On m-sectorial Schrödinger-type operators with singular potentials on manifolds of bounded geometry
|
|
- Corey Bates
- 6 years ago
- Views:
Transcription
1 Comment.Math.Univ.Carolin. 45,1(2004) On m-sectorial Schrödinger-type operators with singular potentials on manifolds of bounded geometry Ognjen Milatovic Abstract. WeconsideraSchrödinger-typedifferentialexpression H V = +V,where isac -boundedhermitianconnectiononahermitianvectorbundle Eofbounded geometryoveramanifoldofboundedgeometry(m, g)withmetric gandpositive C - bounded measure dµ, and V is a locally integrable section of the bundle of endomorphismsof E. Wegiveasufficientconditionfor m-sectorialityofarealizationof H V in L 2 (E). IntheproofweusegeneralizedKato sinequalityaswellasaresultonthe positivityof u L 2 (M)satisfyingtheequation( M + b)u=ν,where M isthescalar Laplacianon M, b >0isaconstantand ν 0isapositivedistributionon M. Keywords: Schrödinger operator, m-sectorial, manifold, bounded geometry, singular potential Classification: Primary 35P05, 58J50; Secondary 47B25, 81Q10 1. Introduction and the main result 1.1 The setting. Let(M, g)beac Riemannianmanifoldwithoutboundary,withmetric ganddimm = n. Wewillassumethat M isconnected. We will also assume that M has bounded geometry. Moreover, we will assume that wearegivenapositive C -boundedmeasure dµ,i.e.inanylocalcoordinates x 1, x 2,...,x n thereexistsastrictlypositive C -boundeddensity ρ(x)suchthat dµ=ρ(x)dx 1 dx 2... dx n. Let EbeaHermitianvectorbundleover M.Wewillassumethat Eisabundle of bounded geometry(i.e. it is supplied by an additional structure: trivializations of E on every canonical coordinate neighborhood U such that the corresponding matrixtransitionfunctions h U,U onallintersections U U ofsuchneighborhoodsare C -bounded,i.e.allderivatives y αh U,U (y),where αisamultiindex, withrespecttocanonicalcoordinatesareboundedwithbounds C α whichdonot dependonthechosenpair U, U ). Wedenoteby L 2 (E)theHilbertspaceofsquareintegrablesectionsof Ewith respect to the scalar product (1.1) (u, v) = u(x), v(x) dµ(x). M Here, denotesthefiberwiseinnerproductin E x.
2 92 O. Milatovic Inwhatfollows, C (E)denotessmoothsectionsof E,and C c (E)denotes smooth compactly supported sections of E. Let : C (E) C (T M E) beahermitianconnectionon E whichis C -boundedasalineardifferential operator, i.e. in any canonical coordinate system U(with the chosen trivializations of E U and(t M E) U ), iswrittenintheform = a α (y) y α, α 1 where αisamultiindex,andthecoefficients a α (y)arematrixfunctionswhose derivatives β y a α (y)foranymultiindex βareboundedbyaconstant C β which does not depend on the chosen canonical neighborhood. We will consider a Schrödinger type differential expression of the form H V = +V, where V isameasurablesectionofthebundleendeofendomorphismsof E. Here : C (T M E) C (E) isadifferentialoperatorwhichisformallyadjointto withrespecttothescalar product(1.1). Ifwetake =d,where d: C (M) Ω 1 (M)isthestandarddifferential, then d d: C (M) C (M)iscalledthescalarLaplacianandwillbedenoted by M. In what follows, we use the notations (1.2) (Re V)(x):= V(x)+(V(x)) 2, (Im V)(x):= V(x) (V(x)) 2i, x M, where i= 1and(V(x)) denotestheadjointofthelinearoperator V(x): E x E x (inthesenseoflinearalgebra). By(1.2),forall x M,(Re V)(x)and(ImV)(x)areself-adjointlinearoperators E x E x,andwehavethefollowingdecomposition: V(x)=(Re V)(x)+i(Im V)(x). Forevery x M,wehavethefollowingdecomposition: (1.3) (Re V)(x)=(Re V) + (x) (Re V) (x). Here(Re V) + (x)=p + (x)(re V)(x),where P + (x):= χ [0,+ ) ((Re V)(x)),and (Re V) (x)= P (x)(re V)(x),where P (x):= χ (,0) ((Re V)(x)). Here χ A denotes the characteristic function of the set A. Wemakethefollowingassumptionon V.
