THE TWO MEMBRANES PROBLEM FOR DIFFERENT OPERATORS 6 1 L.CAFFARELLI,D.DESILVA,ANDO.SAVIN 0 2 Abstract. Westudythetwomembranesproblemfordifferentoperators,pos- n sibly nonlocal. We prove a general result about the Hölder continuity of the a solutionsandwedevelopaviscositysolutionapproachtothisproblem. Then J weobtainC1,γ regularityofthesolutionsprovidedthattheordersofthetwo 1 operators are different. In the special case when one operator coincides with 1 thefractionalLaplacian,weobtaintheoptimalregularityandacharacteriza- tionofthefreeboundary. ] P A . h 1. Introduction t a Inthispaperwestudythetwomembranesproblemfordifferentoperators. Phys- m ically the problem consists in having two elastic membranes made of possibly dif- [ ferent composite materials that are constrained one on top of the other. This is a 1 doubleobstacleprobleminwhicheachmembranecanbeviewedasthe obstaclefor v the other membrane, and the two obstacles interact at the same time. 8 The two membranes problem for the Laplacian was first consideredby Vergara- 4 Caffarelli [VC] in the context of variational inequalities. In this case the situation 5 canbe reducedto the classicalobstacleproblemby lookingatthe verticaldistance 2 0 betweenthemembranes. Thetwomembranesproblemforanonlinearoperatorwas . studied by Silvestre [S1]. He obtained the optimal C1,1 regularity of the solutions 1 0 together with a characterizationof the regularity of the free boundary of the coin- 6 cidence set. The key step is to show that the difference between the two solutions 1 solves an obstacle problem for the linearized operator. : v We also mention that a more general version of the two membranes problem i involving N membranes was considered by several authors (see for example [ARS, X CCVC, CV]). r a The two membranes problem for different operators is more challenging mathe- matically. In the unconstrained parts the membranes solve different equations and therefore their difference solves a fourth order equation rather than a second order equation. For example even in the simplest case of two dimensions and two linear operators,say△and△˜ :=∂ +2∂ ,theoptimalregularityofthesolutionsseems xx yy to be a difficult problem. Inthispaperweconsiderthetwomembranesproblemforthelargeclassofelliptic operators, possibly nonlocal, of order 2s∈(0,2]. The interest in the nonlocal case comesfromtheapplications. ItiswellknownforexamplethattheclassicalSignorini probleminelasticitywhichconsistsinfinding the equilibriumpositionofanelastic bodyrestingonarigidsurface,ismodeledbyanobstacleproblemforthefractional Laplacian∆1/2. Inthecasewhentheelasticbodypressesagainstamembrane,one D.D.issupportedbyNSFgrantDMS-1301535. O.S.issupportedbyNSFgrantDMS-1200701. 