(\Omega)$ defined in an open bounded subset $\Omega$ of
$\mathbb{R}^N$, we prove that a solution $u \in W_0^{1, 1}(\Omega)$ of the
$1$-Laplacian equation ${-}\mathrm{div}{(\frac{\nabla u}{|\nabla u|})} = f$ in
$\Omega$ satisfies $\nabla u = 0$ on a set of positive Lebesgue measure. The
same property holds if $f \not\in L^N(\Omega)$ has small norm in the
Marcinkiewicz space of weak-$L^{N}$ functions or if $u$ is a BV minimizer of
the associated energy functional. The proofs rely on Stampacchia's truncation
method.Comment: Dedicated to Jean Mawhin. Revised and extended version of a note
  written by the authors in 201

Orsina, Luigi

Ponce, Augusto C.

English

arXiv

For every f ∈ LN(Ω) defined in an open bounded subset Ω of RN, we prove that a solution u ∈ W01,1(Ω) of the 1-Laplacian equation -div (∇u/|Δu|)= f in Ω satisfies ∇u = 0 on a set of positive Lebesgue measure. The same property holds if f ∉ LN(Ω) has small norm in the Marcinkiewicz space of weak-LNfunctions or if u is a BV minimizer of the associated energy functional. The proofs rely on Stampacchia's truncation method

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Flat solutions of the 1-Laplacian equation

Abstract. For every f ∈ Ln(Ω) defined in an open bounded subset Ω of Rn, we prove that a solution u ∈ W01,1 (Ω) of the 1-Laplacian equation – div (∇u / |∇u|)= f in Ω satisfies ∇u = 0 on a set of positive Lebesgue measure. The same property holds if f ∈/ Ln(Ω) has small norm in the Marcinkiewicz space of weak–Ln functions or if u is a BV minimizer of the associated energy functional. The proofs rely on Stampacchia’s truncation method