3 On m-sectorial Schrödinger-type operators Assumption A. (i) (Re V) + L 1 loc (EndE),(Re V) L 1 loc (EndE)and(ImV) L1 loc (EndE). (ii) Thereexistsaconstant L >0suchthatforall u L 2 (E)andall x M, (1.4) (Im V)(x) u(x) 2 L (Re V) + (x)u(x), u(x), where (Im V)(x) denotesthenormoftheoperator(im V)(x): E x E x, u(x) denotesthenorminthefiber E x and, denotestheinnerproduct in E x. 1.2Sobolevspace W 1,2 (E). By W 1,2 (E)wewilldenotethecompletionofthe space C c (E)withrespecttothenorm 1definedbythescalarproduct (u, v) 1 := (u, v) + ( u, v) u, v C c (E). By W 1,2 (E)wewilldenotethedualof W 1,2 (E). 1.3 Quadratic forms. In what follows, all quadratic forms are considered in thehilbertspace L 2 (E). 1.By h 0 wedenotethequadraticform (1.5) h 0 (u)= u 2 dµ withthedomaind(h 0 )=W 1,2 (E) L 2 (E). Thequadraticform h 0 isnonnegative,denselydefined(since C c (E) D(h 0 ))andclosed(seesection1.2). 2.By h 1 wedenotethequadraticform (1.6) h 1 (u)= (Re V) + u, u +i (Im V)u, u dµ with the domain (1.7) D(h 1 )= { u L 2 (E): (Re V) + u, u +i (Im V)u, u dµ <+ }. Here, denotesthefiberwiseinnerproductin E x. Inwhatfollows,wewilldenoteby h 1 (, )thecorrespondingsesquilinearform obtainedviapolarizationidentityfrom h 1. Thequadraticform h 1 issectorial.indeed,bytheinequalities (1.8) (Im V)u(x), u(x) (Im V)(x)u(x) u(x) (Im V)(x) u(x) 2 andby(1.4),forall u D(h 1 ),thevaluesof h 1 (u)lieinasectorof Cwithvertex γ=0. Theform h 1 isdenselydefinedsince,by(i)ofassumptiona,wehave
4 94 O. Milatovic Cc (E) D(h 1). Theform h 1 isclosed. Indeed,byTheoremVI.1.11in[6],it sufficestoshowthatthepre-hilbertspaced(h 1 )withtheinnerproduct (u, v) h1 =(Re h 1 )(u, v)+(u, v)= (Re V) + u, v dµ+(u, v), iscomplete.here(, )denotestheinnerproductin L 2 (E)and(Re h 1 )(, )denotes therealpartofthesesquilinearform h 1 (, )(seethedefinitionbelowtheequation (1.9) in Section VI.1.1 of[6]). By(1.7),(1.8)and(1.4),itfollowsthatD(h 1 )isthesetofall u L 2 (E)such that u 2 h 1 <+,where h1 denotesthenormcorrespondingtotheinner product(, ) h1.byexamplevi.1.15in[6],itfollowsthatd(h 1 )iscomplete. 3.By h 2 wedenotethequadraticform (1.9) h 2 (u)= (Re V) u, u dµ with the domain (1.10) D(h 2 )= { u L 2 (E): (Re V) u, u dµ <+ }, where, denotesthefiberwiseinnerproductin E x. Theform h 2 isdenselydefinedbecause, by(i)ofassumptiona,wehave C c (E) D(h 2 ).Moreover,sinceforall x M,theoperator(Re V) (x): E x E x isself-adjoint,itfollowsthatthequadraticform h 2 issymmetric. Wemakethefollowingassumptionon h 2. AssumptionB. Assumethat h 2 is h 0 -boundedwithrelativebound b <1,i.e. (i) D(h 2 ) D(h 0 ), (ii) thereexistconstants a 0and0 b <1suchthat (1.11) h 2 (u) a u 2 + b h 0 (u), forall u D(h 0 ), where denotesthenormin L 2 (E). Remark1.4.If(M, g)isamanifoldofboundedgeometry,assumptionbholdsif (Re V) L p (EndE),where p=n/2for n 3, p >1for n=2,and p=1for n=1.fortheproof,see,forexample,theproofofremark2.1in[7]. 1.5Arealizationof H V in L 2 (E). Wedefineanoperator Sin L 2 (E)bythe formula Su=H V uonthedomain (1.12) { (Re u W 1,2 (E): V) + u, u +i (Im V)u, u dµ <+ and H V u L 2 (E) }.
5 On m-sectorial Schrödinger-type operators Wewilldenotethesetin(1.12)byDom(S). Remark1.6.Forall u D(h 0 )=W 1,2 (E)wehave u W 1,2 (E). From Corollary2.11belowitfollowsthatforall u W 1,2 (E) D(h 1 ),wehave V u L 1 loc (E). Thus H V uin(1.12)isadistributionalsectionof E,andthecondition H V u L 2 (E)makessense. Remark1.7.By(1.4)andby(1.8),thesetDom(S)in(1.12)isequalto { u W 1,2 (E): (Re V) + u, u dµ <+ and H V u L 2 (E) }. Wenowstatethemainresult. Theorem 1.8. Assumethat(M, g)isamanifoldofboundedgeometrywith positive C -boundedmeasure dµ, EisaHermitianvectorbundleofbounded geometryover M,and isac -boundedhermitianconnectionon E.Suppose thatassumptionsaandbhold.thentheoperator Sis m-sectorial. Remark1.9.Theorem1.8extendsaresultofT.Kato;seeTheoremVI.4.6(a)in[6] (seealsoremark5(a)in[5])whichwasprovenfortheoperator +V,where isthestandardlaplacianon R n withthestandardmetricandmeasure,and V L 1 loc (Rn )isasinassumptionsaandbabove(withimv =0).Theorem1.8 also extends the result in[7] which establishes the self-adjointness of a realization in L 2 (E)of H V = +V onmanifold(m, g)with dµ, E,and asinthe hypothesesoftheorem1.8,and V = V 1 + V 2,where0 V 1 L 1 loc (EndE)and 0 V 2 L 1 loc (EndE)arelinearself-adjointbundleendomorphismssatisfying AssumptionsAandB(withImV =0). 2. ProofofTheorem1.8 WeadopttheargumentsfromSectionVI.4.3in[6]tooursettingwiththehelp of a more general version of Kato s inequality(2.1). 2.1 Kato s inequality. We begin with the following variant of Kato s inequality for Bochner Laplacian(for the proof, see Theorem 5.7 in[2]). Lemma2.2. Assumethat(M, g)isariemannianmanifold.assumethat Eisa Hermitianvectorbundleover Mand isahermitianconnectionon E.Assume that w L 1 loc (E)and w L 1 loc (E).Then (2.1) M w Re w,sign w, where sign w(x) = { w(x) if w(x) 0, w(x) 0 otherwise. Remark 2.3. The original version of Kato s inequality was proven in Kato[3].