1 2 L.CAFFARELLI,D.DESILVA,ANDO.SAVIN obtains a two membranes obstacle problem involving a fractional Laplacian and a second order operator. Inthegeneralcase,weprovearesultabouttheHo¨ldercontinuityofthesolutions and we develop a viscosity solution approach. Then we obtain better regularity properties of the solutions providedthat the orders of the two operatorsare different. Heuristically this situation corresponds to the case when one membrane, say the lowermembrane,is more sensitiveto smallinfinitesimal changes. Fromthis we can already deduce a certain initial regularity of the lower membrane. Then, the regularity of the upper membrane can be obtained by solving the obstacle problem in which the obstacle is given by the lower membrane. In order to obtain the optimal regularity we need to repeat these arguments several times. A large part of the paper is devoted to obtaining estimates for various obstacle problems which are optimal with respect to the smoothness of the obstacle. We first discuss the general case of operators that correspond to translation invariant kernels. Then we consider the special case of the fractional Laplacian. As mentioned above in the courseof the paper we alsotreat the obstacle problemfor translationinvariant kernels which is of independent interest. Thepaperisorganizedasfollows. InSection2weformulatethe twomembranes problemandstatepreciselyourresults. InSection3weobtaintheHo¨lderregularity of the minimizing pair. In Section 4 we develop the viscosity approach to the two membranesproblem. InSection5wedealwiththetranslationinvariantkernelsand finally in Section 6 we discuss the case of the fractional Laplacian. The Appendix is devoted to the proof of Schauder estimates for nonlocal equations. 2. Main results 2.1. Notation. Let s ∈ (0,1) and let k(x,y) be a symmetric, measurable kernel proportionalto |x−y|−n−2s, i.e. 0<λ≤k(x,y)|x−y|n+2s ≤Λ, k(x,y)=k(y,x). Given a function u∈L2 we define its Hs seminorm in B , the unit ball, as loc 1 1 (u(x)−u(y))2 kuk2 := dxdy, Hs(B1) 2ˆ ˆ(Rn×Rn)\(CB1×CB1) |x−y|n+2s and if kuk < ∞ we write u ∈ Hs(B ). Here for any set E ⊂ Rn, we denote Hs(B1) 1 by CE its complement in Rn. It is not difficult to check that dx (2.1) u− u ≤Ckuk , dω := . Hs(B1) 1+|x|n+2s (cid:13) B1 (cid:13)L2(Rn,dω) (cid:13) (cid:13) Given tw(cid:13)o functions(cid:13)u,v ∈ Hs(B ) we define the “inner product” of u and v (cid:13) (cid:13) 1 with respect to the kernel k as 1 (2.2) E (u,v):= (u(x)−u(y))(v(x)−v(y)) k(x,y)dxdy. k 2ˆ ˆ(Rn×Rn)\(CB1×CB1) If u minimizes the energy E (u,u) among all functions u ∈ Hs(B ) which are k 1 fixed outside B , say u=u0 ∈Hs(B ) outside B , then 1 1 1 E (u,ϕ)=0, ∀ϕ∈Hs(B ), with ϕ=0 outside B . k 1 1 THE TWO MEMBRANES PROBLEM FOR DIFFERENT OPERATORS 3 The last equality can be written in the sense of distributions as L u=0 in B , k 1 with :=−E (u,ϕ) ∀ϕ∈C∞(B ), k k 0 1 and formally L u can be written as the non-local operator k L u(x)= (u(y)−u(x))k(y,x)dy. k ˆ We wish to include the case when k has order s=1. In this case the quadratic form E (u,u) is given by k (2.3) E (u,u)= (∇u)TA(x)∇udx, A ˆ B1 with A(x) a symmetric n×n matrix satisfying λI ≤ A(x) ≤ ΛI, and the linear operator associated to E is A L (u)=div(A(x)∇u). A Finally, we notice the following scaling property of E after space dilation. Let k u˜(x)=u(rx), be the 1/r dilation of u in the space