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Rend. Istit. Mat. Univ. TriesteVolume 49 (2017), 41–51DOI: 10.13137/2464-8728/16204Flat solutions of the 1-LaplacianequationLuigi Orsina and Augusto C. PonceÀ Jean Mawhin, dont l’enthousiasme et l’amour pour les mathématiques fontencore rêver les nouvelles générationsAbstract. For every f ∈ LN (Ω) defined in an open bounded subsetΩ of RN , we prove that a solution u ∈ W 1,10 (Ω) of the 1-Laplacianequation −div( ∇u|∇u|)= f in Ω satisfies ∇u = 0 on a set of positiveLebesgue measure. The same property holds if f 6∈ LN (Ω) has smallnorm in the Marcinkiewicz space of weak-LN functions or if u is aBV minimizer of the associated energy functional. The proofs rely onStampacchia’s truncation method.Keywords: 1-Laplacian, degenerate elliptic equations, nonlinear elliptic equation,nonexistence of solution.MS Classification 2010: 35J70, 35J25, 35J62, 35J92.1. IntroductionLet Ω ⊂ RN be a smooth bounded open subset. Given a convex functionΦ : RN → R and f ∈ L1(Ω), consider the energy functionalEΦ(u) =∫ΩΦ(∇u)−∫Ωfu,defined on some class of functions u : Ω → R for which the integrands aresummable. Although Φ need not be smooth, one can express the Euler–Lagrange equation of EΦ using the subdifferential of Φ. Indeed, by convexityof Φ, at each point x ∈ RN there exists a subgradient ξ ∈ RN such thatΦ(y) ≥ Φ(x) + ξ · (y − x),for every y ∈ RN ; see [18, Chapter 2]. Denoting the collection of all subgradi-ents ξ at x by ∂Φ(x), one can then write the Euler–Lagrange equation of EΦat some function u as (see [12, Chapter IV] and [22])−divZ = f in the sense of distributions in Ω, (1)42 L. ORSINA AND A.C. PONCEwhere Z is a summable function with values in RN such thatZ ∈ ∂Φ(∇u) almost everywhere in Ω. (2)For example, if Φp(x) = |x|p/p for some exponent p > 1, then Φp is differ-entiable pointwise. Thus, ∂Φp(x) = {|x|p−2x}, and one recovers an equationinvolving the p-Laplace operator:−∆pu = − div (|∇u|p−2∇u) = f.When p = 1, the function Φ1 is not differentiable at 0, and one should be carefulabout the meaning of the quotient∇u/|∇u| that appears in the formal notationof the 1-Laplacian. The correct interpretation is based on the formalism ofsubdifferentials above. Indeed, for Φ1(x) = |x|, one has∂Φ1(x) ={B1(0) if x = 0,{x/|x|} if x 6= 0,(3)where B1(0) denotes the unit open ball in RN .The vector field Z in the Euler–Lagrange equation now satisfies the condi-tions:|Z| ≤ 1 and Z|∇u| = ∇ualmost everywhere in Ω. Observe that, in dimension 1, equation (3) providesone with the maximal monotone graph associated to the sign function.Assuming that f ∈ LN (Ω), the functional EΦ1 associated to Φ1 is well-defined in W 1,10 (Ω), and the Euler–Lagrange equation (1)–(2) is indeed satisfiedby a minimizer. The goal of this paper is to show that one cannot abandon thevector field Z and replace it by the quotient ∇u/|∇u| since the gradient ∇umust vanish on a set of positive Lebesgue measure.Every function u ∈ W 1,1(Ω) such that ∇u 6= 0 a.e. in Ω has a legitimate1-Laplacian ∆1u defined in the sense of distributions as〈∆1u, ϕ〉 := −∫Ω∇u|∇u|· ∇ϕ,for every test function ϕ ∈ C∞c (Ω) with compact support in Ω, but evenfor smooth functions u something strange happens near an interior extremumpoint:Example 1.1. For every N ≥ 1, let u : B1(0)→ R be the function defined byu(x) = 1− |x|2. In the sense of distributions we have, for N = 1,−∆1u = 2δ0,while