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# 244 Class Note for MATH M0070 at Purdue

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Date Created: 02/06/15

ALMOST MONOTONICITY FORMULAS FOR ELLIPTIC AND PARABOLIC OPERATORS WITH VARIABLE COEFFICIENTS NORAYR MATEVOSYAN AND ARSHAK PETROSYAN ABSTRACT In this paper we extend the results of Ca arellieJerisoneKenig Ann of Math 2 155 2002 and Ca arellieKenig Amer J Math 120 1998 by establishing an almost monotonicity estimate for pairs of continue ous functions satisfying ui 20 Lui 271 U39U7 0 in an in nite strip global version or a nite parabolic cylinder localized version where L is a uniformly parabolic operator Lu LAybycu divAm sVu 171 s Vu CE su 7 357i with double Dini continuous A and uniformly bounded b and c We also prove the elliptic counterpart of this estimate This closes the gap between the known conditions in the literature both in the elliptic and parabolic case imposed on ui in order to obtain an almost monotonicity estimate At the end of the paper we demonstrate how to use this new almost mono tonicity formula to prove the optimal 011 regularity in a fairly general class of quasilinear obstacletype free boundary problems 1 INTRODUCTION 11 Background 111 Original Monotonicity Formulas In a seminal paper ACF84 Alt Caffarelli and Friedman have proved the following monotonicity formula if at are two con tinuous functions in the unit ball B1 in R such that uiZO AuiZO airu 0 in B1 then the functional 2 2 00 8007147117 de MM B B T4 WW2 W7 is monotone nondecreasing in 7 E 01 This formula has been of fundamental importance in the regularity theory of free boundaries especially in problems with 2000 Mathematics Subject Classi cation Primary 35R35 Secondary 35K10 35125 Key words and phrases Monotonicity formulas almost monotonicity formulas elliptic ad pare abolic equations variable coef cients divergence form eigenvalue inequalities Gaussian spaces quasilinear obstacle type problems type II superconductors N Matevosyan is partially supported by the WTF Wiener Wissenschafts Forschungs und Technologiefonds project No C106 003 and by award No KUK7117007743 made by King Abdullah University of Science and Technology KAUST A Petrosyan is supported in part by NSF grant DMSAO701015 1 2 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN two phases One of its applications is the ability to produce estimates of the type Canu0l2qui0l2 S 940 S 900 S Cnllulli2091gtHuillizmm which are crucial in establishing the optimal regularity in a number of free boundary problems The parabolic counterpart of the monotonicity formula above has been estab lished by Caffarelli Caf93 if at zs are two continuous functions in the unit strip 51 R X 710l with moderate growth at in nity and such that at 20 A785ui 20 u u 0 in 51 then the functional 0 0 Tuu 37 JR qul2Gzisdzds JR qul2Gzisdzds is monotone nondecreasingi Here Gzt is the heat kernel see more on notations in Section 21 Already Caf93 contains a localized version of this monotonicity formulai It says that if at are de ned only in a parabolic cylinder Q B1 X 71 0 then for any spatial cutoff function 1 E C8 B1 such that 0 S 1 S l illBl l the functional T T u1llu1l is almost monotone in a sense that 2 wmaeww06 www wwn 112 Almost Monotom39cz39ty Formulas In recent years several generalizations of the monotonicity formulas above has been obtained mainly motivated by increasing the range of their applicability These results share the same general trait while relaxing conditions on at they give up the full monotonicity of go or 39i39 but retain an estimate of the type in the elliptic case 8003 S C llullL2B17 lluellesl y which still has a strong potential in applications We call such results almost monotonicity formulas even though in fact no monotonicity is left at alli One notable result is the almost monotonicity formula of Caffarelli and Kenig CKQS which generalizes the parabolic monotonicity formula of Caffarelli Caf93 to the variable coef cient case if L is a uniformly parabolic operator Lu LAJmu I divAzsVu bzs Vu czsu 7 Lu with Dini continuous coef cient matrix Az s and uniformly bounded 121 s and C1 s then for a pair of continuous functions at in Q B1 gtlt 710l satisfying 1amp207 LuiZQ u39u701n Q17 we have an estimate 2 2 2 samamwscwmmm wwwmg for suf ciently small 7 where 1 E C8 B1 is a cutoff function with ill BlZ 1 A similar result can be proved also in the elliptic casei Another notable result is the following estimate of Caffarelli Jerison and Kenig CJKOQ if at are continuous functions in the unit ball satisfying uiZO Auizil uu0 in B1 ALMOST MONOTONICITY FORMULAS 3 then we have 2 much c1Mamballu4li231 for 7 E 012 Even though one only changes the condition Aui 2 0 to Aui 2 71 compared to the original monotonicity formula of ACF84 the proof of this estimate is very involved and is based on a sophisticated iteration schemei Recently a parabolic analogue of this result has been proved by Edquist and Petrosyan EPOS for a pair of functions satisfying uiZO A785ui271 uu0 in 51 A localized version of this formula for functions ui de ned only in Q has also been proved 12 Main Results The main purpose of this paper is to further extend the almost monotonicity formulas above to the pairs of functions satisfying uiZO 11271 uu0 in 51 where L is a uniformly parabolic operator with variable coef cients Theorem I Essentially we accomplish this by combining the techniques from aforementioned papers CK98CJK02 EPOS We further prove the localized version of this estimate Theorem H as well as its elliptic counterpart Theorem HI To be more speci c let L be a uniformly parabolic operator in 51 I R X 71 0 1 1 Lu LAJmu I divAzsVu