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1 Librries nd Lerning Services University of Aucklnd Reserch Repository, ReserchSpce Copyright Sttement The digitl copy of this thesis is protected by the Copyright Act 1994 New Zelnd. This thesis my be consulted by you, provided you comply with the provisions of the Act nd the following conditions of use: Any use you mke of these documents or imges must be for reserch or privte study purposes only, nd you my not mke them vilble to ny other person. Authors control the copyright of their thesis. You will recognize the uthor's right to be identified s the uthor of this thesis, nd due cknowledgement will be mde to the uthor where pproprite. You will obtin the uthor's permission before publishing ny mteril from their thesis. Generl copyright nd disclimer In ddition to the bove conditions, uthors give their consent for the digitl copy of their work to be used subject to the conditions specified on the Librry Thesis Consent Form nd Deposit Licence.

2 Degenerte elliptic second-order differentil opertors with bounded complex-vlued coefficients Tn Duc Do A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in Mthemtics The University of Aucklnd Aucklnd, New Zelnd June 2016

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4 Abstrct This thesis considers core properties for degenerte elliptic second-order differentil opertors in divergence form with bounded complex-vlued coefficients. The min contribution of the thesis is in two prts. In one dimension we chrcterise when the spce of test functions is core for these opertors. In higher dimensions we provide sufficient conditions. For the first prt we consider n opertor of the form A = d dx c d dx + m d dx + w I in L 2 R, where c is bounded Lipschitz continuous complex-vlued function which tkes vlues in sector. We determine for which p [1, the qusi-contrction semigroup generted by A extends consistently to qusi-contrction semigroup on L p R. For those vlues of p we chrcterise when the spce of test functions Cc R is core for the genertor on L p R. For the second prt let c kl W 2,, C for ll k, l {1,..., d}. Let Σ θ be the sector with vertex 0 nd semi-ngle θ in the complex plne. Suppose Cx ξ, ξ Σ θ for ll x nd ξ C d. We consider the divergence form opertor A = l c kl k in L 2. We show tht for ll p in suitble intervl the contrction semigroup generted by A extends consistently to contrction semigroup on L p. For those vlues of p we present condition on the coefficients such tht the spce Cc of test functions is core for the genertor on L p. We lso exmine the opertor A seprtely in the more specil Hilbert spce L 2 setting. In this setting we provide mny more sufficient conditions on the coefficients for Cc to be core for A. Furthermore if ll the functions in the domin DA re smooth enough, we show tht Cc is lwys core for A. i

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6 Acknowledgement Prepring PhD thesis is long journey. I would not hve mde it this fr without the help nd support from my supervisor, my fmily, my friends nd the University of Aucklnd. I thnk Tom ter Elst for being wonderful supervisor, for his wisdom, generosity nd kindness which guided me through difficulties over the whole durtion of my study. I thnk my prents bố Thọ nd mẹ Ln, my two elder sisters Hồng nd Hiền s well s my fincée Hạ Phương for their trust nd unconditionl love which give me more strength nd fith. I thnk Nguyễn Tấn Đạt, Nguyễn Mạnh Tuấn, Hoàng Quốc Việt, Trần Bảo Châu, Nguyễn Xuân Khánh nd Nguyễn Thành Thiên Kim for their continuing friendship tht mkes ech dy more enjoyble dy. Finlly I thnk the University of Aucklnd for the finncil support tht I hve received through University of Aucklnd Doctorl Scholrship. iii

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10 Contents Abstrct Acknowledgement i iii 1 Introduction Core properties for degenerte elliptic opertors Outline of the thesis Preliminries Forms nd opertors ssocited with forms Accretive opertors on Bnch spces Consistent semigroups First-order differentil opertors in L p -spces Second-order differentil opertors in L p -spces Degenerte elliptic opertors in one dimension Introduction The form nd the opertor on L L p extension The genertor on L p Core chrcteristion for p 1, Core chrcteristion for p = Exmples Higher dimensions Introduction The coefficient mtrix C L p extension The opertor B p The core property for A p More sufficient conditions in L Exmples Bibliogrphy 97 Index 100 vii

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12 Chpter 1 Introduction The theme of this thesis is bout core properties for degenerte elliptic second-order differentil opertors in divergence form with bounded complex-vlued coefficients in one nd higher dimensions. More specificlly we will investigte when the spce of test functions is core for those opertors. This chpter consists of two prts. In the first prt I will explin the topic in detil. This will be ccompnied by brief summry of the developments in the field. The second prt provides the outline of the thesis. The ides pursued in subsequent chpters re highlighted. Detils bout collbortive works nd contributions re lso given. 1.1 Core properties for degenerte elliptic opertors The Lplce opertor = d in L 2, where d N, is undoubtedly the most well-known nd ubiquitous type of opertor in mthemticl nlysis. This opertor hs been studied for centuries nd mny nice properties hve been observed to ssocite with it. One mong those properties is tht the mximl domin D of the Lplce opertor consists of ll twice wekly differentible functions in L 2. Tht is, D = W 2, We note tht D is normed spce with the grph norm u u 2 + u 2 for ll u D. Therefore 1.1 must be understood in the sense tht the grph norm of D is equivlent to the Sobolev norm on W 2,2 nd the two spces re equl s vector spces. Since the spce of test functions Cc, which consists of ll infinitely differentible functions on with compct supports, is dense W 2,2, it is lso dense in D with respect to the grph norm. In other words we sy tht Cc is core for the Lplce opertor. This result is of our fundmentl interest. The subject to study in this thesis is to investigte this core property for more generl clss of opertors, known s degenerte elliptic second-order differentil opertors in divergence form with bounded complex-vlued coefficients or degenerte elliptic opertors for short, which includes the Lplce opertor s very specil cse. 1

13 2 CHAPTER 1. INTRODUCTION Degenerte elliptic opertors re extensions of strongly elliptic opertors which in turn re the most direct generlistion of the Lplce opertor. To define strongly elliptic opertors, we consider the second-order differentil opertor in divergence form L of the form L = l c kl k, where c kl W 1,, C for ll k, l {1,..., d}. The opertor L is clled strongly elliptic if there exists µ > 0 such tht Re Cx ξ, ξ µ ξ for ll x nd ξ C d, where Cx = c kl x 1 k,l d for ll x. Let Re C := 1 2 C + C, where C = C T. Then 1.2 is the sme s requiring there exists µ > 0 such tht Re Cx ξ, ξ µ ξ 2 for ll x nd ξ C d. Tht is, ll the eigenvlues of Re Cx must be strictly positive, with the smllest eigenvlue t lest the constnt µ for ll x. Most of the properties possessed by the Lplce opertor re inherited by strongly elliptic opertors, including the core property in prticulr. The theory of strongly elliptic opertors is vst nd firly well-understood. Mny importnt results in this re cn be found in tretises on the subject cf. [GT83], [EE87], [Ev10], [Agm10], [Neč12], etc. s well s in the lrge mount of literture devoted to it. The sitution chnges drsticlly when 1.2 is relxed to Re Cx ξ, ξ for ll x nd ξ C d, in which cse L is sid to be degenerte elliptic. Agin we cn red 1.3 in the sense tht Re Cx ξ, ξ 0 for ll x nd ξ C d. As consequence the mtrix Re C is positive semi-definite when L is degenerte elliptic. But on the contrry to strongly elliptic opertors, some or ll of the eigenvlues of Re C my hve zeros in this cse. The theory of degenerte elliptic opertors is n ctive re of reserch. Mny well-known results for strongly elliptic opertors remin unsolved for degenerte elliptic opertors. We will shortly point out this kind of distinctive difference between the two clsses of opertors in terms of the core properties. Our strting point to formulte the min problems of the thesis is the first representtion theorem, proved independently by Lions cf. [Lio61] nd Kto cf. [Kt80, Theorem VI.2.1]. This theorem presents convenient relistion of n opertor in the setting of the Hilbert spces vi form methods. This pproch is in contrst to the clssicl relistion of n opertor vi detiled description of the domin of the opertor. The difference lies in the fct tht form methods provide more tools nd structures to nlyse n opertor thn the clssicl pproch, t lest in Hilbert spces. To be specific let θ [0, π. Define 2 Σ θ = {r e i ψ : r 0 nd ψ θ}. 1.4