variable. Then E (u,v)=rn−2s E (u˜,v˜) k k˜ where in the double integral on the right we remove the contribution coming from CB ×CB and the kernel k˜(x,y) := rn+2sk(rx,ry) is the rescaling on k, and 1/r 1/r therefore satisfies the same growth conditions as k. 2.2. The two membranes problem – General case, Hölder continuity of the minimizers. We consider the two membranes obstacle problem in B for 1 operatorscorrespondingto twodifferentkernelsk andk asabove,with the order 1 2 s notnecessarilyequaltos . Welookforapairoffunctions(u ,u ),withu ≤u 1 2 1 2 2 1 in B and u , u prescribed outside B , which minimizes the energy functional 1 1 2 1 (2.4) F(u ,u ):=E (u ,u )+E (u ,u )+ u f +u f dx, 1 2 k1 1 1 k2 2 2 ˆ 1 1 2 2 B1 among all (u ,u )∈A. 1 2 Here f ∈L2(B ) and A represents the set of admissible pairs, i 1 A= (u ,u )| u ≤u , u ∈Hsi(B ), u =u0 outside B , 1 2 2 1 i 1 i i 1 with u0i ∈H(cid:8)si(B1), u02 ≤u01 in B1, a given pair of functions. (cid:9) With the convention in the Subsection above, we allow in the definition of the energyF alsothe caseswheneither one orboth ofthe s ’s equalto 1,and weneed i to replace the quadratic form accordingly. Since F is strictly convex, and F(u0,u0) < ∞, we obtain the existence and 1 2 uniqueness of a minimizing pair (u ,u ) by the standard methods of the calculus 1 2 of variations. Proposition 2.1. There exists a unique minimizing pair (u ,u ) ∈ A for the 1 2 functional F in (2.4). Moreover u ∈ L2(Rn,dω ) and ku k ≤ C for a i i i i L2(dωi) constant C depending on the boundary data u0 and on the f ’s. i P i 4 L.CAFFARELLI,D.DESILVA,ANDO.SAVIN We observe that to prove the L2 bound for the minimizing pair, one uses (2.1). Notice that if ϕ≥0 and ϕ∈C∞(B ) then 0 1 (u +ǫϕ,u )∈A and (u ,u −ǫϕ)∈A, 1 2 1 2 which gives (2.5) L u ≤f , L u ≥f in B , k1 1 1 k2 2 2 1 in the sense of distributions, thus L u , L u are Radon measures. k1 1 k2 2 Moreover,if ϕ∈C∞(B ) is not necessarily positive we still have 0 1 (u +ǫϕ,u +ǫϕ)∈A, 1 2 hence (2.6) L u +L u =f +f in B . k1 1 k2 2 1 2 1 Equations (2.5)-(2.6) together with the inequality u ≤ u , can be viewed as the 2 1 Euler-Lagrangecharacterizationof the minimizing pair. Inthispaperweareconcernedwiththeregularityoftheminimizingpair(u ,u ) 1 2 andsomepropertiesofthefreeboundaryΓwhichisdefinedastheboundaryofthe coincidence set, i.e. Γ:=∂{u =u }∩B . 1 2 1 Ourfirstresultisthe followinginteriorHo¨lderregularityofthe minimizing pair. Theorem 2.2. Assume fi ∈ Lqi(B1) with qi > 2nsi. Let (u1,u2) be a minimizing pair. Then u ∈Cα(B ) and i 1 ku k ≤C ku k +kf k , i Cα(B1/2) i L2(dωi) i Lqi(B1) i i X X(cid:0) (cid:1) with α and C depending on n, λ, Λ, s , q . i i To obtain better regularity properties of the minimizing pair we need to require that the kernels k are more regular, as in the next subsection. i 2.3. Translation invariant kernels – Viscosity solutions and higher regularity. We consider the case when k is translation invariant, i.e. k(x,y)=K(x−y), K(y)=K(−y), and satisfies the natural growth condition of the gradient Λ |∇K(y)|≤ . |y|n+1+2s The integro-differential operator associated to this kernel can be written as L w(x):=PV (w(y)−w(x))K(y−x)dy. K ˆ Rn and the value L w(x) is well-defined as long as w ∈L1(Rn,dω) and w is C2s+ǫ at K x. In this case we show that the minimizing pair (u ,u ) satisfies 1 2 (2.7) u ≥u , L u ≤f , L u ≥f in B , 1 2 K1 1 1 K2 2 2 1 (2.8) L u =f on {u >u }, L u = f in B , Ki i i 2 1 Ki i i 1 i i X X THE TWO MEMBRANES PROBLEM FOR DIFFERENT OPERATORS 5 in the viscosity sense, and moreover these inequalities determine uniquely the pair (u ,u ) (see Proposition 4.9). 1 2 When the orders of the operators L are different we improve the result of Ki Theorem 2.2 and obtain the C1,γ regularity of the pair (u ,u ). Notice that the 1 2 twomembranesmay interact,that is {u =u }∩B 6=∅ independently ofthe sign 1 2 1 of f , f . We obtain the following result. 1 2 Theorem2.3. Assumes 1, α =max{1,2s }+ǫ , α =α +2(s −s ) 1 1 0 2 1 2 1 and ku k ≤C ku k +kf k , i Cαi(B1/2) i L1(dωi) i C0,1(B1) i i X X(cid:0) (cid:1) with ǫ and C depending on n, λ, Λ, s . 0 i 2.4. The obstacle problem for operators with translation invariant kernels. In order to obtain Theorem 2.3 we study the obstacle problem for the operator L associated to a translation invariant kernel of order 2s. We obtain the K following result, of independent interest. Assume that u, ϕ are continuous in B , 1 u∈L1(Rn,dω), and (2.9) u≥ϕ in B , 1 (2.10) L u≤f in B , and L u=f in {u>ϕ}∩B , K 1 K 1 with K of order 2s as at the beginning of subsection 2.3. Theorem 2.4. Let u be a solution to (2.9), (2.10), and assume that kukL1(Rn,dω),kϕkCβ(B1),kfkC0,1(B1) ≤1. for some β 6=2s. Then u∈Cα(B ) for α=min{β,max{1,2s}+ǫ } and 1 0 kuk ≤C, Cα(B1/2) where ǫ depends on n, λ, Λ, s, and the constant C may depend also on β. 0 2.5. Fractional laplacian – Optimal regularity and the geometry of the free boundary. In the special case when 1 K(y)= |y|n+2s the operatorL reduces to the fractionalLaplacian∆s and we obtainthe optimal K regularity of the solution. As usual, we can characterize the points on the free boundary Γ:=∂{u=ϕ}∩B . 1 Precisely the set Σ⊂Γ of singular points consists of those y ∈Γ such that (u−ϕ)(x)=o(|x−y|1+s), and Γ\Σ is the set of regular points (or stable points) of the free boundary. 6 L.CAFFARELLI,D.DESILVA,ANDO.SAVIN Theorem 2.5. Let u be a solution to (2.9), (2.10), with kukL1(Rn,dω),kϕkCβ(B1),kfkCβ−2s(B1) ≤1, for some β >1+s. Then u∈C1+s(B ) and 1 kuk ≤C. C1+s(B1/2) Moreover, the free boundary Γ is a C1,γ surface in a neighborhood of each of its regular points. The constants C,γ depend on n, s, and β. Theorem2.5wasobtainedby Caffarelli,SalsaandSilvestrein[CSS]. The main tool in the proof is to establish a version of Almgren’s frequency formula for the “extension” of u to Rn+1. However, Theorem 2.5 is proved in [CSS] in the case when ϕ ∈ C2,1 ( i.e. β = 3). When s = 1/2, Guillen proved Theorem 2.5 in [G]. In Section 6 we show that the Almgren’s monotonicity formula still holds when β >1+s and therefore sharpen the result in [CSS] and obtain Theorem 2.5. Theorem 2.5 yields the following result for the two-membrane problem. When 1 K (y)= 1 |y|n+2s1 we obtain the optimal regularity of the minimizing pair, i.e. u1 ∈ C1,s1 and u2 ∈ C1+2s2−s1 and we can characterize the points on the free boundary Γ:=∂{u =u }∩B , 1 2 1 as in the obstacle problem. Theorem 2.6 (Optimal regularity). Assume that the hypotheses of Theorem 2.3 holdandK isasabove. ThentheconclusionofTheorem 2.3holdswithα =1+s . 