bzs Vu czsu 7 Lu with the following assumptions on the coef cients there exist positive constants A M and a modulus of continuity wp such that at every I s 6 51 we have 12 AlElQ S EAzs S for any 5 E R 1 3 llA175A070ll Swl1l2l5ll2 1 4 lb175ll0175l S M Throughout the paper we will assume that the modulus of continuity wp satis es the double Dini condition 1 7 1w 10 1 1 5 M dpd39r E dp lt 00 0 T 0 P P 0 Note that a Holder modulus of continuity wT CT 0 lt a lt l readily satis es 15 In fact a somewhat weaker condition 13 g 0 L339 2 lt 00 would be suf cient for us see Remark after Proposition Sill Note that 15 follows readily from 15 since 2 39 S 2 W da 12 3 W 0T 4 Likewise let E be a uniformly elliptic operator in B1 16 u Ahcu I divAzVu 121 Vu czu 4 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN such that 17 Al l2SEAz for anyEElR 18 HAW A0ll S MW 19 lb1ll01l S M for positive constants A and u and a modulus of continuity wp satisfying 15 The main results of this paper are the following three theorems We refer to Section 21 for the notation used in the statement of these results Theorem I Parabolic Almost Monotonicity Formula Suppose we have two con tinuous functions ui 17 s in the unit strip S1 that are also in VIECS1 and satisfy uiZO Luizil uu0 in S1 with operator 11 having the properties 12715 Assume also that ui have moderate growth at in nity so that M I uizs2e71232dzds lt 00 S1 Then the functional r I r 4ArAr7 where Air I quil2 d7 5 satis es r S Cwl Mi ME2 for 0 lt r S rm Next we state the localized version of Theorem I Theorem II Localized Parabolic Almost Monotonicity Formula Suppose we have two continuous functions ui 17s in the parabolic halfcylinder Q that are also in V2Qf and satisfy uiZO Luizil uu0 in Q with operator 11 having the properties 12715 Let 1 E C8 B1 be a cuto function such that WEI2 17 SUPP C 334 De ne wi 17 s ui z7 Then the functional r I r 4ArAr7 where Air I leilQ d7 S satis es 2 M CW 1 Humin HuiHitQD for o lt r s Finally7 we state the result in the elliptic case Theorem III Elliptic Almost Monotonicity Formula Suppose we have two con tinuous functions ui 17 s in the unit ball B1 that satisfy uiZO Zuizil uu0 in B1 with operator 16 having the properties 17719 Then the functional 2 gor I r 4arar7 where air z dz ET I satis es ALMOST MONOTONICITY FORMULAS 2 900SCw1HuHi231mulling for0lti m 13 Structure of the Paper Section 2 contains notations as well preliminary results that will be used later in the paper In Section 3 we prove Theorem 1 global parabolic case The technical core of the proof is Proposition 31 followed by Proposition 32734 which allow to establish the iterative estimates in Proposition 35736 ultimately implying Theorem It In Section 4 local parabolic case we prove Theorem Hi The proof is essentially the same only we have to take into account new terms coming from the cutoff function which turn out to be exponentially smalli Section 5 establishes Theorem lll elliptic case as a direct corollary of Theorem Hi In Section 6 we give a variant of the almost monotonicity formula in local parabolic case under additional growth assumption near the origin Finally in Section 7 we give an application of the new almost monotonicity formulas in the elliptic case to a quasilinear obstacle type problem that arises in superconductivity 2i NOTATION AND PRELIMINARIES 21 Notation Throughout the paper we will use the following notations Br y E R lz 7 yl lt 7 spatial ball Br Br 0 Q Br gtlt 7720l ST R X 7720l lower parabolic cylinder in nite strip lTl24t 7 Gzt z E R t gt 0 the heat kernel 1 mm2 6 d7z s Cz is dz ds d75z Cz is dz dl d7 12 Au Zn a lu i1 Vu Vgcu a lup l l Bgcnu the standard Gaussian measure Laplacian spatial gradient For integrals in space and time we use the doubleintegral sign ff regardless of the space dimension while for the integrals in space only we use the singleintegral sign By a modulus of continuity we understand a continuous nondecreasing func tion to 000 A 0 00 such that w0 0i Zillli Convention for Constants We will use the following convention for denoting various constants that appear in this paper unless stated otherwise 6 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN on On Dimensional constants depending only on dimension n c co C CO Universal constants depending only on the structural con stants A n in l2ll4 or l7ll9 and dimension n of Of Constants depending also on f a constant or a function in addition to A n n oflek Cflyujk Constants depending on f1 H fk constants or functions in addition to A n n 22 Notion of Solution We use the notion of a weak solution of the inequality Lu 2 f in the sense of LSUGS More speci cally let 9 be an open subset of R T gt 0 QT Q X 7110 and let V2QT C 7T0L2Q L2 7T0 W1gt2Q be the Banach space with the norm Hum2m sup uzs2dz quFdzdsl 7TltSSO Q QT The corresponding space V02 QT is de ned as the closure of C80 QT in V2QTl Further we say u E V136 QT if u E V2QT for any 9 6 Q and 0 lt T lt T We say that u E V2 HT is a weak solution of Lu 2 f with f E L2QT if Vn AVu 771 Va cu 8577M dzds 2 fndzds QT QT for any 77 6 V02 QT vanishing on Q X This is equivalent to saying that Lu 2 f in Q X 7110 in the sense of distributions Lemma 21 Energy Inequality Let u E V2Qf be a weak solution of Lu 2 f in Then 2 2 2 uVQ4 c llullL2Q4 Wig4 a In the expanded form the Energy Inequality reads as sup uzs2dz qul2dzds S C u2 f2dzdsl 7915ltsgo 334 Q Q We refer to LSU68 for a proof in 23 Inequalities in Gaussian Spaces In this section we collected some inequal ities for functions in Gaussian spaces that we are going to use in this paper Recall that days Cz isdz for s lt 0 In particular dl dy lQ is the standard Gauss ian measurel We say f E