14 1.1. CORE PROPERTIES FOR DEGENERATE ELLIPTIC OPERATORS 3 We consider in L 2 the form 0 u, v = c kl k u l v on the domin D 0 = Cc, where c kl re complex-vlued functions in W 1, for ll k, l {1,..., d} which stisfy the condition Cx ξ, ξ Σ θ 1.5 for ll x nd ξ C d. Under those conditions imposed on the coefficients, the form 0 cn be extended to so-clled closed form. The closure of the form 0 stisfies ll the requirements of the first representtion theorem. Using the theorem we cn ssocite n opertor A with the form in such mnner tht DA D nd u, v = Au, v L2 for ll u DA nd v D. Formlly we cn write A in the form A = l c kl k. Since 1.5 implies 1.3, the opertor A is degenerte elliptic. Furthermore the sme theorem of Lions nd Kto gives tht A is the genertor of C 0 -semigroup S on L 2. Using interpoltion, we will see tht S cn be extended consistently to C 0 -semigroup S p on L p under suitble conditions on the coefficients nd for certin p [1,. Those vlues for p depend on the ngle θ. A very specil cse is when c kl is rel-vlued function for ll k, l {1,..., d}, in which cse we hve θ = 0 nd p cn tke ny vlue in [1,. Let A p be the genertor of S p. The domin DA p is nturlly normed with the grph norm DAp defined by u DAp = u p + A p u p for ll u DA p. If subspce D of DA p is dense in DA p with respect to the grph norm, D is sid to be core for A p. If we know tht DA p is complete nd D is core for A p, then D, DAP = DA p. This explins the significnce of core for n opertor, which lies in the fct tht the domin of A p is sometimes too lrge to work with. However we cn still obtin informtion bout A p by knowing how it cts on core. Since core for n opertor is often smll nd nice subspce of the domin, this is much esier to del with. With this notion of core for A p in mind, we cn now stte the question of our interest: When is C c core for A p? It hs been known for long time tht the spce of test functions Cc is core for A p if the mtrix of coefficients C stisfies the strong ellipticity condition 1.2 with entries belonging to W 1, cf. [ADN59] nd [ER97, Theorem 1.5]. Nevertheless if C is known to stisfy the degenerte ellipticity condition 1.3 only, the sitution will be very different nd usully much more difficult since Cc is no longer core in generl. A common procedure to investigte the core properties in this sitution is to perform pproximtion rguments to trnsit from the degenerte cse bck to the more fmilir strongly elliptic

15 4 CHAPTER 1. INTRODUCTION cse. Next we will describe some previous results obtined by [WD83, Theorem 1], [Ouh05, Theorem 5.2], [CMP98, Theorem 3.5] nd [ERS11, Section 4]. The overll contents re s follows. In 1983 Wong-Dzung considered in [WD83] the opertor B p of the form on the mximl domin B p = l c kl k DB p = {u L p : there exists n f L p such tht for ll φ C c u k c kl l φ = f φ}, where p 1,, the coefficient mtrix C = c kl 1 k,l d stisfies the degenerte ellipticity condition 1.3 nd its entries re rel-vlued C 2 -functions which re possibly nonsymmetric. It ws shown tht Cc is core for B p. It cn be verified tht B p is the sme s the opertor A p obtined bove vi form methods with the sme coefficients s those of B p. In his book [Ouh05] published in 2005, Ouhbz refined the rguments used by WongDzung in [WD83] to prove tht Cc is core for the opertor B 2 in L 2 under weker ssumptions tht C still stisfies the degenerte ellipticity condition but the principl coefficients re merely rel-vlued functions in W 2,. In nother direction, on the unit intervl, Cmpiti, Metfune nd Pllr gve chrcteristion for when Cc 0, 1 is core for the opertor C p defined by on the domin C p = d dx c d dx 1.6 DC p = {u L p 0, 1 : u W 1,p loc 0, 1 nd c u W 1,p 0 0, 1}, where p [1, nd the rel-vlued coefficient c C[0, 1] stisfies tht c0 = c1 = 0 nd cx > 0 for ll x 0, 1 cf. [CMP98, Theorem 3.5]. The techniques used to prove the chrcteristion re intrinsiclly vilble in one dimension only. Up to now extensions of this chrcteristion to higher dimensions remin widely open problems. Next we will exmine the forementioned results more closely by considering the following exmple. Exmple 1.1. Let p [1, nd κ 1. Let C p be defined by 1.6 with coefficient cx = x 2 1 x 2 κ 2 for ll x [0, 1]. Then c W κ, 0, 1. The chrcteristion 1+x 2 1 x 2 [CMP98, Theorem 3.5] gives tht Cc 0, 1 is core for C p if nd only if κ 2 1. p It cn be shown tht this exmple continues to hold on L p R for the sme rnge of p when cx = x 2 1 x 2 κ 2 for ll x R. In this cse we cn lso verify tht C 1+x 2 1 x 2 p is in fct the sme s the opertor A p obtined vi form methods if the coefficient of A p is the