1 1 1 Moreover, the set of regular points of the free boundary Γ is locally a C1,γ surface. 3. The proof of Theorem 2.2 In this section we prove the Ho¨lder regularity of the minimizing pair (u ,u ). 1 2 The parametersλ,Λ,n,s ,s arecalleduniversalandanyconstantdepending only 1 2 on these parameters is called universal as well and it is usually denoted by C,c (though it may change from line to line). Proof of Theorem 2.2. The proof follows from the standard De Giorgi iteration technique. For simplicity we sketch it for f = 0 and s < 1, since the arguments i i carry on without difficulty to the case of nonzero f ’s and when one or both oper- i ators are local. Assume that s ≥s . 2 1 Step 1. Caccioppoli inequality. Let ϕ be a cutoff function supported in B . The 1 key observation is that for ǫ<1, u +ǫϕ2u−,u +ǫϕ2u− ∈A. 1 1 2 2 Using the minimality of th(cid:0)e pair (u1,u2), we let ǫ→(cid:1) 0 and obtain (3.1) E (u ,ϕ2u−)+E (u ,ϕ2u−)≥0. k1 1 1 k2 2 2 Notice that −E (u,ϕ2u−)=E (u−,ϕ2u−)+F (u), k k k THE TWO MEMBRANES PROBLEM FOR DIFFERENT OPERATORS 7 and F (u):=−E (u+,ϕ2u−)=2 ϕ2(x)u+(x)u−(y) k(x,y)dxdy ≥0. k k ˆ ˆ We use the identity (a−b)(p2a−q2b)=(ap−bq)2−ab(p−q)2, thus E (u−,ϕ2u−)=E (ϕu−,ϕu−)−I (u) k k k with I (u)= u−(x)u−(y)(ϕ(x)−ϕ(y))2k(x,y)dxdy ≥0. k ˆ ˆ The identities above give E (ϕu−,ϕu−)+F (u)=−E (u,ϕ2u−)+I (u). k k k k Next we bound above I (u). k Assume that ϕ is the usual cutoff function with ϕ=1 in B and ϕ=0 outside r B for some r ∈(0,1−δ]. When both x and y are in B we use that r+δ/2 r+δ u−(x)u−(y)(ϕ(x)−ϕ(y))2 ≤Cδ−2[(u−(x))2+(u−(y))2]|x−y|2. When x ∈ B and y lies outside B (and symmetrically the other case), we r+δ/2 r+δ use that k(x,y)≤Cδ−n−2sω(y). Thus, we see that I (u) is bounded above by k I (u)≤Cδ−2 (u−)2dx+Cku−k δ−n−2s u−dx. k ˆ L2(dω) ˆ Br+δ Br+δ In this last inequality we used that ku−k ≤ Cku−k . We use these re- L1(dω) L2(dω) lations for u and u in the energy inequality (3.1) together with the fact that 1 2 u− ≥u− in B . We obtain the desired Caccioppoli inequality for u−: 2 1 1 2 (3.2) E (ϕu−,ϕu−)+F (u )≤C δ−n−2 [(u−)2+M u−] dx. k2 2 2 k2 2 0 ˆ 2 0 2 Br+δ with M :=ku−k +ku−k , 0 1 L2(dω1) 2 L2(dω2) and C universal. More generally if v =u +m, we have 0 m 2 (3.3) E (ϕv−,ϕv−)+F (v )≤C δ−n−2 [(v−)2+A v−] dx, k2 m m k2 m 0 ˆ m m m Br+δ and M =k(u +m)−k +k(u +m)−k . m 1 L2(dω1) 2 L2(dω2) Moreover,for all constants m≥0, M ≤M hence m 0 (3.4) E (ϕv−,ϕv−)+F (v )≤C δ−n−2 [(v−)2+M v−] dx. k2 m m k2 m 0 ˆ m 0 m Br+δ Remark 3.1. Sinceu isasubsolutionfortheL operator,v+ :=(u −m)+satisfies 2 k2 m 2 the same inequality (3.3) with the constant M replaced by k(u −m)+k . m 2 L2(dω2) Step 2. The first De Giorgi lemma. We write the first De Giorgi type lemma and provide a sketch of the proof (see also Lemma 3.1 in [CCV].) 