W1gt2Rn days whenever f E L2 Rn days and all its distributional derivatives 8751f E L2R d75 i l n Lemma 22 LogSobolev Inequality For any u E W1gt2Rnd75 we have quogu2d75 S u2d75gt log u2d75gt 4lsl qudes D n n Rquot Rquot For a proof see Gro75ll ALMOST MONOTONICITY FORMULAS 7 Lemma 23 Poincare Inequality in Gaussian Spaces For any 1 E W1gt2Rn days with Rn vdys 0 we have v2d75 2lsl leleysl D Rquot Rquot For a proof of a generalization see Bec89ll Lemma 24 Eigenvalue Inequality in Disjoint Domains For a nonempty open set 9 C R de ne f9 lVledVS f9 f2 d7 7 where the is taken over all nonzero functions f E C0gt1Rn vanishing on R 9 Then for any two nonempty disjoints 9i be two nonempty disjoint open sets in R we have 21 mm 5a 2 2A5R1sl 5 A56 inf This inequality is originally due to Beckner Kenig and Pipher BKP98lll It is closely related to the inequality of Friedland and Hayman FH76 for eigenvalues of disjoint domains on the sphere 3i PARABOLIC FORMULA The purpose of this section is to prove Theorem ll The technical core of the proof is Proposition 31 followed by Propositions 32734 The latter provide the technical background for establishing the iterative inequalities in Propositions 3 57 36 that ultimately imply Theorem ll Sill Initial Reduction First we make few simpli cations 1 Without loss of generality we may assume that 31 A0 0 L Then the difference 32 Bzs I 7 Azs will satisfy 33gt HBWSNSwl1l2l5l12 with a double Dini w as in 15 Besides we may also assume that for any I s E 51 On 34 HBWSN S 7 2 Further we may assume 3 5 czs 0 Indeed if uz s satis es u 2 0 Abcu 2 1 1Since BKPQS is not published we refer to Section 24 in CKQS for the proof of 21 8 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN then Abo u178 6quot ILAb0u173 u178l 2 e MLAJmu 2 7e7 2 ill 3 Finally Without loss of generality we may assume Mi S 1 and prove that Tuu S Cw for 7 S ml In the general case we may replace ui With it uil Mi and use that T 11 11 T 11 111 M2 l M2 to arrive at the conclusion of Theorem I 32 Main Estimates Proposition 31 ch EP08 Propositions 1 1 and 21 Let u 6 C51 V1 CS391 satisfy 1120 LAquZil inSli uzs2e71232dzds S If 51 12 licmm Nam coma lt seems ST Rquot 1177T22dv r2 Suppose also Then for my 0 lt 7 S TM where 97quot CTwT12 OT wan Remark The double Dini condition 15 on wT is to ensures that 97quot above satis es the Dini condition f0167 7 d7 lt 00 Indeed recall that 15 implies 1 5 and notice that the Dini integrability of any function 0T is equivalent to that of 00quot for any a gt 0 since f010397 7 d7 a 01UTD Td7x Also note that in fact the term wT12 is super uous in the formula for 9T in the sense that 97quot can be replaced With 7 2 1 4 2 12 197 CT 0 dp Indeed modifying slightly the proof splitting Rn f x lt s 14 f x gt s 14 one may replace 97quot by 904 Which is easily majorized by Cn1939ri 12 ALMOST MONOTONICITY FORMULAS 9 Proof of Proposition 31 Because of the assumption A00 I we View L as a perturbation of the heat operator A 7 85 and write 36 A 7 85u LAqu divBz 3Vu 7 121 3 Vu 37 A 7 85u22 u Ab0u divBVu 7 qu qul2 2 7u u divBVu 7 uqu qulQ in the sense of distributions Thus the latter inequality means that for any non negative 77 6 C80 51 and 71 lt 32 lt 31 S 0 we have 1 51 l 5 u2ltA Bstzds 7 E u2ltslgtnltslgt 7 u2lts2gtnlts2gtdz 52 n Ru 51 2 71m 7 VunBVu 7 7711qu anu dzdsi 52 W Note also that one can actually take nzs Cz7s in the formula above because of the growth assumption that we impose on u as well as the energy inequality Lemma 21 The same argument justi es also the formal integration by parts in space variables that we are going to use throughout the proof Now the inequality 37 implies that STqul2dv gAfmasxuadwA dv 7 STudivBVud77 STuqudv 1 1 ill I213 14 The rest of the proof consists in a careful estimation of each of the integrals 11 12 13 14 Before starting the estimations we introduce the following convention for the modulus of continuity 67 Z we let T2 1 4 2 12 where C gt 0 is a generic universal constant that may change due course This convention allows to use inequalities of the type 927quot S 9T 07 90 S 9T for 7 lt up Thus the rst inequality assumes that 67 S l for 7 lt TM while the second one should be understood in the sense that 07 6017 S 902 1 To estimate 11 we integrate by parts and use that A 85Gz7s 0 in s lt 0 11 7ST 85u2Gz7sdzds AM u739r22deff2 7 u002 S Rn u 7T22d7 T2i 10 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN 2 To estimate 2 consider the weighted averages of u7 s 7213 usd75 uzsGz7sdzi Rquot Rquot Then for 7T2 S 32 S 31 S 07 using 36 and the equality A 85Gz 7s 0 in s lt 07 we have 77151 m82 7 51 Rn A 7 95uGz7 7Sdzd3 52 51 S l 7 diVBVu quGz7 7sdzds 52 Rquot 51 31 7 32 BVuVCz 7s quGz 78dzds 52 Rquot lt81 7 82gt 51 Em H2ltsgtgtdsy 2 where E2s BVuVCz 7sdz H2s quGz7sdzi Rquot Rquot We next estimate E2s and H2si 220 To estimate E2s7 split it into two parts 1923 zE213E223 msz wasp4 Using the equality VGI78 Gz7s and applying the CauchySchwarz inequality7 we obtain M lt C Wu 7G 173 dz mzmwl lglsl 12 m2 12 SC Vu2d 5 G177sdz lt Rquot l l l wasp4 4l5l2 12 SCe COlsllZ WFW Ru 2 where we have used that MESH4 gGQ 7sdz S Cn672col5l12 Further7 E213 V BVuVCz7s mzmw lt 14 V w sl Rnl ulgs S Ml4 N14247 12 0z 73 1 Now7 note that W R TGI7SdI 721 I VGI78 2RndivzG 2n 3 lE22sl BVuVCz 7s C177s WSW4 W l ALMOST MONOTONICITY FORMULAS 11 Therefore WWI4 2 5 12 lEggsl S CnW AM qul d7 Thus we obtain that wow4 2 5 2 E2ltsgtcnltc W2 W qu For H2s we obtain 12 was s M vulams c lt WWW IRquot IRquot We continue the estimation of Integrating the estimates for lEgsl and lH2sl in 2ii and applying the CauchySchwarz inequality we obtain for 7T2 S s S 0 0 14 12 7213 S m7r2 7 2 On ltC qul2d75gt d3 2 s Ru 77 