16 1.2. OUTLINE OF THE THESIS 5 sme s tht of C p. Hence the core chrcteristion lso pplies to A p. So t lest in one dimension, it tells us in prticulr tht the W 1, R smoothness of the coefficient does not gurntee Cc R to be core for A p, wheres the results of Wong-Dzung [WD83, Theorem 1] nd Ouhbz [Ouh05, Theorem 5.2] sttes W 2, R smoothness is sufficient. Generlising to rbitrry dimensions, we will refer to this phenomenon s the gp in the smoothness of the coefficients regrding the core properties for A p. Bridging the gp, i.e. finding the optiml smoothness of the coefficients for Cc to be core for A p, remins unsolved in higher dimensions up to now. A recent pper published in 2011 tht slightly touched on this direction is tht of ter Elst, Robinson nd Sikor [ERS11], in which they considered mixture of smoothness conditions between W 1, nd W 2, on relvlued coefficients for Cc to be core for A p. In this thesis we will give nswers to the min question posted bove. We emphsise tht here we consider opertors with complex-vlued coefficients in contrst to opertors with rel-vlued coefficients in the forementioned literture. In one dimension we will provide chrcteristion for when Cc is core for the opertor A p. This chrcteristion is n extension of the result in [CMP98] nd the pure second-order cse in [DE15]. In higher dimensions we will provide mny sufficient conditions for when Cc is core for A p. The work is in the spirit of [WD83] nd [Ouh05]. Aprt from the interests in the core properties for degenerte elliptic second-order differentil opertors with bounded coefficients, there is huge literture for sufficient conditions under which the spce of test functions is still core if the coefficients of the opertor re rel-vlued nd unbounded either loclly or t infinity. The detils nd mny interesting results cn be found in [Kt81], [Dv85], [Lis89], [MPPS05], [MPRS10], [CCHL12], [MS14] nd references therein. 1.2 Outline of the thesis In this section we will summrise the content s well s the ides used in the subsequent chpters. References re given to the originl ppers which we bsed our reserch on. Detils bout collbortive works re lso mentioned. Chpter 2: Here we collect ll the bckground knowledge required in subsequent chpters. These include the notions of forms, ccretive opertors, consistent extension of semigroups, first-order nd second-order differentil opertors. Mny well-known results from the literture relted to these notions re presented here for lter use. Chpter 3: In this chpter we consider the one-dimensionl cse. The min result in this chpter is n extension of [CMP98, Theorem 3.5] nd [DE15, Theorem 1.5] which is on its turn n extension of [CMP98, Theorem 3.5]. Let θ [0, π. Let Σ 2 θ be defined s in 1.4. Let c W 1, R be complex-vlued function such tht cx Σ θ for ll x R. Let m W 1, R nd w L R. Suppose m M Re c for some M > 0. We strt with the form 0 defined by 0 u, v = c u v + m u v + w u v R on the domin D 0 = Cc R. Since 0 is closble, we let A be the opertor ssocited with the closure of the form 0. Then A genertes qusi-contrction C 0 -semigroup S

17 6 CHAPTER 1. INTRODUCTION on L 2 R. For certin p [1,, we cn extend S consistently to qusi-contrction C 0 -semigroup S p on L p R. Let A p be the genertor of S p. We will give core chrcteristion for the opertor A p in terms of the degenercy of the coefficient c t its zero points. The pure second-order cse, i.e. if m = w = 0, is joint work with Tom ter Elst cf. [DE15]. Chpter 4: The chpter is motivted by [WD83] nd [Ouh05]. Let d N nd θ [0, π 2. Let c kl W 2, be complex-vlued functions for ll k, l {1,..., d}. Suppose Cx ξ, ξ Σ θ for ll x nd ξ C d, where Cx = c kl x 1 k,l d for ll x nd Σ θ is defined by 1.4. We strt with the form 0 defined by 0 u, v = c kl k u l v on the domin D 0 = Cc. Since 0 is closble, we let A be the opertor ssocited with the closure of the form 0. Then A genertes qusi-contrction C 0 -semigroup S on L 2 R. Let C = R + i B, where R nd B re rel mtrices. Suppose B is symmetric. Then we show tht it is possible to extend S consistently to qusi-contrction C 0 -semigroup S p on L p R for certin p 1,. Let A p be the genertor of S p. We will show tht Cc is core for A p. The cse p = 2 is specil s we nturlly obtin the opertor A from the form. Consequently the ssumption tht B is symmetric is not needed. Suppose B is not symmetric. Then we provide mny sufficient conditions for when Cc is core for A. In prticulr if DA W 1,2, then Cc is lwys core for A regrdless of the symmetry of B.

18 Chpter 2 Preliminries This chpter summrises ll the bckground knowledge required in the subsequent chpters. Forms, ccretive opertors nd semigroups re defined. We will mke cler the reltions between these notions. Mny well-known results will be discussed. The lst two sections present pplictions of previous sections to first-order nd second-order differentil opertors. 2.1 Forms nd opertors ssocited with forms In this section we introduce the powerful tools of form methods. We will consider forms nd their ssocited opertors. The centrl result is the first representtion theorem given in Theorem Let H be Hilbert spce. We emphsise tht the underlying field is C. Definition 2.1. Let D be subspce of H. A function : D D C which is liner in the first vrible nd nti-liner in the second vrible is clled sesquiliner form. We lso refer to s form for short. The subspce D is clled the domin of. To be specific we lso write D = D. For the rest of the section let be form with domin D H. For convenience we will write u = u, u for ll u D. It is worth noting tht the set of vlues {u : u D} determines the form uniquely, thnks to the following proposition. Proposition 2.2 Polristion identity. The following identity u, v = 1 4 holds for ll u, v D. u + v u v + i u + iv i u iv For ll θ [0, π 2 we define Σ θ = {re iψ : r 0 nd ψ θ}, which is sector in the complex plne with semi-ngle θ. We hve the following definitions. Definition 2.3. The form is clled 1. densely defined if D is dense in H. 7

19 8 CHAPTER 2. PRELIMINARIES 2. ccretive if Re u 0 for ll u D. If there exists n ω R such tht Re u + ω u 2 H 0 for ll u D, then is sid to be qusi-ccretive. 3. sectoril if there exists θ [0, π 2 such tht u Σ θ for ll u D. Definition 2.4. Suppose is sectoril. Then the domin D is normed spce with norm defined by u = Re u + u 2 H 1/2 for ll u D. The form is clled closed if D, is complete. A form b which hs the properties tht D Db nd bu = u for ll u D is clled n extension of the form. If there exists closed form b which is n extension of, then is clled closble. In this cse we lso sy tht b is closed extension of. Note tht closed extension of form is in generl not unique if it exists. If is closble, we will denote by the smllest closed extension of. Define the form : D D C by u, v = v, u. The form is clled the djoint form of. If =, the form is clled symmetric. Denote R = nd I = 1 2i. Then = R + i I. We will refer to R s the rel prt of nd I s the imginry prt of. Note tht both R nd I re symmetric forms. Bsed on these notions we cn now describe some typicl properties of sectoril forms. Proposition 2.5 [Kt80, Subsection VI.1.2]. Suppose the form is sectoril. Let u, v D. Then the following hold. 1. Iu tn θ Ru. 2. Ru, v Ru 1/2 Rv 1/2. 3. Iu, v tn θ Ru 1/2 Rv 1/2. 4. u, v 1 + tn θ Ru 1/2 Rv 1/2. More generlly we cn compre two forms with ech other. Theorem 2.6. Let nd b be forms in H. Suppose tht is symmetric nd D = Db = V for some V H. Suppose there exists n M > 0 such tht bu M u for ll u V. Then the following hold. b If b is symmetric, then for ll u, v V. If b is not symmetric, then for ll u, v V. bu, v M u 1/2 v 1/2 bu, v 2M u 1/2 v 1/2

20 2.1. FORMS AND OPERATORS ASSOCIATED WITH FORMS 9 Proof. Let u, v V. By replcing v with e i ψ v for some ψ [0, 2π, we cn ssume without loss of generlity tht bu, v R. Then bu, v = 1 4 bu + v bu v M 4 u + v + u v = M 2 where we used Proposition 2.2 in the lst step. We consider two cses. Cse 1: Suppose u = 0. Then 2.1 gives Replcing v by λ v with λ > 0 gives bu, v M 2 v. bu, v M λ 2 v. u + v, 2.1 Since λ is rbitrry, tking the limit when λ 0 on both sides of the bove inequlity yields bu, v = 0, which implies the clim. Cse 2: Suppose u 0. v Replcing u by u in 2.1 we obtin the desired conclusion. u b Note tht b = Rb + i Ib, where Rb nd Ib re symmetric. Note tht Rbu = bu + b u bu M u 2 for ll u V. It follows from tht Rbu, v M u 1/2 v 1/2 for ll u, v V. Similrly Ibu, v M u 1/2 v 1/2 for ll u, v V. Hence for ll u, v V. bu, v Rbu, v + Ibu, v 2 M u 1/2 v 1/2 Let A be n opertor in H, tht is, A: DA H H. The domin DA of A is nturlly normed with the grph norm DA defined by for ll u DA. Define u DA = u H + Au H ΘA = {Au, u: u DA nd u H = 1}, which is clled the numericl rnge of A. We will denote by ρa the resolvent set of A. We hve the following definitions. Definition 2.7. The opertor A is clled 1. densely defined if DA is dense in H. 2. closed if DA, DA is complete. If there exists closed opertor B in H such tht DA DB nd Bu = Au for ll u DA, then A is clled closble. In this cse we lso sy tht B is closed extension of A.