6 96 O. Milatovic 2.4 Positivity. In what follows, we will use the following lemma whose proof is giveninappendixbof[2]. Lemma 2.5. Assumethat(M, g)isamanifoldofboundedgeometrywitha smooth positive measure dµ. Assume that ( b+ M ) u = ν 0, u L 2 (M), where b >0, M = d disthescalarlaplacianon M,andtheinequality ν 0 meansthat ν isapositivedistributionon M,i.e.(ν, φ) 0forany0 φ C c (M). Then u 0(almost everywhere or, equivalently, as a distribution). Remark2.6.ItisnotknownwhetherLemma2.5holdsif Misanarbitrarycomplete Riemannian manifold. For more details about difficulties in the case of arbitrary complete Riemannian manifolds, see Appendix B of[2]. Lemma2.7. Thequadraticform h:=(h 0 +h 1 )+h 2 isdenselydefined,sectorial and closed. Proof:Since h 0 and h 1 aresectorialandclosed,itfollowsbytheoremvi.1.31 from[6]that h 0 +h 1 issectorialandclosed.by(i)ofassumptionbitfollowsthat D(h 2 ) D(h 0 ) D(h 1 ),andby(1.11),(1.5),and(1.6),thefollowinginequality holds: h 2 (u) a u 2 + b h 0 (u)+h 1 (u), forall u D(h 0 ) D(h 1 ), where denotesthenormin L 2 (E),and a 0and0 b <1areasin(1.11). Thusthequadraticform h 2 is(h 0 +h 1 )-boundedwithrelativebound b <1.Since h 0 + h 1 isaclosedsectorialform,bytheoremvi.1.33from[6],itfollowsthat h=(h 0 + h 1 )+h 2 isaclosedsectorialform. Since Cc (E) D(h 0 ) D(h 1 ) D(h 2 ),itfollowsthat hisdenselydefined. In what follows, h(, ) will denote the corresponding sesquilinear form obtained from h via polarization identity. 2.8 m-sectorial operator H associated with h. Since h is a densely defined, closedandsectorialformin L 2 (E),byTheoremVI.2.1from[6]thereexistsan m-sectorialoperator Hin L 2 (E)suchthat (i) Dom(H) D(h)and h(u, v)=(hu, v), (ii) Dom(H)isacoreof h, (iii) if u D(h), w L 2 (E),and forall u Dom(H)and v D(h), h(u, v)=(w, v) holdsforevery vbelongingtoacoreof h,then u Dom(H)and Hu=w. The operator H is uniquely determined by condition(i). We will also use the following lemma.
7 On m-sectorial Schrödinger-type operators Lemma2.9. Assumethat0 T L 1 loc (EndE)isalinearself-adjointbundle map.assumealsothat u Q(T),where Q(T)={u L 2 (E): Tu, u L 1 (M)}. Then Tu L 1 loc (E). Proof:Byaddingaconstantwecanassumethat T 1(inoperatorsense). Assumethat u Q(T). Wechoose(inameasurableway)anorthogonal basisin eachfiber E x anddiagonalize1 T(x) End(E x ) toget T(x) = diag(c 1 (x), c 2 (x),..., c m (x)), where 0 < c j L 1 loc (M), j = 1,2,..., m and m=dime x. Let u j (x)(j=1,2,..., m)bethecomponentsof u(x) E x withrespectto thechosenorthogonalbasisof E x.thenforall x M Tu, u = m c j (x) u j (x) 2. j=1 Since u Q(T),weknowthat0< Tu, u dµ <+. Since c j >0,itfollows that c j u j 2 L 1 (M),forall j=1,2,...,m. Now,forall x Mand j=1,2,, m (2.2) 2 c j u j =2 c j u j c j + c j u j 2. Therighthandsideof(2.2)isclearlyin L 1 loc (M).Therefore c ju j L 1 loc (M). But(Tu)(x)hascomponents c j (x)u j (x)(j=1,2,...,m)withrespecttochosenbasesof E x.therefore Tu L 1 loc (E),andthelemmaisproven. Corollary2.10. If u D(h 1 ),then((re V) + + i(im V))u L 1 loc (E). Proof:Let u D(h 1 ).Then (Re V) + u, u L 1 (M),and,hence,byLemma2.9 weget(re V) + u L 1 loc (E).By(1.4)weobtain (Im V) u 2 L 1 (M).Sincefor all x Mwehave 2 (Im V)(x)u(x) 2 (Im V)(x) u(x) (Im V)(x) + (Im V)(x) u(x) 2, and since, by AssumptionA, (Im V) L 1 loc (M), it followsthat (ImV)u L 1 loc (E),andthecorollaryisproven. Corollary2.11. If u D(h),then V u L 1 loc (E). Proof: Let u D(h) = D(h 0 ) D(h 1 ). ByCorollary2.10it followsthat ((Re V) + + i(im V))u L 1 loc (E).SinceD(h) D(h 2)andsince(Re V) (x) 0 asanoperator E x E x,bylemma2.9wehave(re V) u L 1 loc (E). Thus V u L 1 loc (E),andthecorollaryisproven.
8 98 O. Milatovic Lemma The following operator relation holds: H S. Proof:Wewillshowthatforall u Dom(H),wehave Hu=H V u. Let u Dom(H).Byproperty(i)ofSection2.8wehave u D(h);hence,by Corollary2.11weget V u L 1 loc (E).Then,forany v C c (E),wehave (2.3) (Hu, v)=h(u, v)=( u, v)+ V u, v dµ, where(, )denotesthe L 2 -innerproduct. Thefirstequalityin(2.3)holdsbyproperty(i)fromSection2.8,andthesecond equality holds by definition of h. Hence,usingintegrationbypartsinthefirsttermontherighthandsideof thesecondequalityin(2.3)(see,forexamplelemma8.8from[2]),weget (2.4) (u, v)= Hu V u, v dµ, forall v C c (E). Since V u L 1 loc (E)and Hu L2 (E),itfollowsthat(Hu V u) L 1 loc (E),and (2.4)implies u=hu V u(asdistributionalsectionsof E).Therefore, u+vu=hu, andthisshowsthat Hu=H V uforall u Dom(H). Nowbydefinitionof SitfollowsthatDom(H) Dom(S)and Hu=Sufor all u Dom(H).Therefore H S,andthelemmaisproven. Lemma2.13. C c (E)isacoreofthequadraticform h 0+ h 1. Proof:Itsufficestoshow(seeTheoremVI.1.21in[6]andtheparagraphabove theequation(1.31)insectionvi.1.3of[6])that C c (E)isdenseintheHilbert spaced(h 0 + h 1 )=D(h 0 ) D(h 1 )withtheinnerproduct (u, v) h0 +h 1 := h 0 (u, v)+(re h 1 )(u, v)+(u, v), where h 0 (, )denotesthesesquilinearformcorrespondingto h 0 viapolarization identityand(re h 1 )denotestherealpartofthesesquilinearform h 1 (, ). Let u D(h 0 + h 1 )and(u, v) h0 +h 1 =0forall v C c (E).Wewillshowthat u=0. We have (2.5) 0 = h 0 (u, v)+(re h 1 )(u, v)+(u, v) = (u, v)+ (Re V) + u, v dµ+(u, v).