3 mm r2 090 qul2d7gt12i Integrating one more time gives 12 udy msds g T272442 r4 Omen qul2d7gt 12 S T2m739r2 7 4 Cn l 4 67 2 quFdV ST S T2m739r2 Cn39r4 97quot quleyi ST That is 12 S T2m739r2 CnT4 97quot qude ST 3 We now estimate 13 7 udiVBVud7 VuBVuG uBVuVC 131 132 T ST T 320 Repeating the arguments as before we obtain Hgll S wlsl14qul2GzisC quFG lTlSl5l14 MENU WV Wude 060 ST where in the estimation of the second integral over we have used that Cz7s S eicolsllZeimzg for 2 lsl14 and the following corollary from the Energy Inequality Lemma 2 1 quFe xggdzds g C u2e E215dzds g CM C ST 51 12 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN for 7 S 1 recall the assumption M 1i To estimate 32 we introduce vzs uz7 s 7 7213 and split the integral into 132 UBVUVG msBVuVC 1321 1322 ST ST Notice that V1 Vui 11 1321 is estimated as in CK987 p404 Let E3s UBV UVG177S 7 UBVUGI78 1Rquot Rn 1 E318 E328 WEN 1 WSW4 Lao Applying CauchySchwarz and then Poincare inequalities7 we obtain 12 m2 12 E31sSClt v2Gz7sgt vvvjaa s 1Rquot l1l2 5 14 M 12 m2 12 Sow2 vvraws vvriaws Rn wasp4 l3l2 SCne COlsllZ leFe gczgdz where7 we have used that Cz is S Cnlsl nQeimz4 for any 13 6 5391 2 Gzis S Cne COlsllZeimz8 for 2 lsl14771lt s lt 0 3 But then 0 E31sdsCeCOT quFe EZgdzdsSCe COT 2 ST 77 where we have used the Energy Inequality7 as in Step 3i 31160 Further Es2ltsgt wow4 Rn v2 75012 Rn vvramrs 12 anls14 vvraws Rn provided we use following clairni Claim vQ d 5 lt C leFd 5 R 8 2 7 7 n Ru 7 ALMOST MONOTONICITY FORMULAS 13 Proof We have 2 7 2 v Cz s 7 2 v zVGz s lsl 1Rquot Qn v2Gz7s4 vszGz7s n W S 2n v2Gz7s 4 v2Gzisgtl2 3 vvFGa 7801 Then we use the Poincare inequality Lemma 2 to nish the proof D Collecting the estimates for E31s and E32s integrating and using that e cT S CT4 we obtain mm 3 CT4anT12 qum ST 17 Further write 1322 02 mSE2sds 7r and recall the estimates from 2ii 72iiioz mltsgt mm r 0760 ltST mm W for 77 lt s S 0 The proof of the estimate above contains also the following inequality 0 12 lE2slds S LL90 qul2d7gt 7T2 S Combining these estimates and using the CauchySchwarz again we obtain uml 090 quleV 07360 qul2d7gt 12 Cnme wo lt S T qul2d7gt 12 S Cn39r4 Cnm7r22t97quot Cn67 2 90 quley 2 Thus we have Hgggl S 07 4 Cnt97 m739r22 LL90 quleVi ST Collecting the estimates in 3ii73iii we also obtain Hgl S 07 4 Cnt939rm7r22 LL90 quleVi Sr 14 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN 4 Finally we estimate I4 7 uquGz7sdzdsl ST We use approach similar to the estimation of 13 Let vz s uzs 7 7218 as before and split I4 7 vavG 7 msvaG 1141 142 5 s 420 Using CauchySchwarz and Poincare inequalities we obtain m 94 71vade g M Way2 lel2d7gtU g CTAT wva CTAT qum 42392390 Further we have 0 12 Hal S M 7213 qul2d75gt d3 7T2 Rn 12 S M sup msr qul2d7gt 424 ST Recalling the estimate mltsgt mm r 0760 ltST mm W for 7T2 lt s S 0 we therefore obtain um M W72 T3 4247 12 owe quFdV S Cm7r22r 07 4 07 2 7 67 2 qule7 ST 2 g CmHWT 04 90 qum Thus collecting the estimates in 4li 74lii we conclude m 3 04 9Tm77quot22 90 qum 5 Combining all the estimates in 10740 we arrive atT WWW 1 l 0560 ult7r2gt2dwg 04 s Rquot T2m739r2 LL90 ST quleVl ALMOST MONOTONICITY FORMULAS 15 Dividing by 1 Cnt9T7 and assuming that T lt 7 is small enough7 we obtain 17 Cn6T quFdV S u7T22defr2 CT4 CnT2m7T2i ST 1Rquot Finally7 applying the Holder inequality mm Anult42gt2d gtlQ we complete the proof of Proposition Sill D We will also need the following simple corollary from Proposition Sill Proposition 31 fu is as in Proposition 31 then foT 0 lt T lt TM we also have quFdV S COT4 C0 inf us2d75 ST SE747 277 2 Rn C quleV S COT4 quVi D ST T SWST The proof of the next two propositions is based on the key Proposition 31 combined with the logSoboleV inequality see Lemma 22 It is not much different from the case when L A 7 85 detailed proof of which can be found in EPOS Therefore the proofs are omitted Proposition 32 cf EPOS7 Propositions 1 2 and 22 Let uz7 s be as in PTopo sitioTL 31 and Q I u gt 0 Suppose quFdV 1T4 lt 00 Q ST 4 1T qule7 2 7 msT4 256 foT some 0 lt T S TM Then 9 ST2 ST4l 2 607 2 gt 07 pTovided a gt 10 foT su cieutly laTge a0 HeTe 7E d7 foT E C R X 7000 D Proposition 33 cf EPOS7 Propositions 1 3 and 23 Let uz7 s be as in PTopo sitioTL 31 and Q I u gt 0 Suppose quleV 1T4 lt 007 Q ST foT some 0 lt T S TM Suppose also theTe exists 7 gt 0 such that l9 7 Sr2 SrM lt 1 MST2 Sr4l Then theTe exists 6 Bquot lt 1 such that quFdVS quleV Q STA msT pTovided a gt 10 foT su cieutly laTge 10 D and 16 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN In the next proposition we let Am ammo go Ma 0 P for a universal constant co gt 0 to be determined and T T74fl T14l Notice that since we assume the double Dini condition on wT 9T satis es the Dini condition and therefore gT is nite and converges to 0 as T A 0l Proposition 34 cf EP08 Propositions 1 4 and 24 Let at be as in TheoTem I with Mi S l and At 39i39 as de ned above Then theTe exists a univeTsal constant C0 such that Aim 2 COT4 foT all p E TT 0 lt T S TM then 1 1 VAN VAN imp 2 700T 03 foT all p E TT RemaTk We may replace fli by Ai in this proposition since the factor e009 is bounded away from 0 and 00 Yet we must take the derivative of 39i39 to compensate for having 9T in Proposition Sill PToof We start with the same remark as in the proof of Lemma 24 in CJKOZ The functions Ai are continuous nondecreasing functions hence quot is the sum of a nonnegative singular measure and an absolutely continuous part and we need to obtain the bound on quot at the points T that are Lebesgue points for the integrands of Ail Thus we assume that T is such that Bio R NunHardy lt co and that A T 2TBi We then have 20090 7 g 2T 2T lt74 20090 2T27EE 2T2 Next