21 10 CHAPTER 2. PRELIMINARIES 3. ccretive if Re Au, u 0 for ll u DA. If A is ccretive nd 1 ρ A, then A is sid to be m-ccretive. If there exists n ω R such tht Re Au, u + ω u 2 H 0 for ll u DA, then A is clled qusi-ccretive. If there exists n ω R such tht Re Au, u + ω u 2 H 0 for ll u DA nd ω + 1 ρ A, then A is sid to be qusi-m-ccretive. 4. sectoril if ΘA Σ θ for some θ [0, π. If A is sectoril nd m-ccretive, then A 2 is sid to be m-sectoril. Note tht closed extension of n opertor is in generl not unique if it exists. If A is closble, we will denote by A the smllest closed extension of A. Closble opertors nd ccretive opertors re relted to ech other in the following mnner. Proposition 2.8 [Ouh05, Lemm 1.47]. Suppose A is densely defined nd ccretive. Then A is closble. The following definition is of our min interest. Definition 2.9. A subspce D DA is clled core for A if D is dense in DA with respect to the grph norm. The next theorem provides useful criterion for proving the core properties for ccretive opertors. It will be used extensively in Chpter 4. Theorem 2.10 [Ouh05, Theorem 1.50]. Let A be ccretive. Assume tht S is n m- ccretive opertor stisfying the following two conditions. 1. DS DA. 2. There exists constnt β R such tht Re Au + β u, Su 0 for ll u DS. Then the closure A is m-ccretive. Furthermore DS is core for A. Next we will describe some correspondences between forms nd opertors. We strt first with forms constructed from sectoril opertors. Theorem 2.11 [Kt80, Theorem VI.1.27]. Suppose A is sectoril nd u, v = Au, v with D = DA. Then is closble. Conversely we hve the following well-known result. Theorem 2.12 The first representtion, [Kt80, Theorem VI.2.1]. Suppose is densely defined, closed nd sectoril form. Let A be defined in the following mnner. Let u, w H. We sy tht u DA nd Au = w if u D nd u, v = w, v H for ll v D. Then A is m-sectoril. The opertor A in Theorem 2.12 is clled the opertor ssocited with the form. If the form is not closble, we cn still ssocite with it n m-sectoril opertor. This is the content of the following theorem.

22 2.2. ACCRETIVE OPERATORS ON BANACH SPACES 11 Theorem 2.13 [AE12, Theorem 3.2]. Suppose is densely defined nd sectoril form. Let A be defined in the following mnner. Let u, w H. We sy tht u DA nd Au = w if there exists sequence u n n N in D such tht sup n N Re u n <, lim n u n = u in H nd lim n u n, v = w, v H for ll v D. Then A is m-sectoril. It is remrkble fct tht m-sectoril opertors re genertors of holomorphic semigroups. For bckground informtion on semigroups we refer to [EN00], [Ng86], [Pz83], [Gol85] nd [Yos80]. Theorem 2.14 [Kt80, Theorem IX.1.24]. Let θ [0, π. Suppose A is n m-sectoril 2 opertor with ΘA Σ θ. Then A genertes contrction holomorphic semigroup with ngle π θ Accretive opertors on Bnch spces In Section 2.1 we discussed ccretive opertors on Hilbert spces. In this section we will extend the notion of ccretivity of opertors to Bnch spces. We re prticulrly interested in ccretive opertors in L p -spces. Let X be Bnch spce. Let A, DA be n opertor in X. We view the domin DA s normed spce with the grph norm DA defined by u DA = u X + Au X for ll u DA. The notion of core for n opertor, which is the definition of our min interest, extends verbtim to this more generl context. For the ske of clrity nd completeness, we lso repet it here. Definition A subspce D DA is clled core for A if D is dense in DA with respect to the grph norm. Next we will consider ccretive opertors. We denote by X the dul spce of X. Define F u = {f X : u, f = u 2 X = f 2 X } for ll u X. By the Hhn-Bnch theorem the set F u is non-empty for ll u X. A prticulr cse is when X = L p with d N, in which cse we hve F u = u p 2 u 1 [u 0] } for ll u X \ {0}. We hve the following definition. { u 2 p p Definition The opertor A is clled 1. densely defined if DA is dense in X. 2. closed if DA, DA is complete. If there exists closed opertor B, DB in X such tht DA DB nd Bu = Au for ll u DA, then A is clled closble. In this cse we lso sy tht B is closed extension of A. 3. ccretive if for every u DA there exists n f F u such tht Re Au, f 0. If A is ccretive nd 1 ρ A, then A is clled m-ccretive. If there exists n

23 12 CHAPTER 2. PRELIMINARIES ω R such tht for every u DA there exists n f F u which stisfies Re ω u + Au, f 0, then A is clled qusi-ccretive. If there exists n ω R such tht ω + 1 ρ A nd for every u DA there exists n f F u which stisfies Re ω u + Au, f 0, then A is sid to be qusi-m-ccretive. It is esy to verify tht this definition coincides with Definition 2.7 if X is Hilbert spce. The importnce of m-ccretive opertors is due to the following well-known theorem. Theorem 2.17 Lumer-Phillips theorem, [LP61]. Let A be densely defined opertor in X. Then A is m-ccretive if nd only if A is the genertor of contrction C 0 -semigroup. When X is reflexive, the density requirement of the opertor s domin in Theorem 2.17 cn be removed, s stted by the following theorem. Theorem 2.18 [Kt59, Corollry 2], [Kt80, Theorem III.5.29], [EN00, Corollry II.3.20]. Suppose X is reflexive. Let S be n m-ccretive opertor in X. Then S is densely defined. Moreover, S is lso densely defined. 2.3 Consistent semigroups In this section we will explore the reltions between consistent C 0 -semigroups on L p -spces. Let d N. Let p 1, p 2 [1,. Let S p 1, S p 2 be C 0 -semigroups on L p1 nd L p2 respectively. Definition We sy tht S p 1 nd S p 2 re consistent if S p 1 t u = S p 2 t u for ll t > 0 nd u L p1 L p2. Theorem Suppose S p 1 nd S p 2 re bounded C 0 -semigroups which re consistent. Let A p1 nd A p2 be the genertors of S p 1 nd S p 2 respectively. Then A p1 u = A p2 u for ll u DA p1 DA p2 nd DA p1 DA p2 = {u DA p1 L p2 : A p1 u L p2 } = {u DA p2 L p1 : A p2 u L p1 } = I + A p1 1 L p1 L p2 = I + A p2 1 L p1 L p Moreover, DA p1 DA p2 is core for A p1 in L p1 nd for A p2 in L p2. Proof. First let u DA p1 DA p2 nd φ L 1 L. Then 1 A p1 u, φ = lim t 0 t p I S 1 t u, φ 1 = lim t 0 t p I S 2 t u, φ = A p2 u, φ. It follows tht A p1 u = A p2 u L p1 L p2. In prticulr this implies DA p1 DA p2 {u DA p1 L p2 : A p1 u L p2 }. Conversely let u DA p1 L p2 be such tht A p1 u L p2. Then 1 t I Sp 2 t u = 1 t I Sp 1 t u = 1 t t 0 S p 1 s A p1 u ds = 1 t t 0 S p 2 s A p1 u ds