9 On m-sectorial Schrödinger-type operators Hereweusedintegrationbypartsinthefirsttermontherighthandsideofthe second equality. Since u D(h 0 +h 1 ) D(h 1 ),itfollowsthat (Re V) + u, u +i (ImV)u, u L 1 (M). Hence (Re V) + u, u L 1 (M). ByLemma2.9weget(Re V) + u L 1 loc (E).From(2.5)wegetthefollowingdistributionalequality: (2.6) u=( (Re V) + 1)u. From(2.6)wehave u L 1 loc (E).ByLemma2.2andby(2.6),weobtain (2.7) M u Re u,signu = ((Re V) + +1)u,signu u. Thelastinequalityin(2.7)holdssince(Re V) + (x) 0asanoperator E x E x. Therefore, (2.8) ( M +1) u 0. ByLemma2.5,itfollowsthat u 0.So u=0,andthelemmaisproven. Lemma2.14. C c (E)isacoreofthequadraticform h=(h 0+ h 1 )+h 2. Proof: Sincethequadraticform h 2 is(h 0 + h 1 )-bounded,thelemmafollows immediately from Lemma ProofofTheorem1.8 ByLemma2.12wehave H S,soitisenoughtoshowthatDom(S) Dom(H). Let u Dom(S).BydefinitionofDom(S)inSection1.5,wehave u D(h 0 ) D(h 2 )and u D(h 1 ).Hence u D(h). Forall v C c (E)wehave h(u, v)=h 0 (u, v)+h 1 (u, v)+h 2 (u, v)=(u, v)+ V u, v dµ=(h V u, v). Thelastequalityholdssince H V u=su L 2 (E). ByLemma2.14itfollows that C c (E)isaformcoreof h. Nowfromproperty(iii)ofSection2.8wehave u Dom(H)with Hu=H V u.thisconcludestheproofofthetheorem. References [1] Aubin T., Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, Berlin, [2] Braverman M., Milatovic O., Shubin M., Essential self-adjointness of Schrödinger type operators on manifolds, Russian Math. Surveys 57(2002), no. 4, [3] Kato T., Schrödinger operators with singular potentials, Israel J. Math. 13(1972),
10 100 O. Milatovic [4] Kato T., A second look at the essential selfadjointness of the Schrödinger operators, Physical reality and mathematical description, Reidel, Dordrecht, 1974, pp [5] Kato T., On some Schrödinger operators with a singular complex potential, Ann. Scuola Norm.Sup.PisaCl.Sci(4)5(1978), [6] Kato T., Perturbation Theory for Linear Operators, Springer-Verlag, New York, [7] Milatovic O., Self-adjointness of Schrödinger-type operators with singular potentials on manifolds of bounded geometry, Electron. J. Differential Equations, No. 64(2003), 8pp (electronic). [8] Reed M., Simon B., Methods of Modern Mathematical Physics I, II: Functional Analysis. Fourier Analysis, Self-adjointness, Academic Press, New York e.a., [9] Shubin M.A., Spectral theory of elliptic operators on noncompact manifolds, Astérisque no. 207(1992), [10] Taylor M., Partial Differential Equations II: Qualitative Studies of Linear Equations, Springer-Verlag, New York e.a., Department of Mathematics, University of Toledo, Toledo, OH , USA Ognjen.Milatovic@utoledo.edu (Received September 9, 2003)
SELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS
More informationTitle: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on
Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on manifolds. Author: Ognjen Milatovic Department Address: Department
More informationTHE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS
THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily
More informationOn m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry
On m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry Ognjen Milatovic Department of Mathematics and Statistics University of North Florida Jacksonville, FL 32224 USA. Abstract
More informationA PROPERTY OF SOBOLEV SPACES ON COMPLETE RIEMANNIAN MANIFOLDS
Electronic Journal of Differential Equations, Vol. 2005(2005), No.??, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A PROPERTY
More informationSectorial Forms and m-sectorial Operators
Technische Universität Berlin SEMINARARBEIT ZUM FACH FUNKTIONALANALYSIS Sectorial Forms and m-sectorial Operators Misagheh Khanalizadeh, Berlin, den 21.10.2013 Contents 1 Bounded Coercive Sesquilinear
More informationGeneralized Schrödinger semigroups on infinite weighted graphs
Generalized Schrödinger semigroups on infinite weighted graphs Institut für Mathematik Humboldt-Universität zu Berlin QMATH 12 Berlin, September 11, 2013 B.G, Ognjen Milatovic, Francoise Truc: Generalized
More informationTHE ESSENTIAL SELF-ADJOINTNESS OF SCHRÖDINGER OPERATORS WITH OSCILLATING POTENTIALS. Adam D. Ward. In Memory of Boris Pavlov ( )
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 46 (016), 65-7 THE ESSENTIAL SELF-ADJOINTNESS OF SCHRÖDINGER OPERATORS WITH OSCILLATING POTENTIALS Adam D. Ward (Received 4 May, 016) In Memory of Boris Pavlov
More informationWave equation on manifolds and finite speed of propagation
Wave equation on manifolds and finite speed of propagation Ethan Y. Jaffe Let M be a Riemannian manifold (without boundary), and let be the (negative of) the Laplace-Beltrami operator. In this note, we
More informationPeriodic Schrödinger operators with δ -potentials
Networ meeting April 23-28, TU Graz Periodic Schrödinger operators with δ -potentials Andrii Khrabustovsyi Institute for Analysis, Karlsruhe Institute of Technology, Germany CRC 1173 Wave phenomena: analysis
More informationON THE SPECTRUM OF THE DIRICHLET LAPLACIAN IN A NARROW STRIP
ON THE SPECTRUM OF THE DIRICHLET LAPLACIAN IN A NARROW STRIP LEONID FRIEDLANDER AND MICHAEL SOLOMYAK Abstract. We consider the Dirichlet Laplacian in a family of bounded domains { a < x < b, 0 < y < h(x)}.