by Proposition 31 we have 12 2lt1ecn6ltrgtgtAltTgtscr4Cr2 mwrw meow Rquot Rquot for a universal constant C Before we proceed observe that u7T2 cannot vanish identically on R if the constant C0 in the statement of Proposition 34 is suf ciently large lndeed otherwise we would have AltTgt S CT4 from Propo sition 31 which would be a contradiction Similarly u 7T2 cannot vanish identically on Rnl Then 2 9 TM742 gt 0 ALMOST MONOTONICITY FORMULAS 17 are nonempty If we now denote Ai T2A T2 5 the normalized eigenvalues of 9 as de ned in Lemma 24 we will have 38 A A 21 Next by the de nition of Ai we have A1 f2 w s r2 1W1 w 9 2 932 Combining this with Proposition 31 we obtain 39 217 Cn6TAiT S CT4 CT3BiTi TQBiTAii T To complete the proof we consider the following possibilities 1 T2B T 2 2AT or T2B T 2 2A Then T 1 23 23 7 7 gt i 0 T lt 4 2009T 2T 2T S 0 A A 2 T2B7 S 2AT and A 2 1 or T2B T S 2AT and A 2 1 Then by 39 if A7 2 COT4 is suf ciently large we have 217 Cn6TA T S CT2A T T2BTi It follows then M gt E 42coeltrgt WEN 0 7 T AT 2m 1 ZCT 742509T474cn6r 7 7 2 AT Tlt H gt AT if we choose co 2 2 3 TQBi T S 2Ai7 and Ai S 1 Then by 39 if Ai7 2 COT4 are suf ciently large we have 2Ai17 Cn6TAiT S CT2Ai7 TQBiTi If we use now that A A 2 1 and choose co 2 Zen then lt30 1 C T 33 T23 lt30 Ult WM AM2 AW 2 1 1 25090 4m MO 7 67W 02 m T 1 1 7CT 7 7 2 1n the same way we prove the estimate for 3930 for any p E TT and the proof is complete D 18 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN As we mentioned before7 the propositions above constitute the technical core of the proof of Theorem I The rest of the proof is purely arithmetic in nature and is exactly the same as in CJKOZ We consider a geometric sequence of radii r 4 16 k 012 and de ne Aquot Aiud a 44 where flir ecogrAi r are as in Propositions 34 so that we have 446 44mm Am Ag One may also treat as the correctly rescaled versions of because of the the following property if we consider the parabolic scaling uTz7 s I r 2urzr2s7 then A1ur quTde r74 quleV r 4Aru 51 ST The proofs of the next two propositions are the same as those of CJKOZ7 Lem mas 28297 based on Propositions 32734 instead of CJKOZ7 Lemmas 2123 and therefore are omitted Proposition 35 cf EPOS7 Propositions 15 and 25 Let uibe as in Theorem I with Mi S 1 There exists a universal constant C0 such that I 2 Co for k 2 km then C0 C0 7 xbi xb Proposition 36 cf EPOS7 Propositions 16 and 26 Let ui be as in Theorem I with Mi S 1 There exists a universal constant 50 gt 0 such that I 2 C0 and 441211 2 for k 2 km then ALA S 17 80A D 44A k1 D Ag g Am 1 6k with 6k Proof of Theorem I Recalling the initial reduction steps in Subsection 317 we need to show that ruu S Cw for r S rm Now7 Propositions 35736 provide the same iteration scheme starting from k kw as in CJKOZ which implies that lt1gtT uu g Cw 1 A444 Amalie T g 4 Finally7 note that by Proposition 31 Ai 44 S C7 which completes the proof of the theorem D 4 LOCALIZED PARABOLIC FORMULA In this section we prove Theorem H The proof is essentially the same as in the global case7 ie Theorem L but has to take into account additional error terms coming from the cutoff function Those new error terms7 as we will see7 are actually exponentially small7 since they are basically the integrals over the tails of Gaussian function Cz for 2 12 Proposition 41 cf Proposition 31 Let u E CQl V2Q1 satisfy u207 Ab0u2 1 in Q ALMOST MONOTONICITY FORMULAS 19 uzs2dzds S 1 Q Let 1 E C B1 be such that 0 lt 1 S1 MEI2 1 and de ne wzs uzs1Jz7 13 6 Suppose also Then 2 12 17 an 9T lede S C T4 CnT2 w77 22dey T gt ST W l w397T22d77T2 for my 0 lt 7 S TM where 7 2 1 4 2 12 97quot CTwT12 0 wan i Proof Let wz7 s Then for L LAN we have 41 A 7 85w Lu divBVw 7 bVw u uL LJ 2V AVu divBVw 7 bVw 2 71 divBVw 7 bVw u 2V12AVu 42 A 7 85w22 2 w7l divBVw 7 bVw uL LJ 2V AVul lel2 2 7w w divBVw 7 waw wuLiJ 2wV AVu leF in the sense of distributions Then 42 implies lelev STm 7asgtltw2gtdvAdev7Adeivltvagtdv ST wawd7ATwu 2wV AVWd7 1 1 i1112131415 The terms 11 i i i 7 I4 are estimated very similarly to the global case7 however7 there are some minor differences and we prefer to provide more details The term I5 is new7 but we are going to see that its contribution is exponentially smalll 1 Exactly as in the global case7 we have IlS w7T22d77T2l 2 To estimate 2 consider the weighted averages of w7 s 7213 Rnwsd75 nwzsGz7sdzl 20 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN Then for 7T2 S 32 S 31 S 0 using 41 and that A 85Gz 7s 0 we have 51 m31 7m32 7 A 7 85wzGz7s 52 Rquot 51 S l 7 diVBVw bVw 7 uL LJ 7 2V AVuG 52 n 51 7 31 7 32 7 BVwVG bVwG 7 um 7 2V AVugtG 52 Rquot 7 817 82gt 51 Em H2ltsgt mews where E2s I AM BVwVGz 7s H2s I AM bVwGz7s K2s I 7 Rnu 1J 2V AVuGz 7s 220 Arguing exactly as in the global case we obtain wow4 2 5 12 lE2sl SCn ltCW Ranwl d7 12 H2ltsgt0lt WWW 7 R Further to estimate K2 3 integrate by parts the rst term K2s 7 u diVAV ubV 2V AVuG Rn VuAVibG uAVibVG 7 udeJG 7 2V AVuG Ru and note that the cutoff function 1 appears in the latter integral only in the form of V112 which vanishes on Blgi Therefore using the Energy Inequality and the fact that Cz 7s and lVGz 7sl are bounded above by e COlsl for 2 12 we easily obtain that K2 8 S C e COlsli l w We continue the estimation ofmsi Using the estimates on lEg sl lH2sl and lK2sl in 2ii and applying the CauchySchwarz inequality for 7T2 S s S 0 we obtain 0 14 12 7213 S m739r2 01172 On ltC Rn lel2d75gt d3 7T2 S m7T2 CWZ 090 lel2d7gt12i ALMOST MONOTONICITY FORMULAS 21 Integrating one more time gives 0 12 I2 wdy msds S T2m739r2 C T4 Cnr2t97quot leleey ST 42 ST S T2m739r2 C T4 Cn39r4 67 2 lele7 ST S T2m739r2 C T4 97quot lede ST That is7 I2 S T2m739r2 C T4 97quot leleyi