24 2.3. CONSISTENT SEMIGROUPS 13 for ll t > 0. Therefore lim t 0 1 t I Sp 2 t u = A p1 u in L p2. It follows tht u DA p2. This proves the first two equlities in 2.2. Secondly, if u DA p1 DA p2 then I + A p1 u = I + A p2 u L p1 L p2. Consequently DA p1 DA p2 I + A p1 1 L p1 L p2. Conversely if u L p1 L p2 then I + Ap1 1 u, φ = 0 e t S p 1 t u, φ dt = 0 e t S p 2 t u, φ dt = I + A p2 1 u, φ for ll φ L 1 L. Hence I + A p1 1 u = I + A p1 1 u DA p1 DA p2. The lst two equlities in 2.2 now follows. Lstly we note tht S p1 leves DA p1 invrint nd S p2 leves DA p2 invrint. As consequence S p 1 t DAp1 DA p2 DA p1 DA p2 for ll t > 0. But DA p1 DA p2 = I + A p1 1 L p1 L p2 is dense in L p1. Hence DA p1 DA p2 is core for A p1 in L p1 by [EN00, Proposition 1.7]. The sttement for A p2 is proved similrly. Lemm Suppose S p 1 nd S p 2 re consistent. Suppose further tht there exists n M > 0 nd ω R such tht S p 1 t u M e ωt u 2.3 for ll t > 0 nd u L 1 L. Then DA p1 DA p2 L is core for A p1 in L p1. Proof. Without loss of generlity we ssume tht S p 1 nd S p 2 re bounded nd ω = 0. By hypothesis I + A p1 1 : L p1 DA p1 is bounded nd bijective. Since L 1 L is dense in L p1, we hve I + A p1 1 L 1 L is dense in DA p1. We will show tht I + A p1 1 L 1 L DA p1 DA p2 L, 2.4 from which the clim follows. Indeed we hve L 1 L L p1 L p2. Therefore I + A p1 1 L 1 L I + A p1 1 L p1 L p2 = DA p1 DA p2, where the lst equlity follows from 2.2. Next let u L 1 L. Then I + A p1 1 u = 0 e t S p 1 t u dt M u e t dt = M u, 0 0 e t S p 1 t u dt where we used 2.3 in the third step. It follows tht I + A p1 1 L 1 L L. Hence 2.4 holds.

25 14 CHAPTER 2. PRELIMINARIES 2.4 First-order differentil opertors in L p -spces In this section we present some properties of first-order differentil opertors with Lipschitz coefficients in L p -spces. Let d N. Let b l W 1,, R for ll l {1,..., d}. For ll p 1, consider the first-order differentil opertor Y p,0 of the form on the domin Y p,0 u = l b l u l=1 DY p,0 = W 1,p. It is cler tht Y p,0 Z for ll p 1,, where Z is the opertor in L q defined by Zu = b l l u on the domin DZ = Cc nd q is the dul exponent of p. Hence Y p,0 is closble for ll p 1,. Let Y p be the closure of Y p,0 in L p for ll p 1,. Theorem 2.22 [Rob91, Theorem V.4.1]. Let p 1,. Then the opertor Y p genertes qusi-contrction C 0 -semigroup S p on L p. Moreover, S p 1 nd S p 2 re consistent for ll p 1, p 2 1,. Let p 1,. We hve the following core property for Y p. Proposition Let p 1,. Then the spce C c is core for Y p. Proof. Since W 1,p DY p, there exists n M > 0 such tht l=1 u DYp M u W 1,p 2.5 for ll u W 1,p. Let u DY p. Let ε > 0. Since Y p = Y p,0, there exists v W 1,p such tht u v DYp < ε. Also the spce 2 C c is dense in W 1,p. Therefore there exists φ Cc such tht v φ W 1,p ε. It follows tht 2M u φ DYp u v DYp + v φ DYp ε 2 + M v φ W 1,p ε 2 + M ε 2M = ε, where we used 2.5 in the second step. This justifies the clim. For ech n N let J n be the usul mollifier with respect to suitble function in Cc. It is useful tht functions in DY p cn be pproximted by smooth functions which re formed by using the mollifiers. This is the content of the next proposition. Proposition 2.24 [ERS11, Proposition 2.1]. Let p 1,. Then for ll u DY p we hve lim n Y pj n u = Y p u in L p.

26 2.5. SECOND-ORDER DIFFERENTIAL OPERATORS IN L p -SPACES Second-order differentil opertors in L p -spces This section dels with second-order differentil opertors in divergence form in L p -spces. In prticulr we re mostly interested in those opertors which re strongly elliptic. We will consider ccretive properties, elliptic regulrity nd semigroup extensions of these opertors. Let d N. Let c kl W 1, nd m k, w L for ll k, l {1,..., d}. In wht follows we denote Cx = c kl x 1 k,l d for ll x. Definition Let p [1, ]. An opertor L with suitble domin in L p of the form Lu = l c kl k u + m k k u + w u k=1 is clled strongly elliptic if there exists µ > 0 such tht for ll x nd ξ C d. If 2.6 is replced by Re Cx ξ, ξ µ ξ Re Cx ξ, ξ 0 for ll x nd ξ C d, then we sy tht L is degenerte elliptic. Suppose C stisfies the strong ellipticity condition 2.6. Consider the form u, v = c kl k u l v + m k k u v + w u v on the domin D = W 1,2. Then is closed. Since the coefficient mtrix C stisfies the strong ellipticity condition, is lso qusi-sectoril. Using the first representtion theorem, Theorem 2.12, we cn ssocite with the form n qusi-m-sectoril opertor A. Note tht W 2,2 DA nd Au = k=1 l c kl k u + m k k u + w u for ll u W 2,2. Let S be the holomorphic qusi-contrction semigroup generted by A. The following theorem is consequence of [Aus96, Theorem 4.8]. Theorem For ech t > 0 let K t D R 2d be the distributionl kernel of S t. Then 1. The kernel K t is Hölder continuous function for ll t > There exist constnts c, ω, κ, β > 0 such tht k=1 K t x, y c β x y 2 e t e ωt, td/2