8 L.CAFFARELLI,D.DESILVA,ANDO.SAVIN Lemma 3.2 (L∞ bound). Assume v :=u +m satisfies (3.4) for all 0≤m≤1 m 2 and some M > 0. There exists ǫ depending on the universal parameters and M 0 0 0 such that if ku−k ≤ǫ (M ), 2 L2(B1) 0 0 then u− ≤1 in B . 2 1/2 Proof. We apply (3.4), with (j ≥2) 1 m=m :=1−2−j, r =r := +2−j, δ =δ :=2−j. j j j 2 Using that F (v ) ≥ 0 together with Sobolev inequality we get (1/2∗ = 1/2− k2 mj s /n) 2 2/2∗ (3.5) (v− )2∗ ≤C δ−n−2 [(v− )2+M v− ] dx ˆBrj mj ! 0 j ˆBrj+δj mj 0 mj :=R . j Call, a := (v− )2 j ˆ mj Brj and A :={v <0}∩B . j mj rj Applying Holder’s inequality to the left-hand-side of (3.5) and using the notation above we get (3.6) aj ≤|Aj|2ns2Rj ≤|Aj|2ns2(C02Mjaj−1+M0a1j−/21|Aj|1/2), for some large M. Since on A , v <−2j, we easily obtain that j mj−1 a ≥|A |2−2j. j−1 j Thus, (3.6) gives (for some positive σ and with C¯ depending on the universal constants and M ) 0 a ≤C¯2Mja1+σ. j j−1 Standard De Giorgi iteration gives that if a is small enough (depending on C¯) 2 a →0 as j →∞ and from this we deduce our claim. (cid:3) j Our minimization problem remains invariant after multiplication with a constant. Thus, after multiplication with a small constant we may apply Lemma 3.2 and obtain the L∞ bound for u in B . 2 1/2 Step 2. The second De Giorgi lemma and the Ho¨lder continuity of u . In order 2 to obtain the Ho¨lder continuity of u , we need to iterate the next Lemma 3.3, and 2 this is point where we need s ≥s . 2 1 Notice that ingeneralthe minimization problemis not invariantafter a dilation in the space variable. Indeed, if u˜ (x)=u (ρx) then i i (u˜ ,u˜ ) minimizes the energy ρ2(s2−s1)E (u˜ ,u˜ )+E (u˜ ,u˜ ). 1 2 k1 1 1 k2 2 2 Thusifρ≤1theargumentsaboveapplyandtheCaccioppoliinequality(3.2)holds for u˜ with 2 M˜ =ρ2(s2−s1)ku˜−k +ku˜−k ≤M˜ :=ku˜−k +ku˜−k . 1 L2(dω1) 2 L2(dω2) 0 1 L2(dω1) 2 L2(dω2) THE TWO MEMBRANES PROBLEM FOR DIFFERENT OPERATORS 9 Notice also that (3.7) ku˜ikL2(Rn\B1/ρ,dωi) ∼ρsikuikL2(Rn\B1,dωi). Lemma 3.3 (Oscillation decay). Assume that u satisfies (3.3), for all constants 2 m. Suppose that for some R≥1, u ≥u in B , 1 2 R and |u2|≤1 in BR, ku−1kL2(Rn\BR,dω1) ≤µ, ku−2kL2(Rn\BR,dω2) ≤µ, with µ universal. Then in B either u ≤1−µ or u ≥−1+µ. 1 2 2 Proof. Let us assume first that |u |≤1 in Rn. 2 Assume that 1 (3.8) |{u >0}∩B |≥ |B |. 2 1 1 2 We willshowthatthereis auniversalconstantη suchthatu ≥−1+η inB . Let, 2 1 v :=2j(u +(1−2−j)), A :={v <0}∩B . j 2 j j 1 We aim to show that there is a large enough j such that (3.9) |A |≤δ j+1 0 with δ universal to be made precise later. 