for N ≥ 2,−∆1u =N − 1|x|.FLAT SOLUTIONS OF THE 1-LAPLACIAN EQUATION 43In the previous example, the topological singularity of the vector field−x/|x| is detected by its divergence, and the 1-Laplacian does not belong toLN (Ω). We show that this is a general fact that holds for Sobolev functions,not necessarily smooth:Theorem 1.2. There exists no function u ∈ W 1,10 (Ω) such that ∇u 6= 0a.e. in Ω and∆1u ∈ LN (Ω).In Example 1.1 above for N ≥ 2, one sees that the right-hand side belongsto the Marcinkiewicz spaceMN (Ω) of weak-LN functions f in Ω equipped withthe seminorm‖f‖MN (Ω) = supA⊂Ω1|A|N−1N∫A|f |,where |A| denotes the Lebesgue measure of A and the supremum is computedwith respect to every Borel subset of Ω. In the case of the example, the functionf = (N − 1)/|x| satisfies‖f‖MN (B(0;1)) = Nω1/NN , (4)where ωN denotes the volume of the unit ball in RN .A variant of the proof of Theorem 1.2 based on Peetre–Alvino’s imbeddingof W 1,1(RN ) in the Lorentz space LNN−1 ,1(RN ) shows that this quantity (4) iscritical for the existence of flat levels of solutions involving the 1-Laplacian:Theorem 1.3. Let N ≥ 2. There exists no function u ∈ W 1,10 (Ω) such that∇u 6= 0 a.e. in Ω,∆1u ∈MN (Ω) and ‖∆1u‖MN (Ω) < Nω1/NN .Theorems 1.2 and 1.3 are related to the degenerate limit behavior of solu-tions of the p-Laplacian equation as p tends to 1 that has been studied by severalauthors; see e.g. [9, 20, 21], starting with the pioneering work of Kawohl [15],and also clarify the need for relying on the vector field Z in replacement of∇u/|∇u|.Example 1.4. For any 0 < r < 1, let u : B1(0) → R be the function definedbyu(x) ={1− |x|2 if |x| ≥ r,1− r2 if |x| < r.Then, u ∈ W 1,10 (B1(0)). If Z : B1(0) → B1(0) is any smooth extension of thefunctionx ∈ B1(0) \Br(0) 7−→ −x|x|∈ RN ,44 L. ORSINA AND A.C. PONCEthen u and Z satisfy the Euler–Lagrange equation (1)–(2) for some functionf ∈ L∞(B1(0)).Observe that the Sobolev space W 1,10 (Ω) is not the natural setting for look-ing for minimizers of EΦ1 , due to the lack of reflexivity of L1(Ω;RN ). This isin contrast to minimization problems in W 1,p(Ω) for 1 < p < +∞ which canbe investigated using techniques based on the uniform convexity of the space;see [11].Let us assume that EΦ1 is bounded from below for some given f ∈ LN (Ω).This is the case for example if the norm ‖f‖LN (Ω) is small, depending onthe Sobolev constant; see e.g. [16]. One can now take a minimizing sequence(un)n∈N in W1,10 (Ω) such thatlimn→∞EΦ1(un) = infW 1,10 (Ω)EΦ1 .Each function un, extended by zero to RN , is an element ofW 1,1(RN ). Since thesequence (∇un)n∈N is bounded in L1(RN ;RN ), we can extract a subsequence(∇unk)k∈N converging weakly to some finite vector-valued measure in RN sup-ported in Ω. Applying the Rellich–Kondrashov compactness theorem, we de-duce that there exists u ∈ BV (RN ) such that u = 0 in RN \ Ω, andlimk→∞EΦ1(unk) ≥∫RN|Du| −∫Ωfu.The limit function u is a minimizer of the extended functionalEΦ1(v) :=∫RN|Dv| −∫Ωfv, (5)over the class of functions v ∈ BV (RN ) such that v = 0 in RN \ Ω. Such afunctional provides a relaxed formulation of the minimization problem for whicha solution exists; see [14]. In the spirit of Theorems 1.2 and 1.3, minimizersof (5) must have flat level sets:Theorem 1.5. Let f ∈ LN (Ω) and let u ∈ BV (RN ) with u = 0 in RN \Ω be aminimizer of the extended functional EΦ1 . Then, u ∈ L∞(RN ) and the set ofextremal