27 16 CHAPTER 2. PRELIMINARIES for ll t > 0 nd x, y nd K t x, y K t x + h, y + K t x, y K t x, y + h c h κ e β x y 2 t d/2 t 1/2 t e ωt, + x y for ll t > 0 nd x, y, h with 2 h t 1/2 + x y. The next theorem gurntees tht S cn be extended consistently to holomorphic qusicontrction semigroups on L p -spces. Theorem 2.27 [Ouh05, Theorem 6.16]. Let T be bounded holomorphic semigroup on sector Σ θ on L 2, where θ [0, π 2. For ll z Σ θ let K z D R 2d be the distributionl kernel of T z. Suppose there exist c, β > 0 such tht K t x, y c t d/2 e β x y 2 t for ll t > 0 nd for.e. x, y R 2d. Then for every ψ [0, θ there exist c ψ, β ψ > 0 such tht K z x, y c ψ Re z d/2 e β ψ x y z for ll z Σ ψ nd for.e. x, y R 2d. In ddition, T extends consistently to bounded holomorphic semigroup on the sector Σ θ on L p for ll p [1,. Hence S extends consistently to holomorphic semigroup S p on L p for ll p [1,. Let A p be the genertor of S p for ll p [1,. Theorem Let p [1,. Then DA DA p L is core for A p. Proof. For ech t > 0 let K t D R 2d be the distributionl kernel of S t. By Theorem 2.26 there exist constnts c, ω, β > 0 such tht K t x, y G t x y for ll t > 0 nd x, y, where G t x = c β x 2 e t e ωt td/2 for ll t > 0 nd x. Let t > 0 nd u L 1 L. Then S p t ux = S t ux = K t x, y uy dy R d K t x, y uy dy G t x y uy dy for.e. x. We hve G t u G t 1 u G t 1 = c e ωt t d/2 e β x 2 t dx = c e ωt β x 2 e dx = M e ωt,

28 2.5. SECOND-ORDER DIFFERENTIAL OPERATORS IN L p -SPACES 17 where It follows tht M = c β x 2 e dx <. S p t u M e ωt u. Since S nd S p re consistent, the spce DA DA p L is core for A p by Lemm It is well-known tht elements of DA p possess certin regulrity properties when p 1,, s stted in the next theorem. Theorem 2.29 Elliptic regulrity. Let p 1,. Then DA p = W 2,p. This theorem is folklore. For n explicit reference see [ER97, Theorem 1.5]. Before ending this section we will consider n integrtion by prts formul for pure second-order differentil opertors. For the rest of this section let c kl W 1, for ll k, l {1,..., d}. Tht is, we no longer require tht C = c kl 1 k,l d stisfies the strongly elliptic condition 2.6. Let p 1,. Define the pure second-order differentil opertor B p : DB p L p L p by B p u = l c kl k u on the domin DB p = {u L p : there exists n f L p such tht u k c kl l φ = f φ for ll φ Cc }. Note tht W 2,p DB p. In Chpters 3 nd 4 we will be interested in proving tht opertors of this type re ccretive or qusi-ccretive under further restrictions. A convenient tool we would like to use in doing so is the method of integrtion by prts. The following theorem sttes tht we cn indeed perform this method on B p. Theorem Let u W 2,p. Then B p u u p 2 u = u p 2 C u, u [u 0] [u 0] + p 2 i p 2 [u 0] [u 0] u p 4 C Re u u, Re u u u p 4 C Re u u, Im u u. 2.7

29 18 CHAPTER 2. PRELIMINARIES Proof. This follows immeditely from the proof of [MS08, Proposition 3.5]. We emphsise tht we do not require c kl = c lk for ll k, l {1,..., d} in the bove theorem cf. [MS08, Theorem 3.1] nd [MS08, Proposition 3.5]. Let u W 2,p. If p [2, then u p 2 u W 1,q, where q is the dul exponent of p. In this cse 2.7 is consequence of the integrtion by prts. Therefore the significnce prt of Theorem 2.30 is when p 1, 2, in which cse the smoothness of u p 2 u is not obvious due to the singulrity of u p 2 ner the zeros of u. Next we will extend Theorem 2.30 to the cse p = 1 in one dimension. Let, b [, ] with < b. Let c W 1, R, C. Proposition Let u W 2,1, b. Then b c u u u 2 + δ 2 1/2 = for ll δ R \ {0}. b c u 2 u 2 + δ 2 1/2 b c u u Re u u u 2 + δ 2 3/2 c u u u 2 + δ 2 1/2 b 2.8 Proof. First note tht u W 1,, b by the Sobolev imbedding theorem. Let u n n N Cc R be such tht lim n u n = u in W 2,1, b. Without loss of generlity we my ssume tht lim n u n x = ux nd lim n u nx = u x for.e. x, b s well s there exists n M > 0 such tht u n L,b M u L,b nd u n L,b M u L,b for ll n N. Let δ R \ {0}. It follows from integrtion by prts tht b c u n u n u n 2 + δ 2 1/2 = b c u n 2 u n 2 + δ 2 1/2 b c u n u n Re u n u n u n 2 + δ 2 3/2 c u n u n u n 2 + δ 2 1/2 b. 2.9 Since W 1,1, b L, b by the Sobolev imbedding theorem nd lim n u n = u in W 2,1, b, we hve lim n u n = u in L, b. Next note tht lim n u n 2 + δ 2 1/2 x = u 2 + δ 2 1/2 x for.e. x, b nd u n 2 + δ 2 1/2 δ 1 for ll n N. Therefore we hve un u n 2 + δ 2 1/2 u u 2 + δ 2 1/2 L,b un u u n 2 + δ 2 1/2 L,b + u un 2 + δ 2 1/2 u 2 + δ 2 1/2 L,b u n u L,b 1 δ + u u n u L,b L,b, δ 2 where the lst step follows from the men vlue theorem pplied to the function x x 2 + δ 2 1/2 on R. Hence lim n u n u n 2 + δ 2 1/2 = u u 2 + δ 2 1/2 in L, b.

30 2.5. SECOND-ORDER DIFFERENTIAL OPERATORS IN L p -SPACES 19 On the other hnd we hve lim n c u n = c u in L 1, b s c W 1, R, u W 2,1, b nd lim n u n = u in W 2,1, b. Therefore b lim n c u n u n u n 2 + δ 2 1/2 = b c u u u 2 + δ 2 1/2. We lso hve W 1,1, b C[, b] if, b R by the Sobolev imbedding theorem nd lim x ± vx = 0 for ll v W 1,1 R by [Bre11, Corollry 8.9]. Hence lim c n u n u n u n 2 + δ 2 1/2 b = c u u u 2 + δ 2 1/2 b. Next we consider the second term on the right hnd side of 2.9. lim n u n = u in L 1, b nd We note tht u n u 2 L 2,b = b u n u 2 u n u L,b b M + 1 u L,b u n u L1,b. u n u It follows tht lim n u n = u in L 2, b. Furthermore by ssumption u n L,b M u L,b for ll n N. Therefore lim n u n u n = u u in L 2, b. It follows tht b lim n c u n u n Re u n u n u n 2 + δ 2 3/2 = b c u u Re u u u 2 + δ 2 3/2 by the Lebesgue dominted convergence theorem. The clim now follows. Corollry Let u W 2,1, b. Then b Proof. By Proposition 2.31 we hve b u u u 2 + δ 2 1/2 = b Im u u 2 u 3 1 [u 0] <. u 2 u 2 + δ 2 1/2 u u u 2 + δ 2 1/2 b b for ll δ R \ {0}. Tking the rel prts on both sides gives Re b u u u 2 + δ 2 1/2 = = b u 2 u 2 + δ 2 1/2 b Re u u u 2 + δ 2 1/2 b b u u 2 u 2 + δ 2 3/2 Re u u u 2 + δ 2 1/2 b b b u u Re u u u 2 + δ 2 3/2 Re u u 2 u 2 + δ 2 3/2 Re u u 2 u 2 + δ 2 3/2 Im u u 2 u 2 + δ 2 3/2 Re u u u 2 + δ 2 1/2 b.