0 Assume by contradiction that |A |>δ j+1 0 and let us choose δ <<δ so that 0 δ 0 (3.10) |A ∩B |≥ . j+1 1−δ 2 By Caccioppoli inequality (3.2) for v we obtain j (3.11) F (v )≤Cδ−n−2 k2 j wherewehaveusedthatv− ≤1inRn,andthatu ≥u ,sothatthecorresponding j 1 2 constant M in (3.2) is bounded by a universal constant M¯. j On the other hand, F (v ):=2 ϕ2(x)v+(x)v−(y) k (x,y)dxdy ≥ k2 j ˆ ˆ j j 2 c v+(y)dy v−(x)dx≥ ˆ j ˆ j B1 Aj+1∩B1−δ c(2j −1)|A ∩B ||B |≥2jcδ . j+1 1−δ 1 0 In the third inequality above we used that 1 v− ≥ on A j 2 j+1 and (3.8). Thus, (3.10) is violated if j is large enough. Denote such j by ¯j. 10 L.CAFFARELLI,D.DESILVA,ANDO.SAVIN Now we can apply Lemma 3.2 to v¯j+1 and choose δ0 = ǫ0(2M¯) where M¯ is the universal constant that bounds all the M ’s (as observed above.) We obtain the j conclusion with η =2−(¯j+1). Now assume that |u |≤1 in B and u ≥u in B , for R≥1. Let also 2 R 1 2 R ku−i kL2(Rn\BR,dωi) ≤ǫ. Then,for ǫ smallenoughthe argumentabovestill holdsfor the fixed¯j. Indeedone can still guarantee that M¯j ≤2M¯ for ǫ small enough. Finally,if (3.8)doesnothold,thenwecanworkwiththe Caccioppoliinequality for (u −m)+ and obtain that u separates from the top (see Remark 3.1). (cid:3) 2 2 FinallywecaniterateLemma3.3andobtaintheinteriorCα Holdercontinuityof u . Indeed,afteramultiplicationbyaconstantwemayassumethatku k are 2 i L2(dωi) sufficiently small and |u | ≤1 in B . Then we perform an initial dilation of size 2 1/2 R , and we may apply Lemma 3.3. Notice that the hypotheses are satisfiedthanks 0 to (3.7). Moreover it is easy to check that the hypotheses hold for the sequence of Ho¨lder rescalings m−1 2 u (R−mx)+const, m=1,2,.. 2−µ 2 0 (cid:18) (cid:19) provided that R is chosen sufficiently large, and we may apply Lemma 3.3 indefi- 0 nitely. Step 3. The second De Giorgi lemma and the Holder continuity of u . Next we 1 obtain the Ho¨lder continuity of u by thinking that u ∈ Cα is a fixed obstacle 1 2 lying above, and u minimizes E (u ,u ) among admissible functions. 1 k1 1 1 Notice that since |u | ≤ 1 and u ≥ u we can obtain an L∞ bound for u 2 1 2 1 by applying the (standard) first De Giorgi lemma to (u −1)+. Indeed in the set 1 u >1, u solves the equation L u =0. 1 1 k1 1 The Ho¨lder continuity of u follows by iterating the following version of the 1 oscillation decay lemma. Lemma 3.4. Assume that for some R≥1 |u1|≤1 in BR, ku1kL2(Rn\BR,dω) ≤µ, oscB1u2 ≤1/4. Then in B either u ≤1−µ or u ≥−1+µ. 1 1 1 The proofofLemma3.4is a variationofthe proofabove. Indeed, ifu (0)≥−1 2 2 then the conclusion is obvious since u ≥u ≥−3. 1 2 4 If u (0) ≤ −1, we distinguish two cases. When |{u > 0}∩B | > 1/2, we use 2 2 1 1 that L u ≤0 hence we apply De Giorgitechnique to conclude that u ≥−1+µ. k1 1 1 Otherwise, since u ≤ −1 in B , u is not constrained in the set {u > 0} 2 4 1 1 1 and L u = 0 there. Again, we can apply De Giorgi technique and conclude k1 1 u ≤1−µ. (cid:3) 1 4. Translation invariant kernels and viscosity solutions In this section we investigate further properties of the minimizing pair (u ,u ) 1 2 when the kernelsk are more regular. More precisely, fromnow on we assume that i the kernel k used in the definition of the energy E in (2.2) is translation invariant k i.e. k(x,y)=K(x−y).

The two membranes problem for different operators

Similar The two membranes problem for different operators