More informationWentzell Boundary Conditions in the Nonsymmetric Case
Math. Model. Nat. Phenom. Vol. 3, No. 1, 2008, pp. 143-147 Wentzell Boundary Conditions in the Nonsymmetric Case A. Favini a1, G. R. Goldstein b, J. A. Goldstein b and S. Romanelli c a Dipartimento di
More informationHeat kernels of some Schrödinger operators
Heat kernels of some Schrödinger operators Alexander Grigor yan Tsinghua University 28 September 2016 Consider an elliptic Schrödinger operator H = Δ + Φ, where Δ = n 2 i=1 is the Laplace operator in R
More informationQUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE. Denitizable operators in Krein spaces have spectral properties similar to those
QUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE BRANKO CURGUS and BRANKO NAJMAN Denitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces.
More informationThe Kato square root problem on vector bundles with generalised bounded geometry
The Kato square root problem on vector bundles with generalised bounded geometry Lashi Bandara (Joint work with Alan McIntosh, ANU) Centre for Mathematics and its Applications Australian National University
More informationThe spectral zeta function
The spectral zeta function Bernd Ammann June 4, 215 Abstract In this talk we introduce spectral zeta functions. The spectral zeta function of the Laplace-Beltrami operator was already introduced by Minakshisundaram
More informationRough metrics, the Kato square root problem, and the continuity of a flow tangent to the Ricci flow
Rough metrics, the Kato square root problem, and the continuity of a flow tangent to the Ricci flow Lashi Bandara Mathematical Sciences Chalmers University of Technology and University of Gothenburg 22
More informationThe Kato square root problem on vector bundles with generalised bounded geometry
The Kato square root problem on vector bundles with generalised bounded geometry Lashi Bandara (Joint work with Alan McIntosh, ANU) Centre for Mathematics and its Applications Australian National University
More informationTHE L 2 -HODGE THEORY AND REPRESENTATION ON R n
THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some
More informationSELF-ADJOINTNESS OF DIRAC OPERATORS VIA HARDY-DIRAC INEQUALITIES
SELF-ADJOINTNESS OF DIRAC OPERATORS VIA HARDY-DIRAC INEQUALITIES MARIA J. ESTEBAN 1 AND MICHAEL LOSS Abstract. Distinguished selfadjoint extension of Dirac operators are constructed for a class of potentials
More informationREMARKS ON THE MEMBRANE AND BUCKLING EIGENVALUES FOR PLANAR DOMAINS. Leonid Friedlander
REMARKS ON THE MEMBRANE AND BUCKLING EIGENVALUES FOR PLANAR DOMAINS Leonid Friedlander Abstract. I present a counter-example to the conjecture that the first eigenvalue of the clamped buckling problem
More informationMULTIPLE SOLUTIONS FOR AN INDEFINITE KIRCHHOFF-TYPE EQUATION WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2015 (2015), o. 274, pp. 1 9. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIOS
More informationPOINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO
POINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO HART F. SMITH AND MACIEJ ZWORSKI Abstract. We prove optimal pointwise bounds on quasimodes of semiclassical Schrödinger
More informationPERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 207 222 PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS Fumi-Yuki Maeda and Takayori Ono Hiroshima Institute of Technology, Miyake,
More informationABSOLUTELY CONTINUOUS SPECTRUM OF A TYPICAL SCHRÖDINGER OPERATOR WITH A SLOWLY DECAYING POTENTIAL
ABSOLUTELY CONTINUOUS SPECTRUM OF A TYPICAL SCHRÖDINGER OPERATOR WITH A SLOWLY DECAYING POTENTIAL OLEG SAFRONOV 1. Main results We study the absolutely continuous spectrum of a Schrödinger operator 1 H
More informationL -uniqueness of Schrödinger operators on a Riemannian manifold
L -uniqueness of Schrödinger operators on a Riemannian manifold Ludovic Dan Lemle Abstract. The main purpose of this paper is to study L -uniqueness of Schrödinger operators and generalized Schrödinger
More informationSUBELLIPTIC CORDES ESTIMATES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 000-9939XX0000-0 SUBELLIPTIC CORDES ESTIMATES Abstract. We prove Cordes type estimates for subelliptic linear partial
More informationGORDON TYPE THEOREM FOR MEASURE PERTURBATION
Electronic Journal of Differential Equations, Vol. 211 211, No. 111, pp. 1 9. ISSN: 172-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GODON TYPE THEOEM FO
More informationRadial Symmetry of Minimizers for Some Translation and Rotation Invariant Functionals
journal of differential equations 124, 378388 (1996) article no. 0015 Radial Symmetry of Minimizers for Some Translation and Rotation Invariant Functionals Orlando Lopes IMECCUNICAMPCaixa Postal 1170 13081-970,
More informationSOLVABILITY RELATIONS FOR SOME NON FREDHOLM OPERATORS
SOLVABILITY RELATIONS FOR SOME NON FREDHOLM OPERATORS Vitali Vougalter 1, Vitaly Volpert 2 1 Department of Mathematics and Applied Mathematics, University of Cape Town Private Bag, Rondebosch 7701, South
More informationOn the Virial Theorem in Quantum Mechanics
Commun. Math. Phys. 208, 275 281 (1999) Communications in Mathematical Physics Springer-Verlag 1999 On the Virial Theorem in Quantum Mechanics V. Georgescu 1, C. Gérard 2 1 CNRS, Département de Mathématiques,
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More informationOn the Solvability Conditions for a Linearized Cahn-Hilliard Equation
Rend. Istit. Mat. Univ. Trieste Volume 43 (2011), 1 9 On the Solvability Conditions for a Linearized Cahn-Hilliard Equation Vitaly Volpert and Vitali Vougalter Abstract. We derive solvability conditions
More informationInverse scattering problem with underdetermined data.