31 20 CHAPTER 2. PRELIMINARIES For the rest of the proof we use the convention tht u u u 1 x = 0 for ll x [, b] such tht ux = 0. Since u u u 2 + δ 2 1/2 u for ll δ R \ {0} nd u L 1, b, we hve lim δ 0 b u u u 2 + δ 2 1/2 = b u u u 1 by the Lebesgue dominted convergence theorem. It follows from Ftou s lemm tht 0 b lim inf δ 0 = Re Im u u 2 u 3 1 [u 0] lim inf δ 0 b This verifies the clim. Re b b Im u u 2 u 2 + δ 2 3/2 u u u 2 + δ 2 1/2 + Re u u u 2 + δ 2 1/2 b u u u 1 + Re u u u 1 b <. Proposition Suppose cx [0, for ll x R. Let u W 2,1, b. Then the limit exists in [0, nd Re b L = lim δ 0 b c u u u 1 1 [u 0] = L + b c u 2 δ 2 u 2 + δ 2 3/ c Im u u 2 u 3 1 [u 0] c Re u u u 1 b, where we use the convention tht u u u 1 x = 0 for ll x [, b] such tht ux = 0. Proof. Let δ R \ {0}. Tking the rel prt both sides of 2.8 gives Note tht b Re b c u u u 2 + δ 2 1/2 = c Re u u 2 u 2 +δ 2 3/2 = Therefore Re b b c u u u 2 + δ 2 1/2 = b c u 2 u 2 + δ 2 1/2 b c Re u u 2 u 2 + δ 2 3/2 c Re u u u 2 + δ 2 1/2 b. c u u 2 u 2 +δ 2 3/2 b b + b c u 2 δ 2 u 2 + δ 2 3/2 c Im u u 2 u 2 +δ 2 3/2. c Im u u 2 u 2 + δ 2 3/2 c Re u u u 2 + δ 2 1/2 b.

32 2.5. SECOND-ORDER DIFFERENTIAL OPERATORS IN L p -SPACES 21 Clerly lim c Re u u u 2 + δ 2 1/2 b = c Re u u u 1 b, δ 0 where we use the convention tht u u u 1 x = 0 for ll x [, b] such tht ux = 0. It follows from the Lebesgue dominted convergence theorem tht b lim Re c u u u 2 + δ 2 1/2 = Re δ 0 By the monotone convergence theorem we lso hve b lim δ 0 Hence by Corollry 2.32 the limit exists in [0, nd Re b s climed. c Im u u 2 u 2 + δ 2 3/2 = L = lim δ 0 b c u u u 1 1 [u 0] = L + b b b c u 2 δ 2 u 2 + δ 2 3/2 c u u u 1 1 [u 0]. c Im u u 2 u 3 1 [u 0]. c Im u u 2 u 3 1 [u 0] c Re u u u 1 b It is possible tht the limit L defined s in 2.10 is strictly positive for some u W 2,1, b, s shown by the following exmple. Exmple Let = 0, b = 1 nd c = 1 R. Let ux = x for ll x 0, 1. Let L be defined s in Then 1 L = lim δ 0 0 δ 2 1/ δ dx = lim x 2 + δ 2 3/2 δ dx = x /2 0 1 dx = 1. x /2

33 22 CHAPTER 2. PRELIMINARIES

34 Chpter 3 Degenerte elliptic opertors in one dimension 3.1 Introduction The subject to study in this chpter is degenerte elliptic second-order differentil opertors with bounded complex-vlued coefficients in one dimension. We will give core chrcteristion for these opertors. The results re extensions of those in [CMP98, Theorem 3.5] nd [DE15, Theorem 1.5]. The following ssumptions will be mde throughout the chpter without being mentioned explicitly. If further ssumptions re imposed, they will be rticulted in the sttements in which they re required. Let θ [0, π. Define 2 Σ θ = {r e i ψ : r 0 nd ψ θ}. Let c W 1, R, C be such tht cx Σ θ for ll x R. Let m L R, C nd M m > 0. Suppose m M m Re c. 3.1 Let w L R, C. Consider the form 0 defined by 0 u, v = R c u v + m u v + w u v 3.2 on the domin D 0 = Cc R. We will show in Lemm 3.7 in Section 3.2 tht 0 is closble. Let be the closure of the form 0. It follows from the first representtion theorem [Kt80, Theorem VI.2.1] tht there exists n qusi-m-sectoril opertor A ssocited with the form. Formlly we cn write A = d d d c + m dx dx dx + w I. Let S be the C 0 -semigroup generted by A. If A is strongly elliptic, tht is, if inf Re c > 0, then S extends consistently to C 0 -semigroup on L p R for ll p [1, by [AMT98, Theorem 2.21]. We prove in Section 3.3 the following extension. 23

35 24 CHAPTER 3. DEGENERATE ELLIPTIC OPERATORS IN ONE DIMENSION Proposition 3.1. Let p 1, nd m W 1, R, C. Suppose I 1 2 p < cos θ or II 1 2 p cos θ nd m = 0. Then S extends consistently to qusi-contrction C 0 -semigroup on L p R. The cse p = 1 follows directly from [AE12, Corollry 3.13iii]. Proposition 3.2. Suppose c, m W 1, R, R. Then S extends consistently to qusicontrction C 0 -semigroup on L 1 R. Let p [1,. Suppose S extends consistently to C 0 -semigroup S p on L p R. Let A p be the genertor of S p. Clerly Cc R DA p. Our im is to chrcterise when Cc R is core for A p. For this we need to introduce more nottion. Define P = [Re c > 0] nd N = [Re c = 0]. Let {I k : k K} be the set of connected components of P, with I k I k for ll k, k K with k k. Write I k = k, b k for ll k K, with k, b k [, ]. Let E l = { k : k K} R, E r = {b k : k K} R nd E = E l E r be the set of ll finite left endpoints, finite right endpoints nd finite endpoints respectively. The intersection with R is needed to del with unbounded I k nd to obtin tht E R. For ll k K define m k = k +b k 2 if k R nd b k R, k + 1 if k R nd b k =, b k 1 if k = nd b k R, 0 if k = nd b k =. Next define the function Z : R R by { mk 1 if x I Zx = x Re c k nd k K, if x N. The min theorem of this chpter is s follows. Theorem 3.3. Let p 1, nd m W 1, R, C. Suppose I 1 2 p < cos θ or II 1 2 p cos θ nd m = 0. Suppose tht for ll x 0 E there exists δ 0 > 0 such tht m <. c x 0 δ 0,x 0 +δ 0 P Then the spce Cc R is core for A p if nd only if Z x δ,x+δ L q x δ, x + δ for ll x E nd δ > 0, where q is the dul exponent of p.