ST 3 We now estimate 13 7 wdiVBVwd7 VwBVwG wBVwVG ST ST ST 1 131 132 320 Repeating the arguments as before7 we obtain Igllg wlsl14lel2GIisC vwFG wasp4 wasp4 SwT12 lel2d7C e C T7 ST where in the estimation of the second integral over we have used that C177s S e COlsllZ7 for lel14 lelezds S C11 u2dzds S C11 ST 1 for 7 S 12 recall the assumption M l by the Energy Inequality Lemma 2 1 and that To estimate 32 we introduce 111 3 wzs 7 7218 and split the integral into 132 UBVUVG msBVwVG 1321 1322 ST ST Notice that V1 Vwi 3110 Repeating the arguments in the global case7 we estimate l1321l S C T4 CnWT12 lvledV ST 17 Further7 to estimate 13227 we write it as 1322 02msE2sdsi 7r Arguing as in the global case7 we obtain from 2ii that lE2slds g 0mm WWW 127 22 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN which combined with the uniform bound from 2iii mltsgt mm Ow 0760 ltS lel2d7gt 12 for 7T2 lt s S 0 gives T Hgggl S C T4 Cnt97 m739r22 LL90 leleyi Now collecting the estimates in 3ii73iii we also obtain T llgl S C T4 Cnt97 m739r22 LL90 leleyi Sr 4 Next we estimate I4 7 wawGz7sdzdsi ST As before we denote vz s wz s 7 7213 and split the integral I4 7 vavG 7 msbVUG 1141 142 T 5 Same arguments as in the global case prove that um 3 CT lede ST Hal S Cm77quot2239r C T4 90 lede Sr which implies that 14 S C T4 9Tm77quot22 67 leleyi 5 Finally we estimate I5 wuLiJ 2wV AVud7i ST Integrating by parts in space we obtain I5 wu divAV1JG wubViJG 2wV AVuG ST 7unAV1JG 7 quAVdJG 7 quViJVG wubViJG 2wV AVuGi ST Every term in the last integrand contains V112 which vanishes on Blgi Then using the Energy Inequality and the estimate on G and lVGl for 2 12 we easily obtain that l15l 0500 6 Combining all the estimates in 10750 and using that m T22 S n w 7T22d7 T2 ALMOST MONOTONICITY FORMULAS 23 we arrive at leley S Uri n60 w7r22deff2 LN4 ST Rquot r2m7r2 090 S wme Dividing by 1 Cn67 and assuming that r lt rw is small enough we obtain Oia UDAlVMWWS AnMw45Wf waQ mP t as claimed This completes the proof of Proposition 41 D Proof of Theorem U Using Proposition 41 one may prove the analogues of Propo sitions 32736 with obvious changes possibly adding the dependence of constants on But then we argue as in the proof of Theorem I to complete the proof D 5 ELLIPTIC FORMULA In this section we prove Theorem llli Even though it possible to give a direct proof by working on spheres as in CJKOZ instead of Gaussian spaces we prefer to obtain the elliptic almost monotonicity formula from the localized parabolic one Theorem 11 Proof of Theorem I Let ui satisfy the assumptions of Theorem llli Adding a dummy variable 8 by setting ummaweq we see that ii satisfy also the assumptions of Theorem H with the operator Lu Z 7 85M divAIVu bzVu czu 7 Lu Moreover we claim that 51 arui S CnAir1Mli r lt 12 or in the expanded form WWW 2 BT I ni2 dz S CnST Cz7sdzdsi Since 1 1 on Blg it is enough to show that 7 2 L2 S Cztdt for z 6 Br lxln 0 Making a substitution t M27 we obtain 7 2 7 2 Cz tdt Cn t 2e lxl24 dt 0 0 C mz C 1 n 7n2eii47df 2 n 7 rmed47 0 W W 0 Cn M714 Hence 51 followsi Consequently we obtain that r 11 11 S Cn r 24 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN Fixing a cutoff function 1 and applying Theorem H we obtain 2 Tvud ui S Cw1llalliZQfllaill 12ltQfgtgt 2 ct 1 Hulli231 H Maw for 7 lt Twi The theorem is proved D 6 A VARIANT OF THE FORMULA Under assumptions on the growth of functions ui near the origin following CJK02 Theorem 38 it is possible to prove versions of Theorems lilll where T and 907 retain more monotonicity properties eg that the limit 0 existsi Here we state only the result in the localized parabolic case We also assume that we are in the normalized case 313 3 as well as under the assumption 5 0 which does not limit the generality Theorem IV Almost Monotonicity Formula with Growth Assumption Let ui it wi Ali and 39i39 be as in Theorem U Assume additionally that uglys S 0lrl2 MW far 178 6 Q for a Dihi modulus of continuity 0T so that fol dp lt Then NT S 1 a lgt CanawaWL 0 lt T S S Tan 7 1 2 T 00312 T 90 017 7co7 0397 0 poJrO Tap 7 2 14 2 12 97quot C0TwT12 Map 0 P and M llullL2Q lluillLHQf where Remark It is easy to see that the inequality above implies the existence of the limit OJr limrnoar T Proof As before without loss of generality we may assume M S 1 otherwise we could consider uil To simplify notations in this proof we are going to deviate from our convention for constants and denote by C a generic constant depending it a w in addition to n and the structural constants in 1 271i4 which we would normally denote Cw wi Notice that if T 0 then the estimate in the theorem is trivially satisfied Therefore we will assume that T gt 0 Further as in the proof of Proposition 34 we may assume 7 to be a Lebesgue point for Bi which yields that im Mm Bio Rn vwiwrw lt oo Next as in Proposition 34 we introduce Am ecwWAim gm T ifbp 0quot T 4A HA 7 e2009l 7quoti ALMOST MONOTONICITY FORMULAS 25 Then we have quot 6 1 1 T BM Bio 0 2T Altrgt Am 71 S C T T2BT T2B7T Vlt LN Am Aim 4 20090 7 27 To proceed we are going to assume that 0012 2 Ti This does not limit the generality since we can replace 0p with 0p p2 without affecting the form of a39r in the statement of the theoremi 1 We now claim that the additional growth assumption on u implies that 62 Ai39r S 007 122 7 lt mi lndeed let w be either 10 or 107 Then by Proposition 41 we have leley S 07 4 C w2z 7T2GIT2d1 ST Rquot 07 4 C w2Gz 7T2d1 C w2Gz 7T2d1 lxlltT12 lElgtT12 S 07 4 007 122 0670 T S CUT122i This implies 62 2 We next claim that satis es 0 T12 2 a T12 63 Tgt70 70 M 0ltTltm To this end let 9 wi 7T2 gt 0 Denoting by Ai TQA Tg 5 the normalized eigenvalues of 9 as de ned in Lemma 24 we have 