Math. Methods in Natur. Phenom. (MMNP), 9, N5, (2014), 244-253. Inverse scattering problem with underdetermined data. A. G. Ramm Mathematics epartment, Kansas State University, Manhattan, KS 66506-2602,
More informationIsoperimetric Inequalities for the Cauchy-Dirichlet Heat Operator
Filomat 32:3 (218), 885 892 https://doi.org/1.2298/fil183885k Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Isoperimetric Inequalities
More informationHOMEOMORPHISMS OF BOUNDED VARIATION
HOMEOMORPHISMS OF BOUNDED VARIATION STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN Abstract. We show that the inverse of a planar homeomorphism of bounded variation is also of bounded variation. In higher
More informationAn Example of Embedded Singular Continuous Spectrum for One-Dimensional Schrödinger Operators
Letters in Mathematical Physics (2005) 72:225 231 Springer 2005 DOI 10.1007/s11005-005-7650-z An Example of Embedded Singular Continuous Spectrum for One-Dimensional Schrödinger Operators OLGA TCHEBOTAEVA
More informationEXPONENTIAL ESTIMATES FOR QUANTUM GRAPHS
Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 162, pp. 1 12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXPONENTIAL ESTIMATES FOR QUANTUM GRAPHS
More informationarxiv: v1 [math.ap] 5 Nov 2018
STRONG CONTINUITY FOR THE 2D EULER EQUATIONS GIANLUCA CRIPPA, ELIZAVETA SEMENOVA, AND STEFANO SPIRITO arxiv:1811.01553v1 [math.ap] 5 Nov 2018 Abstract. We prove two results of strong continuity with respect
More informationOn projective classification of plane curves
Global and Stochastic Analysis Vol. 1, No. 2, December 2011 Copyright c Mind Reader Publications On projective classification of plane curves Nadiia Konovenko a and Valentin Lychagin b a Odessa National
More informationBASIC MATRIX PERTURBATION THEORY
BASIC MATRIX PERTURBATION THEORY BENJAMIN TEXIER Abstract. In this expository note, we give the proofs of several results in finitedimensional matrix perturbation theory: continuity of the spectrum, regularity
More informationarxiv:math/ v1 [math.ap] 22 May 2006
arxiv:math/0605584v1 [math.ap] 22 May 2006 AN ANALOGUE OF THE OPERATOR CURL FOR NONABELIAN GAUGE GROUPS AND SCATTERING THEORY A. SEVOSTYANOV Abstract. We introduce a new perturbation for the operator curl
More informationSpectral Gap and Concentration for Some Spherically Symmetric Probability Measures
Spectral Gap and Concentration for Some Spherically Symmetric Probability Measures S.G. Bobkov School of Mathematics, University of Minnesota, 127 Vincent Hall, 26 Church St. S.E., Minneapolis, MN 55455,
More informationOn Schrödinger equations with inverse-square singular potentials
On Schrödinger equations with inverse-square singular potentials Veronica Felli Dipartimento di Statistica University of Milano Bicocca veronica.felli@unimib.it joint work with Elsa M. Marchini and Susanna
More informationOn a generalized Sturm theorem
On a generalized Sturm theorem Alessandro Portaluri May 21, 29 Abstract Sturm oscillation theorem for second order differential equations was generalized to systems and higher order equations with positive
More informationATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS. Emerson A. M. de Abreu Alexandre N.
ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS Emerson A. M. de Abreu Alexandre N. Carvalho Abstract Under fairly general conditions one can prove that
More informationAPPROXIMATION OF MOORE-PENROSE INVERSE OF A CLOSED OPERATOR BY A SEQUENCE OF FINITE RANK OUTER INVERSES
APPROXIMATION OF MOORE-PENROSE INVERSE OF A CLOSED OPERATOR BY A SEQUENCE OF FINITE RANK OUTER INVERSES S. H. KULKARNI AND G. RAMESH Abstract. Let T be a densely defined closed linear operator between
More informationOn some weighted fractional porous media equations
On some weighted fractional porous media equations Gabriele Grillo Politecnico di Milano September 16 th, 2015 Anacapri Joint works with M. Muratori and F. Punzo Gabriele Grillo Weighted Fractional PME
More informationMem. Differential Equations Math. Phys. 36 (2005), T. Kiguradze
Mem. Differential Equations Math. Phys. 36 (2005), 42 46 T. Kiguradze and EXISTENCE AND UNIQUENESS THEOREMS ON PERIODIC SOLUTIONS TO MULTIDIMENSIONAL LINEAR HYPERBOLIC EQUATIONS (Reported on June 20, 2005)
More informationA LIOUVILLE THEOREM FOR p-harmonic
A LIOUVILLE THEOREM FOR p-harmonic FUNCTIONS ON EXTERIOR DOMAINS Daniel Hauer School of Mathematics and Statistics University of Sydney, Australia Joint work with Prof. E.N. Dancer & A/Prof. D. Daners
More informationThe effect of Group Velocity in the numerical analysis of control problems for the wave equation
The effect of Group Velocity in the numerical analysis of control problems for the wave equation Fabricio Macià École Normale Supérieure, D.M.A., 45 rue d Ulm, 753 Paris cedex 5, France. Abstract. In this
More informationESTIMATES OF DERIVATIVES OF THE HEAT KERNEL ON A COMPACT RIEMANNIAN MANIFOLD
PROCDINGS OF H AMRICAN MAHMAICAL SOCIY Volume 127, Number 12, Pages 3739 3744 S 2-9939(99)4967-9 Article electronically published on May 13, 1999 SIMAS OF DRIVAIVS OF H HA KRNL ON A COMPAC RIMANNIAN MANIFOLD
More informationApplied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.
Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R
More informationStability of the Magnetic Schrödinger Operator in a Waveguide
Communications in Partial Differential Equations, 3: 539 565, 5 Copyright Taylor & Francis, Inc. ISSN 36-53 print/153-4133 online DOI: 1.181/PDE-5113 Stability of the Magnetic Schrödinger Operator in a
More informationSMOOTHING ESTIMATES OF THE RADIAL SCHRÖDINGER PROPAGATOR IN DIMENSIONS n 2
Acta Mathematica Scientia 1,3B(6):13 19 http://actams.wipm.ac.cn SMOOTHING ESTIMATES OF THE RADIAL SCHRÖDINGER PROPAGATOR IN DIMENSIONS n Li Dong ( ) Department of Mathematics, University of Iowa, 14 MacLean
More informationProceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005
Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER
More informationThe Mollifier Theorem
The Mollifier Theorem Definition of the Mollifier The function Tx Kexp 1 if x 1 2 1 x 0 if x 1, x R n where the constant K is chosen such that R n Txdx 1, is a test function on Rn. Note that Tx vanishes,
More informationNOTES ON NEWLANDER-NIRENBERG THEOREM XU WANG
NOTES ON NEWLANDER-NIRENBERG THEOREM XU WANG Abstract. In this short note we shall recall the classical Newler-Nirenberg theorem its vector bundle version. We shall also recall an L 2 -Hörmer-proof given
More informationEdge-cone Einstein metrics and Yamabe metrics
joint work with Ilaria Mondello Kazuo AKUTAGAWA (Chuo University) MSJ-SI 2018 The Role of Metrics in the Theory of PDEs at Hokkaido University, July 2018 azuo AKUTAGAWA (Chuo University) (MSJ-SI 2018 at
More informationTraces for star products on symplectic manifolds
Traces for star products on symplectic manifolds Simone Gutt sgutt@ulb.ac.be Université Libre de Bruxelles Campus Plaine, CP 218 BE 1050 Brussels Belgium and Université de etz Ile du Saulcy 57045 etz Cedex
More informationA SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY
A SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY PLAMEN STEFANOV 1. Introduction Let (M, g) be a compact Riemannian manifold with boundary. The geodesic ray transform I of symmetric 2-tensor fields f is
More information1.4 The Jacobian of a map
1.4 The Jacobian of a map Derivative of a differentiable map Let F : M n N m be a differentiable map between two C 1 manifolds. Given a point p M we define the derivative of F at p by df p df (p) : T p
More informationSUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS
SUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS A. RÖSCH AND R. SIMON Abstract. An optimal control problem for an elliptic equation
More informationA Walking Tour of Microlocal Analysis
A Walking Tour of Microlocal Analysis Jeff Schonert August 10, 2006 Abstract We summarize some of the basic principles of microlocal analysis and their applications. After reviewing distributions, we then
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik Harmonic spinors and local deformations of the metric Bernd Ammann, Mattias Dahl, and Emmanuel Humbert Preprint Nr. 03/2010 HARMONIC SPINORS AND LOCAL DEFORMATIONS OF
More informationEXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS
Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC
More informationReproducing formulas associated with symbols
Reproducing formulas associated with symbols Filippo De Mari Ernesto De Vito Università di Genova, Italy Modern Methods of Time-Frequency Analysis II Workshop on Applied Coorbit space theory September
More informationASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS. Tian Ma. Shouhong Wang
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 11, Number 1, July 004 pp. 189 04 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS Tian Ma Department of
More informationA collocation method for solving some integral equations in distributions
A collocation method for solving some integral equations in distributions Sapto W. Indratno Department of Mathematics Kansas State University, Manhattan, KS 66506-2602, USA sapto@math.ksu.edu A G Ramm
More informationA nodal solution of the scalar field equation at the second minimax level
Bull. London Math. Soc. 46 (2014) 1218 1225 C 2014 London Mathematical Society doi:10.1112/blms/bdu075 A nodal solution of the scalar field equation at the second minimax level Kanishka Perera and Cyril
More informationAnalysis in weighted spaces : preliminary version
Analysis in weighted spaces : preliminary version Frank Pacard To cite this version: Frank Pacard. Analysis in weighted spaces : preliminary version. 3rd cycle. Téhéran (Iran, 2006, pp.75.
More informationTrotter s product formula for projections
Trotter s product formula for projections Máté Matolcsi, Roman Shvydkoy February, 2002 Abstract The aim of this paper is to examine the convergence of Trotter s product formula when one of the C 0-semigroups
More informationMULTI-VALUED BOUNDARY VALUE PROBLEMS INVOLVING LERAY-LIONS OPERATORS AND DISCONTINUOUS NONLINEARITIES
MULTI-VALUED BOUNDARY VALUE PROBLEMS INVOLVING LERAY-LIONS OPERATORS,... 1 RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Tomo L (21), pp.??? MULTI-VALUED BOUNDARY VALUE PROBLEMS INVOLVING LERAY-LIONS
More informationA note on W 1,p estimates for quasilinear parabolic equations
200-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 2 3. http://ejde.math.swt.edu or http://ejde.math.unt.edu
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE
ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE ALAN CHANG Abstract. We present Aleksandrov s proof that the only connected, closed, n- dimensional C 2 hypersurfaces (in R n+1 ) of
More informationMULTIPLE POSITIVE SOLUTIONS FOR KIRCHHOFF PROBLEMS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 015 015), No. 0, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE POSITIVE SOLUTIONS
More informationDeterminant of the Schrödinger Operator on a Metric Graph
Contemporary Mathematics Volume 00, XXXX Determinant of the Schrödinger Operator on a Metric Graph Leonid Friedlander Abstract. In the paper, we derive a formula for computing the determinant of a Schrödinger
More informationSCHRÖDINGER OPERATORS WITH PURELY DISCRETE SPECTRUM
SCHRÖDINGER OPERATORS WITH PURELY DISCRETE SPECTRUM BARRY SIMON Dedicated to A. Ya. Povzner Abstract. We prove +V has purely discrete spectrum if V 0 and, for all M, {x V (x) < M} < and various extensions.