36 3.2. THE FORM AND THE OPERATOR ON L 2 25 This theorem hs the following obvious corollry. Corollry 3.4. Let p 1, nd m W 1, R, C. Suppose c W 1, R, C nd c is Hölder continuous of order 1 1. Moreover, ssume tht p I θ [0, rccos 1 2 p or II m = 0 nd θ = rccos 1 2. Suppose tht for ll x 0 E there exists δ 0 > 0 such tht m <. c Then the spce C c R is core for A p. p x 0 δ 0,x 0 +δ 0 P If p = 1 then the condition on m c is superfluous in Theorem 3.3. Theorem 3.5. Suppose c, m W 1, R, R. Then the spce C c R is core for A 1. The following is brief summry of the subsequent sections. In Section 3.2 we give detiled description of the form nd its ssocited opertor A. In Section 3.3 we prove Proposition 3.1. We will determine the opertor A p in Section 3.4. The min theorems, Theorems 3.3 nd 3.5, re proved in Sections 3.5 nd 3.6 respectively. Finlly we consider three intriguing exmples to illustrte Theorem 3.3 in Section The form nd the opertor on L 2 Let 0 be s in Section 3.1. We will now show tht 0 is indeed closble. Lemm 3.6. There exist n ω > 0 nd θ [0, π such tht 2 0u + ω u 2 2 Σ θ u D 0. for ll Proof. Let u D 0. Then Re 0 u Re c u 2 m u u w u 2 1 Re c u 2 εm m Re c u 2 4ε + w u 2 2 = 1 εm m 1 Re c u 2 4ε + w u 2 2 for ll ε > 0. Choosing ε = 1 2M m in the bove inequlity gives Re c u 2 2 Re 0 u + M m + 2 w u It follows tht Im 0 u tn θ Re c u 2 + m u u + w u 2

37 26 CHAPTER 3. DEGENERATE ELLIPTIC OPERATORS IN ONE DIMENSION tn θ + M m 1 Re c u w u tn θ + M m 2 Re 0 u + M m + 2 w u w u 2 2 = 2tn θ + M m Re 0 u + tn θ + M m M m + 2 w w u 2 2. Tht is, 0 u + ω u 2 2 Σ θ where 1 ω = tn θ + M m M m + 2 w + 1 2tn θ + M m 4 + w nd θ is such tht The lemm now follows. Proposition 3.7. The form 0 is closble. Proof. Define the opertor A 0 in L 2 R by tnθ = 2tn θ + M m. A 0 u = c u + m u + w u on the domin DA 0 = Cc R. Then 0 u, v = A 0 u, v nd A 0 u, u + ω u 2 2 Σ θ for ll u, v Cc R, where ω nd θ re s in Lemm 3.6. It now follows from [Kt80, Theorem VI.1.27] tht 0 is closble. Our next im is to derive comprehensive description of the closure of 0. If Ω R is open nd u W 1,1 loc Ω, then we denote by u L 1,loc Ω the derivtive. If v L 1,loc R then we sy tht v CΩ if v Ω hs continuous representtive which extends continuously to Ω. For ll u L 1,loc R with u P W 1,1 P define Du: R C by { u x if x P, Dux = 0 if x N. In order to void clutter we write Define the form by nd L 2 R W 1,2 loc P = {u L 2R: u P W 1,2 loc P}. D = {u L 2 R W 1,2 loc P: Re c Du L 2 R} u, v = R c Du Dv + m Du v + w u v c for ll u, v D. Note tht c Du cos θ Re c Du nd m Du Mm Re c Du for ll u D. Hence c Du nd m Du belong to L 2 R if u D. The domin D is nturlly normed with 1/2, u Re u ω u 2 2 where ω is s in Lemm 3.6. The following proposition shows tht we cn replce this norm by simpler one.

38 3.2. THE FORM AND THE OPERATOR ON L 2 27 Proposition 3.8. The norm u where ω is s in Lemm 3.6. Re u ω u 2 2 1/2 on D is equivlent to 1/2, u Re c Du 2 + u 2 2 Proof. Let u D. Then Re u ω u 2 2 Re c u 2 + R 1 + M m m u u ω + w u Re c u ω + w u 2 2 M Re c u 2 + u 2 2, where M = 1 + M m ω + w. Also by 3.3 we hve Re c u 2 + u Re u M m + 2 w u 2 2 This completes the proof M m + 2 w Re u ω u 2 2. Due to Proposition 3.8 we define the norm D on D by u 2 D = Re c Du 2 + u 2 2. Note tht if u D, then u 1 P, u 1 N D nd Moreover, u 1 N D = u 1 N L2 R. R u 2 D = u 1 P 2 D + u 1 N 2 D. 3.4 Lemm 3.9. The form is the closure of 0. Moreover, C c P+L 2 N is dense in D. Proof. We first show tht is closed. Let u n n N be Cuchy sequence in D. Then u n n N is Cuchy sequence in L 2 R, so u = lim u n exists in L 2 R. Similrly v = lim Re c Du n exists in L 2 R. Let τ Cc P. Then 1 u, τ L2 P = limu n, τ L2 P = limu n, τ L2 P = v, τ. Re c L 2 P So u W 1,2 loc P nd u = 1 Re c v P in L 2,loc P. Then Re c Du = v L 2 R. It is now esy to verify tht lim u n = u in D. So is closed. Secondly we show tht for ll u D with u = u 1 P nd for ll ε > 0 there exists v C c P such tht u v D < ε. Since u 1 Ik D for ll u D nd k K nd u 2 D = k K u 1 I k 2 for ll u D with u = u 1 P, it suffices to show tht

39 28 CHAPTER 3. DEGENERATE ELLIPTIC OPERATORS IN ONE DIMENSION for ll u D, k K nd ε > 0 with u = u 1 Ik, there exists v Cc I k such tht u v D < ε. If I k is bounded nd c tkes rel-vlued, this is the content of [CMP98, Lemm 2.6]. Now suppose tht I k = k, b k is unbounded with k R nd b k =. The proof for other cses like bounded I k is similr. Without loss of generlity we my ssume tht u is rel-vlued. Since lim n u n = u in D by [GT83, Lemm 7.6], we my ssumed tht u is bounded. For ll n N with n k + 3 define χ n : R R by 0 if x k + 1 n, nx k 1 n if k + 1 < x n k + 2, n χ n x = 1 if k + 2 < x n, n 1 x 2n if n < x 2n, n 0 if x > 2n. Then χ n W 1,2 R W 1, R, 0 χ n 1 nd R Re c χ n 2 c x k n 2 + k + 1 n, k+ 2 n So χ n u D nd χ n u 2 D R n,2n c n 2 2 c + c. u Re c χ n 2 u Re c χ n 2 Du 2 R R 2 u 2 D + 4 c + 2 c u 2 for ll n N with n k +3. Pssing to subsequence if necessry, the sequence χ n u n N is wekly convergent in D. But lim χ n u = u in L 2 R. So lim χ n u = u in D. Let V = {v D: supp v is compct nd supp v I k }. We proved tht u is in the wek closure of V in D. Since V is convex, the wek closure equls the norm closure nd u is therefore in the norm closure of V in D. By regulrising elements of V we see tht u is n element of the norm closure of Cc I k in D. Thirdly, the density of Cc P+L 2 N in D now follows from 3.4 nd the previous step. Finlly, let u D nd ε > 0. Since Cc R is dense in L 2 R, there exists v 1 Cc R such tht u v 1 L2 R < ε. Therefore u v 1 1 N D = u v 1 1 N L2 N u v 1 L2 R < ε. By the bove there exists v 2 Cc P such tht u v 1 1 P v 2 D < ε. Using 3.4 it follows tht u v 1 v 2 2 D = u v 1 1 P v 2 2 D + u v 1 1 N 2 D 2ε 2. So C c R is dense in D nd is the closure of 0. Let A be the opertor ssocited with. possible, s shown in the following lemm. Then the detiled descriptions of A re