64 A7 21 Now by the de nition of Ai we also have Al 21 s r2 ZlVdev Q a i and therefore using Proposition 41 we obtain that 65 217 Cn6TAiT S 07 4 CT3BTi T2BiTii We may rewrite the preVious inequality as 66 3 2 217 Eng 7 07 7 0T3 AiT ART ART Using 66 we next obtain estimates on 3930 by considering three possibilities 220 T2B 7 2 214470 or 7 2B7 7 2 2A7 Then from 61 we have quot 7 bT g lt74 20090 272 23 gt 0 26 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN T2B7 S 2AT and A7 2 1 or 7 2B77 S 2A7T and A7 2 1 Then by 66 we have B CT4 CT2 2 7 gt 7 7 7 7 T S 21 Cn6T A Then assuming co gt Zen from 61 we obtain 7 2amp0 B B7 7 27 7 7 27 39i39 T 7 T T A 2 5090 T A7 CT T4 7 2 Z i i T A7 AJr 20090 C6009 T 2 ce7A ltgt T T T 0T122 0012 gt 7 7 7 77 i S C T C T T 7 2Bi T S 2A7 T and A7 S 1 Then by 66 we have 4 2 T2 2 2Ai 1 CT CT 7 Cn67 7 Ti 7 Using now that A7 7 2 1 and assuming co 2 Zen from 61 we obtain T 7 72 7 2 7 509TA 7 T2 7 2 7 5097A7l gt 70 i 72 4 7 T 7 2 A7 m Z l C 2cogT gt ii7im7A47 T 700T122 7 Thus we see that the inequality 63 holds in all cases 066090 0 NAT Z for T lt Twi 3 We next Claim that 67 iw 7 002712 7 0017 2 0 0 lt T lt m where 7 0 12 7 0122 mWA w mA7ELw lndeed 67 is equivalent to 2 2 12 MT 2 700 7 2037 T 7 0027 which follows easily from 63 ALMOST MONOTONICITY FORMULAS 27 4 Taking now 0 lt 7 S p S TM we obtain M 3 0 0020 12 0010 02 Cam 2 Camp S lt30 CWWW 010 Note that 20 27 S 2 UT1201T S 0012 010quot and therefore introducing ampT I 0012 017quot M 06m Squaring and using that a b2 S a21 ampp b2l 1ampp with a i p and b Campp we obtain we have ltigtltrgt 1ampltpgtltigtltpgt cap 0 lt T s p g Now recalling that e2009r r and using that 6260907 S 1 Cogp for 0 lt p S TM provided TM is so small that 90 lt l we arrive at lt1gtTS 1 1 00507 CoQlt gtllt1gtlt gt 0503 This implies the theorem with ap l C0ampp Cogp which clearly has the required formi D 7 AN APPLICATION The almost monotonicity formulas proved in the previous sections can be applied to various free boundary problems such as the ones considered in CJKOQ with more general assumptions on the governing operator In this section however we give an application of the almost monotonicity formula to a quasilinear obstacle type problem 71 divaqul2Vu fzuVuXQ 72 qul 0 on Q in a certain domain D in R where Q is an apriori unknown open set This kind of free boundary problems appear eg in mean eld models describing type H superconductors see BBC94li If the operator is uniformly elliptic and f is bounded then from the general theory of quasilinear equations 71 alone would imply that u 6 qt D for some 0 lt B lt 1 However to study the ner regularity properties of the free boundary F 89 N D one in fact will need the optimal Cielregularity of u see CSS04 where a simpli ed equation with a E 1 is considered 28 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN We will make the following assumptions on the functions a R A R and f z D X R X R A R 73gt a e 01 0 00gt lot 74 aqaq 2a qq 6 A0 lAO for any 4 2 0 with A0 gt 07 75 lVgcfl lefl S M uniformly for 1210 6 D X R X R Note that 74 is the uniform ellipticity condition on the quasilinear operator diVaqul2Vu7 while 75 means that f is uniformly bounded and Lipschitz continuous with respect to all its variables Theorem 71 Let u E W1gt2 B1 L B1 be a weak distributional solution of 7172 with assumptions 7375 Then u E Clt clB1 and llull01 1Bl2 S C 632704771va llulleBlgt with Ca Hallow0RltnA0MHuHm31gt39 This theorem generalizes that of Shahgholian ShaOS for equations of the type Au fzum7 qul 0 on 90 See also the work of Uraltseva UraOl for a similar result in a twophase membrane problem We explicitly remark here that the result in Theorem 71 is new even in the case a E 1 when fzuVu depends nontriVially on I u7 Wu The key idea and the connection with the almost monotonicity formulas is seen in the following lemma Lemma 72 Let u be as in Theorem 71 Then for any direction e the functions wi LZui maxi eu0 satisfy wi 2 07 diVAzVwi bzVui czwi 2 7M w u 07 where Az aquzl2I 2aquzl2Vuz Vuz7 171 foXIquLVuWDy 01 azfXIquLVuWN Remark Note that from equation 71 we have u E C11 5B1 and therefore A E o 0043B17 so the double Dini continuity condition on A is satis ed Also7 the loc condition 75 implies the uniform boundedness of b and c Furthermore7 since the exponent and llullcl Bl2 depend only on n7 A0 M and lluHLngl the structural constants in 12715 depend only on the latter constants and the OLD norm of a on 07 HVuHinBgA Proof of Lemma 72 The idea of the proof is as follows in the open set 9 Beu gt 0 C 9 we may differentiate the equation to obtain 76 Be diVaqul2Vu diV aqul2VBeu Qaqul2VuVBeuVu emf Beuazf Vaeuvpf7 which implies that 77 LAM Beu eVgcf 2 7M in ALMOST MONOTONICITY FORMULAS 29 Then by using Katols inequality Kat72 we conclude that 78 LAbcBeu 2 7M in D In fact 7 677i7 is quite easy to justify in the sense of distributions and we therefore have for w 8877 7AIva77 121an czw77 2 7M 77 D D for any 77 E W3 2Q 77 2 0 To justify 78 we choose w 77 X where 7 6 C80 D 7 2 0 and X E C00 R is such that Xt20 xt0fort l xtlfort22i Then we have AIVwVw x g 7 x 3 imzwwvw bltzgtvww ammo D ziMng zeMDw Now using that AzVwVw 2 0 X 2 0 7 2 0 we may throw away the rst term in the above integral to obtain D X 7AIva b1vw 2 MD Noticing that xw5 A XQ are as 5 A 0 and using the dominated convergence theorem we obtain that 7Altzgtvwvw bltzgtvww cltzgtww 2 7M w 9 D which is equivalent to LAbCw 2 7Mi The proof of the lemma is complete D Proof of Theorem 71 First without loss of generality we assume that lluHLngl S 1 otherwise we replace u with ul Hulle 31 which satis es an equation of the type 71 with rescaled a and Next from the CalderonZygmund