More informationsatisfying the following condition: If T : V V is any linear map, then µ(x 1,,X n )= det T µ(x 1,,X n ).
ensities Although differential forms are natural objects to integrate on manifolds, and are essential for use in Stoke s theorem, they have the disadvantage of requiring oriented manifolds in order for
More informationSpectral Theory of Orthogonal Polynomials
Spectral Theory of Orthogonal Polynomials Barry Simon IBM Professor of Mathematics and Theoretical Physics California Institute of Technology Pasadena, CA, U.S.A. Lecture 1: Introduction and Overview Spectral
More informationGENERATORS WITH INTERIOR DEGENERACY ON SPACES OF L 2 TYPE
Electronic Journal of Differential Equations, Vol. 22 (22), No. 89, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GENERATORS WITH INTERIOR
More informationTransparent connections
The abelian case A definition (M, g) is a closed Riemannian manifold, d = dim M. E M is a rank n complex vector bundle with a Hermitian metric (i.e. a U(n)-bundle). is a Hermitian (i.e. metric) connection
More informationGlobal regularity of a modified Navier-Stokes equation
Global regularity of a modified Navier-Stokes equation Tobias Grafke, Rainer Grauer and Thomas C. Sideris Institut für Theoretische Physik I, Ruhr-Universität Bochum, Germany Department of Mathematics,
More informationTraces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains
Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains Sergey E. Mikhailov Brunel University West London, Department of Mathematics, Uxbridge, UB8 3PH, UK J. Math. Analysis
More informationTHE TWO-PHASE MEMBRANE PROBLEM REGULARITY OF THE FREE BOUNDARIES IN HIGHER DIMENSIONS. 1. Introduction
THE TWO-PHASE MEMBRANE PROBLEM REGULARITY OF THE FREE BOUNDARIES IN HIGHER DIMENSIONS HENRIK SHAHGHOLIAN, NINA URALTSEVA, AND GEORG S. WEISS Abstract. For the two-phase membrane problem u = λ + χ {u>0}
More informationarxiv: v2 [math.fa] 17 May 2016
ESTIMATES ON SINGULAR VALUES OF FUNCTIONS OF PERTURBED OPERATORS arxiv:1605.03931v2 [math.fa] 17 May 2016 QINBO LIU DEPARTMENT OF MATHEMATICS, MICHIGAN STATE UNIVERSITY EAST LANSING, MI 48824, USA Abstract.
More informationLecture 4: Harmonic forms
Lecture 4: Harmonic forms Jonathan Evans 29th September 2010 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 1 / 15 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 2 / 15
More information23 Elements of analytic ODE theory. Bessel s functions
23 Elements of analytic ODE theory. Bessel s functions Recall I am changing the variables) that we need to solve the so-called Bessel s equation 23. Elements of analytic ODE theory Let x 2 u + xu + x 2
More informationOperators with numerical range in a closed halfplane
Operators with numerical range in a closed halfplane Wai-Shun Cheung 1 Department of Mathematics, University of Hong Kong, Hong Kong, P. R. China. wshun@graduate.hku.hk Chi-Kwong Li 2 Department of Mathematics,
More informationTHE BEST CONSTANT OF THE MOSER-TRUDINGER INEQUALITY ON S 2
THE BEST CONSTANT OF THE MOSER-TRUDINGER INEQUALITY ON S 2 YUJI SANO Abstract. We consider the best constant of the Moser-Trudinger inequality on S 2 under a certain orthogonality condition. Applying Moser
More informationHIGHER INTEGRABILITY WITH WEIGHTS
Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 355 366 HIGHER INTEGRABILITY WITH WEIGHTS Juha Kinnunen University of Jyväskylä, Department of Mathematics P.O. Box 35, SF-4351
More informationON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 167 176 ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES Piotr Haj lasz and Jani Onninen Warsaw University, Institute of Mathematics
More informationON THE SPECTRUM OF NARROW NEUMANN WAVEGUIDE WITH PERIODICALLY DISTRIBUTED δ TRAPS
ON THE SPECTRUM OF NARROW NEUMANN WAVEGUIDE WITH PERIODICALLY DISTRIBUTED δ TRAPS PAVEL EXNER 1 AND ANDRII KHRABUSTOVSKYI 2 Abstract. We analyze a family of singular Schrödinger operators describing a
More informationApplications of Mathematics
Applications of Mathematics Liselott Flodén; Marianne Olsson Homogenization of some parabolic operators with several time scales Applications of Mathematics, Vol. 52 2007, No. 5, 431--446 Persistent URL:
More informationhere, this space is in fact infinite-dimensional, so t σ ess. Exercise Let T B(H) be a self-adjoint operator on an infinitedimensional
15. Perturbations by compact operators In this chapter, we study the stability (or lack thereof) of various spectral properties under small perturbations. Here s the type of situation we have in mind:
More informationAlgebraic Sum of Unbounded Normal Operators and the Square Root Problem of Kato.
REND. SEM. MAT. UNIV. PADOVA, Vol. 110 (2003) Algebraic Sum of Unbounded Normal Operators and the Square Root Problem of Kato. TOKA DIAGANA (*) ABSTRACT - We prove that the algebraic sum of unbounded normal
More informationA Class of Shock Wave Solutions of the Periodic Degasperis-Procesi Equation
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.9(2010) No.3,pp.367-373 A Class of Shock Wave Solutions of the Periodic Degasperis-Procesi Equation Caixia Shen
More informationA symmetry-based method for constructing nonlocally related partial differential equation systems
A symmetry-based method for constructing nonlocally related partial differential equation systems George W. Bluman and Zhengzheng Yang Citation: Journal of Mathematical Physics 54, 093504 (2013); doi:
More informationCohomology of the Mumford Quotient
Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use a Witten
More information