points {x ∈ RN : |u(x)| = ‖u‖L∞(RN )}has positive Lebesgue measure.We deduce in this case that the absolute continuous part Dau of the measureDu vanishes a.e. on a set of positive measure since Dau = 0 a.e. on level sets{u = α} for every α ∈ R [4, Proposition 3.73]. The counterpart of Theorem 1.5involving the condition ‖f‖MN (Ω) < Nω1/NN is true but uninteresting since EΦ1FLAT SOLUTIONS OF THE 1-LAPLACIAN EQUATION 45is nonnegative and 0 is the unique minimizer. This follows from a standardapplication of Alvino’s version of the Sobolev inequality in Lorentz spaces.Renormalized solutions to equations involving the 1-Laplacian have beenintroduced in the spirit of the relaxed minimization problem above, but ingeneral such solutions merely have bounded variation or do not satisfy thehomogeneous Dirichlet boundary condition [1, 5, 6, 8, 10,19].Example 1.6 (Remark 3.10 in [19]). For every N < r ≤ R, the functionu = (1−N/r)χBr(0) is a renormalized solution of the Dirichlet problem{−∆1v = h− v in BR(0),v = 0 on ∂BR(0),with bounded datum h = χBr(0). Note that if r < R then ur is a BV functionwith compact support in BR(0), while if r = R then ur is a W1,1 functionwhich does not vanish on the boundary.In the next section, we prove Theorems 1.2, 1.3 and 1.5. This paper is arevised and extended version of a note written by the authors in 2012 that wasonly available at the arxiv.org website.2. Proofs of the main resultsProof of Theorem 1.2. Assume by contradiction that there exists a functionu ∈W 1,10 (Ω) such that ∇u 6= 0 almost everywhere in Ω and f := ∆1u ∈ LN (Ω).Then, ∫Ω∇u|∇u|· ∇ϕ =∫Ωfϕ,for every ϕ ∈ C∞c (Ω). Note that ∇u/|∇u| ∈ L∞(Ω) and u ∈ LNN−1 (Ω) by theGagliardo-Nirenberg-Sobolev imbedding. By density of C∞c (Ω) in W1,10 (Ω) wededuce that ∫Ω∇u|∇u|· ∇v =∫Ωfv, (6)for every v ∈W 1,10 (Ω).We proceed using Stampacchia’s truncation method. For this purpose, forevery κ > 0 let Gκ : R→ R be the function defined byGκ(t) =t+ κ if t < −κ,0 if −κ ≤ t ≤ κ,t− κ if t > κ.(7)46 L. ORSINA AND A.C. PONCESince u ∈W 1,10 (Ω), we have Gκ(u) ∈W1,10 (Ω). Hence,∇u|∇u|· ∇Gκ(u) = G′κ(u)|∇u| = |∇Gκ(u)|.Applying identity (6) with test function Gκ(u), we get∫Ω|∇Gκ(u)| =∫ΩfGκ(u).Since Gκ vanishes on the interval [−κ, κ], by the Hölder inequality we have∫ΩfGκ(u) =∫{|u|>κ}fGκ(u) ≤ ‖f‖LN ({|u|>κ})‖Gκ(u)‖LNN−1 (Ω).Thus, ∫Ω|∇Gκ(u)| ≤ ‖f‖LN ({|u|>κ})‖Gκ(u)‖LNN−1 (Ω).By the Gagliardo-Nirenberg-Sobolev inequality,‖Gκ(u)‖LNN−1 (Ω)≤ C∫Ω|∇Gκ(u)|,for some constant C > 0 depending only on the dimension N . Hence,(1− C‖f‖LN ({|u|>κ}))‖Gκ(u)‖LNN−1 (Ω)≤ 0. (8)Let T := ‖u‖L∞(Ω) if u is essentially bounded, or T := +∞ otherwise. Wehavelimκ↗T‖f‖LN ({|u|>κ}) = ‖f‖LN ({|u|=T}).We observe that the set {|u| = T} has zero Lebesgue measure. This is indeedthe case when T = +∞ since u is finite a.e. When T < +∞, we observe that∇u = 0 a.e. on the level set {|u| = T}; since by assumption ∇u 6= 0 a.e. in Ω,the set {u = T} must have zero Lebesgue measure. This implies thatlimκ↗T‖f‖LN ({|u|>κ}) = ‖f‖LN ({|u|=T}) = 0.In particular, there exists 0 < κ < T such that C‖f‖LN ({|u|>κ}) < 1. Wededuce from (8) that‖Gκ(u)‖LNN−1 (Ω)≤ 0.Therefore, |u| ≤ κ a.e. in Ω. Hence, T = ‖u‖L∞(Ω) ≤ κ, and this contradictsthe choice of κ. The proof of the theorem is complete.FLAT SOLUTIONS OF THE 1-LAPLACIAN EQUATION 47To prove Theorem 1.3, we rely on Peetre’s imbedding of Sobolev functionsin Lorentz spaces, with the best constant