estimates it follows that u E WlifB1 for any 1 lt p lt 00 Fixing a p gt n this implies that at any Lebesgue point 10 of D277 the function u is twice differentiable see eg Eva98 Theorem 585 Then we x such 10 E Blg and de ne 88771 for a unit vector 6 orthogonal to Vuzo if Vuzo 0 take arbitrary unit 6 Without loss of generality we may assume IO 0 Our aim is to obtain a uniform estimate for 8 78u0 8757mm j 1Hi n By construction 700 0 and w is differentiable at 0 Hence we have the Taylor expansion E I 0lzl 5 Vw0i Now ifE 0 then 817w0 0 for allj 1H n and there is nothing to estimate lf 5 f 0 consider the cone c6 z e R sz 2 lEHIlZL 30 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN which has a property that Cg mT C w gt 0 7C6 mT C w lt 0 for sufficiently small T gt 0 Consider also the rescalings 1441 M I 6 Eli T Note that 1441 A 1001 2 5 z uniformly in B1 and Vwr A Vwo in L1 B17 p gt n The latter follows from the equality l leTz 7 Elpdz 7 7 Vw0lpdz 31 T BT where the righthand side goes to zero as T A 07 since 10 0 is a Lebesgue point for With Then for on gt 0 we have C m4 7 le0zl2dz le0zl2dz n c mB1 lxln ic mBl lxln Q 2 2 hm leTzl dz leTzl dz THO c mB1 ic mBl 1 l lezl2dz lezl2dz 1m 7 THO T4 c mBT lxln ic mBT lIln Q 2 S 2 gliminfi W BT BT THO T4 M714 M714 W W2 where in the last step we have use the inclusion ng BT C iw gt 0 for small T gt 0 Summarizing7 we obtain that le0l4 onlimigwmwtw where so is as in Theorem llll Now7 by Lemma 7 wi satisfy the conditions of Theorem III in Blg instead of B1 and therefore we have 2 lirriigifg0Twwi S C lt1 HwHi2312gt S C Recall that we assume llulleUgl S l which gives that HwHL2Bl2 S C The latter implies that lVaeuzol S C7 which doesn7t yet give the desired estimate on lD2uzol since 6 is subject to the condition 6 Vuzo 07 unless Vuzo 0 If Vuzo 07 we may choose the coordinate system so that Vuzo is parallel to ell Then7 taking 6 62 l l r en in the estimate above7 we obtain lamyuao 0 2392mn j12mnl To obtain the estimate in the missing direction 61 we notice that since u E W2 17B17 the equation divaqul2Vu fz7 u7 Vu is also satisfied in the strong loc sense aqul26ij 2a qul2Bxlw9 7 u 6111 u WWu CIu 996 where 91 frvur7VuIXnI ALMOST MONOTONICITY FORMULAS 31 In particular we may assume that this equation is satis ed at 101 Then at 10 the secondorder term has the form aquIol2AuIo 2a quIol2quIolanlxluWo and thus using 714 we obtain 1 n Aolaxlxlum S lyzol 70 Z lamum MquIol Mlurol 62 This implies the missing estimate and completes the proof of the theoremi ACF84 Bec89 BKPQS BBC94 Caf93 ems CJKOZ 03304 EP08 Eva98 FH76 Kat72 Gro75 LSU68 Sha03 l61111u10l S 0 REFERENCES Hans Wilhelm Alt Luis A Ca arelli and Av39ner Friedman Variational problems with two phases and their free boundaries Trans Amer Math Soc 282 1984 no 2 4317 461 MR732100 85h49014 William Beckner A generalized Poincare inequality for Gaussian measures Proc Amer Math Soc 105 1989 no 2 3977400 MR954373 89m42027 W Beckner C Kenig and J Pipher A convexity property of eigenvalues with applica7 tions 1998 unpublished H Berestycki A Bonnet and S J Chapman A semieelliptic system arising in the theory of typeJI superconductivity Comm Appl Nonlinear Anal 1 1994 no 3 1721 MR1295490 95e35192 Luis A Ca arelli A monotonicity formula for heat functions in disjoint domains Boundary value problems for partial differential equations and applications RMA Res Notes Appl Math vol 29 Masson Paris 1993 pp 530 MR1260438 95e35096 Luis A Caffarelli and Carlos E Kenig Gradient estimates for variable coe icient pare abolic equations and singular perturbation problems Amer J Math 120 1998 no 2 3917439 MR1613650 99b35081 Luis A Ca arelli David Jerison and Carlos E Kenig Some new monotonicity theorems with applications to free boundary problems Ann of Math 2 155 2002 no 2 36 404 MR1906591 2003f35068 Luis Ca arelli Jorge Salazar and Henrik Shahgholian Freeeboundary regularity for a problem arising in superconductivity Arch Ration Mech Anal 171 2004 no 1 115 128 MR2029533 2004m82156 Anders Edquist and Arshak Petrosyan A parabolic almost monotonicity formula Math Ann 341 2008 no 2 4297454 MR2385663 Lawrence C Evans Partial di erential equations Graduate Studies in Mathematics vol 19 American Mathematical Society Providence RI 1998 MR1625845 99e35001 S Friedland and W K Hayman Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions Comment Math Helv 51 1976 no 2 1337161 MR0412442 54 568 Tosio Kato Schrodinger operators with singular potentials Proceedings of the Interna7 tional Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces Jerusalem 1972 1972 pp 1357148 1973 MR0333833 48 12155 Leonard Gross Logarithmic Sobolev inequalities Amer J Math 97 1975 no 4 10617 1083 MR0420249 54 8263 O A Ladyienskaja V A Solonnikov and N N Ural ceva Lineinye i hvazilineinye uravneniya parabolicheshogo tipa Izdat Nauka Moscow 1968 Russian English transl Linear and quasilinear equations of parabolic type Translations of Mathe matical Monographs Vol 23 American Mathematical Society Providence RI 1967 MR0241822 39 3159b MR0241821 39 3159a Henrik Shahgholian C1111 regularity in semilinear elliptic problems Comm Pure Appl Math 56 2003 no 2 27amp281 MR1934623 2003h35087 32 NORAYR MATEVOSYAN AND ARSHAK PETROSYAN UraOl N N Uraltseva Twoiphase obstacle problem J Math Sci New York 106 2001 no 3 307373077 Rmction theory and phase transitions MR1906034 2003e35331 DEPARTMENT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS UNIVERSITY OF CAMr BRIDGE CAMBRIDGE CB3 OWA UK Eimail addness n matevosyanedamtp cam ac uk DEPARTMENT OF MATHEMATICS PURDUE UNIVERSITY WEST LAFAYETTE IN 47907 USA Eimail addness arshakaath purdue edu

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