computed by Alvino. We recall thatthe Lorentz space Lp,1(RN ) for 1 ≤ p < ∞ can be defined as the vector spaceof measurable functions g in RN such that‖g‖Lp,1(RN ) :=∫ ∞0|{|g| > t}|1/p dt < +∞.Using an equivalent definition to this one, Lorentz [17] established the dualitybetween Lp,1(RN ) and Mpp−1 (RN ) for p > 1 by proving an estimate whichamounts to ∫Rd|fg| ≤ ‖f‖Mpp−1 (RN )‖g‖Lp,1(RN ),for every g ∈ Lp,1(RN ) and f ∈Mpp−1 (RN ), where‖f‖Mpp−1 (RN ):= supA⊂Ω1|A|1p∫A|f | ;see [17, Theorem 5] and the computation of the Lorentz norm in [7, Section 2].Here one should not rely on the quasi-norm supt>0{t |{|f | > t}|p−1p}, whichgives a quantity that is only equivalent to ‖f‖Mpp−1 (RN ).Peetre [23] proved by interpolation that W 1,1(RN ) ⊂ LNN−1 ,1(RN ) andAlvino [2] later showed using rearrangements that the inequality‖v‖LNN−1 ,1(RN )≤ γ1‖∇v‖L1(RN )holds with the best constant given by γ1 := 1/(Nω1/NN ).Proof of Theorem 1.3. Proceeding as in the previous proof, by the duality be-tween LNN−1 ,1 and MN one gets∫RN|∇Gκ(u)| =∫ΩfGκ(u) ≤ ‖f‖MN (RN )‖Gκ(u)‖LNN−1 ,1(RN ),where the functions f and u have been extended by zero to RN ; this does notchange their seminorms. Using Alvino’s improvement of the Sobolev inequalitywith v = Gκ(u), it follows that(1− γ1‖f‖MN (RN ))‖Gκ(u)‖LNN−1 ,1(RN )≤ 0.Under the assumption of the theorem we have ‖f‖MN (RN ) < 1/γ1, hence thequantity in parenthesis is positive. We deduce that ‖Gκ(u)‖LNN−1 ,1(RN )= 0 forevery κ > 0, and then u = 0 a.e. in Ω, but this is not possible.48 L. ORSINA AND A.C. PONCEThe proof of Theorem 1.5 relies on a property of BV function related to thechain rule. For this purpose, given κ > 0 denote by Tκ : R→ R the truncationfunction at levels ±κ:Tκ(t) =−κ if t < −κ,t if −κ ≤ t ≤ κ,κ if t > κ.Observe that, for every t ∈ R,t = Tκ(t) +Gκ(t), (9)where Gκ is the function defined by (7). Since Tκ and Gκ are Lipschitz con-tinuous, it is straightforward to verify using an approximation argument thatTκ(u) and Gκ(u) both belong to BV (RN ) for every u ∈ BV (RN ). In addition,by the identity above we haveDu = D(Tκ(u)) +D(Gκ(u)).One then verifies that∫RN|Du| =∫RN|D(Tκ(u))|+∫RN|D(Gκ(u))|, (10)where, for a given vector-valued measure µ,∫RN|µ| = sup{∫RNΦ · µ : Φ ∈ C∞c (RN ;RN ) and |Φ| ≤ 1 in RN}.Indeed, the inequality ≤ in (10) follows from the triangle inequality in RN .The reverse inequality ≥ can be deduced from Vol’pert’s chain rule for BVfunctions [3]. A more elementary approach is based on an approximation of uusing the sequence of smooth functions (ρn∗u)n∈N, where (ρn)n∈N is a sequenceof mollifiers in C∞c (RN ). In this case, one observes that∫RN|D(ρn ∗ u)| →∫RN|Du|as n→∞; see [13, Theorem 5.3]. On the other hand, there exist a subsequence(ρnj ∗ u)j∈N and finite positive measures σ1 and σ2 such that|D(Tκ(ρnj ∗ u))|∗⇀ σ1 in M(RN ;RN ),|D(Gκ(ρnj ∗ u))|∗⇀ σ2 in M(RN ;RN ),as j → ∞, where σ1 ≥ |D(Tκ(u))| and σ2 ≥ |D(Gκ(u))|. This implies thereverse inequality in (10).FLAT SOLUTIONS OF THE 1-LAPLACIAN EQUATION 49Proof of Theorem 1.5. Since u minimizes EΦ1 , and Tκ(u) is also an admissiblefunction in the minimization class, we haveEΦ1(u) ≤ EΦ1(Tκ(u)).Thus, ∫RN[|Du| − |D(Tκ(u))|]≤∫RNf(u− Tκ(u)).We deduce from (10) and (9) that∫RN|D(Gκ(u))| ≤∫RNfGκ(u).We can now pursue the strategy of the proof of Theorem 1.2 to get the conclu-sion. Indeed, the Sobolev and Hölder inequalities imply that(1− C‖f‖LN ({|u|>κ}))‖Gκ(u)‖LNN−1 (Ω)≤ 0.For every 0 < κ < ‖u‖L∞(RN ), where we do not exclude the possibility that‖u‖L∞(RN ) = +∞, we have ‖Gκ(u)‖LNN−1 (Ω)> 0. Thus,‖f‖LN ({|u|>κ}) ≥1C.Since u is finite a.e., this inequality cannot hold for every κ > 0. Therefore,we must have ‖u‖L∞(RN ) < ∞ and so u is essentially bounded. Letting κ →‖u‖L∞(RN ), we deduce that {|u| > κ} has positive measure.AcknowledgementsThe authors would like to thank Francesco Petitta for bringing to their atten-tion problems related to the 1-Laplacian and Stefan Steinerberger for suggestinga counterpart of their original result in the setting of weak-LN functions. Thesecond author (ACP) was supported by the Fonds de la Recherche scientifique–FNRS under research grants 1.5.199.10F and J.0026.15. He warmly thanks theDipartimento di Matematica of “Sapienza” Università di Roma for the invita-tion and hospitality.References[1] F. Alter, V. Caselles, and A. Chambolle, A characterization of convexcalibrable sets in RN , Math. Ann. 332 (2005), no. 2, 329–366.[2] A. Alvino, Sulla diseguaglianza di Sobolev in spazi di Lorentz, Boll. Un. Mat.Ital. A (5) 14 (1977), 148–156.50 L. ORSINA AND A.C. PONCE[3] L. Ambrosio and G. Dal Maso, A general chain rule for distributional deriva-tives, Proc. Amer. Math. Soc. 108 (1990), no. 3, 691–702.[4] L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation andfree discontinuity problems, Oxford Mathematical Monographs, The ClarendonPress, Oxford University Press, New York, 2000.[5] F. Andreu, C. Ballester, V. Caselles, and J. M. Mazón, The Dirichletproblem for the total variation flow, J. Funct. Anal. 180 (2001), no. 2, 347–403.[6] F. Andreu, A. Dall’Aglio, and S. Segura de León, Bounded solutionsto the 1-Laplacian equation with a critical gradient term, Asymptot. Anal. 80(2012), no. 1-2, 21–43.[7] H. Bahouri and A. Cohen, Refined Sobolev inequalities in Lorentz spaces, J.Fourier Anal. Appl. 17 (2011), no. 4, 662–673.[8] G. Bellettini, V. Caselles, and M. Novaga, Explicit solutions of the eigen-value problem −div(Du|Du|)= u in R2, SIAM J. Math. Anal. 36 (2005), no. 4,1095–1129.[9] M. Cicalese and C. 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Papageorgiou, Minimization of nonsmoothintegral functionals, Internat. J. Math. Math. Sci. 15 (1992), no. 4, 673–679.[23] J. Peetre, Espaces d’interpolation et théorème de Soboleff, Ann. Inst. Fourier(Grenoble) 16 (1966), no. 1, 279–317.Authors’ addresses:Luigi Orsina“Sapienza” Università di RomaDipartimento di MatematicaP.le A. Moro 200185 Roma, ItalyE-mail: orsina@mat.uniroma1.itAugusto C. PonceUniversité catholique de LouvainInstitut de Recherche en Mathématique et PhysiqueChemin du cyclotron 2, bte L7.01.021348 Louvain-la-Neuve, BelgiumE-mail: Augusto.Ponce@uclouvain.beReceived December 1, 2016Accepted March 17, 2017

For every $f\inL^N(\Omega)$ defined in an open bounded subset $\Omega$ of $\mathbb{R}^N$, we prove that a solution $u\inW^{1,1}_0(\Omega)$ of the 1-Laplacian equation -div$(\frac{\nabla u}{|\nabla u|})=f$ in $\Omega$ satisfies $\nabla u=0$ on a set of positive Lebesgue measure. The same property holds if $f\not\in L^N(\Omega)$ has small norm in the Marcinkiewicz space of weak-$L^N$ functions or if $u$ is a BV minimizer of the associated energy functional. The proofs rely on Stampacchia’s truncation method

Flat solutions of the 1-Laplacian equation

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