Concentration Analysis and Applications to PDE: ICTS Workshop, Bangalore, January 2012 PDF
Preview Concentration Analysis and Applications to PDE: ICTS Workshop, Bangalore, January 2012
Trends in Mathematics Adimurthi K. Sandeep Ian Schindler Cyril Tintarev Editors Concentration Analysis and Applications to PDE ICTS Workshop, Bangalore, January 2012 Trends in Mathematics TrendsinMathematicsisaseriesdevotedtothepublicationofvolumesarising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the communityasrapidlyaspossiblewithoutcompromisetoqualityandtoarchive theseforreference. ProposalsforvolumescanbesubmittedusingtheOnlineBookProjectSubmis- sionFormatourwebsitewww.birkhauser-science.com. Materialsubmittedforpublicationmustbescreenedandpreparedasfollows: All contributions should undergo a reviewing process similar to that carried outbyjournals andbe checkedforcorrectuse oflanguagewhich,as a rule,is English.Articleswithoutproofs,orwhichdonotcontainanysignificantlynew results,shouldberejected.Highqualitysurveypapers,however,arewelcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction.Any version of TEX is acceptable, but the entire collectionoffilesmustbeinoneparticulardialectofTEXandunifiedaccording tosimpleinstructionsavailablefromBirkha¨user. Furthermore,in orderto guaranteethe timely appearanceofthe proceedingsit isessentialthatthefinalversionoftheentirematerialbesubmittednolaterthan oneyearaftertheconference. Forfurthervolumes: http://www.springer.com/series/4961 Concentration Analysis and Applications to PDE ICTS Workshop, Bangalore, January 2012 Adimurthi K. Sandeep Ian Schindler Cyril Tintarev Editors Editors Adimurthi Ian Schindler K. Sandeep CeReMath Centre for Applicable Mathematics Université Toulouse I Tata Institute of Fundamental Research Toulouse, France Bangalore Karnataka, India Cyril Tintarev Department of Mathematics Uppsala University Uppsala, Sweden ISBN 978-3-0348-0372-4 ISBN 978-3-0348-0373-1 (eBook) DOI 10.1007/978-3-0348-0373-1 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2013953867 Mathematics Subject Classification (2010): 35-02, 35-06, 58-02, 58-06, 46-02, 46-06 © Springer Basel 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com) Contents Adimurthi, K. Sandeep, I. Schindler and C. Tintarev Introduction ........................................................ vii H. Bahouri On the Elements Involved in the Lack of Compactness in Critical Sobolev Embedding ...................................... 1 P. Caldiroli and R. Musina A Class of Second-order Dilation Invariant Inequalities .............. 17 P. Esposito, A. Pistoia and J. V´etois Blow-up Solutions for Linear Perturbations of the Yamabe Equation ............................................... 29 G. Mancini and K. Sandeep Extremals for Sobolev and Exponential Inequalities in Hyperbolic Space ................................................. 49 A. Pistoia The Ljapunov–Schmidt Reduction for Some Critical Problems ....... 69 F. Robert and J. V´etois A General Theorem for the Construction of Blowing-up Solutions to Some Elliptic Nonlinear Equations via Lyapunov– Schmidt’s Finite-dimensional Reduction ............................. 85 C. Tintarev Concentration Analysis and Cocompactness ......................... 117 G. Vaira A Note on Non-radial Sign-changing Solutions for the Schro¨dinger–PoissonProblem in the Semiclassical Limit ............. 143 Concentration AnalysisandApplicationstoPDE TrendsinMathematics,vii–x ⃝c 2013SpringerBasel Introduction Adimurthi, Kunnath Sandeep, Ian Schindler and Cyril Tintarev The ICTS mini-program School and workshop on cocompact imbeddings, profile decompositions, and their applications to PDE1 took place January 3–12,2012 in Bangalore, at the Centre for Applicable Mathematics of Tata Institute of Fun- damental Research, funded by International Centre for Theoretical Sciences with someadditionalsupportfromEuropeanMathematicalSociety.The mini-program consisted of a 4-day conference (January 5–8) and a series of mini-courses, given by the conference speakers before and after the meeting. The focus of the mini- programwas varietyof concentrationphenomena that occur in partialdifferential equations as well as general theory of concentration analysis. The choice of venue reflectsextensiveuseofconcentrationmethodsinIndianmathematics(Adimurthi, Prashanth, Srikanth, Sandeep, Yadava) and the significant role of Indian mathe- maticians in the global mathematical community. This collection of papers con- sistsmostlyofsurveyarticlesbyconferencespeakers,representingseveralresearch trends in the area. Concentrationanalysisis a discipline that straddles partialdifferential equa- tions, function space theory,geometric analysis,harmonic analysis and functional analysis. The main objective of concentration analysis is to provide, in settings without a priori available compactness, a manageable structural description for the functional sequences intended to approximate solutions of partial differential equations.Useofconcentrationargumentsbecameanimportantapproachtofind- ing solutions of PDE in the early 1980’s due to the works of Uhlenbeck, Brezis, Coron,Lieb,Nirenberg,Aubin,Struwe,andP.-L.Lions.Theoriginalmethoddealt with convergence issues for sequences in Sobolev spaces, whose restrictions to ar- bitrarily small neighborhoods of certain sets in ℝ𝑁 (concentration sets) retained positive energies preventing compactness, and, conversely, compactness of such sequences followed from the absence of concentration. Perhaps the most famous problematthattime,wherethe concentrationphenomenaplayedthe crucialrole, wastheprescribedscalarcurvature(orYamabe)problem.Ina1984paperonglobal compactness Michael Struwe gave a representation of bounded Palais–Smale se- quencesforsemilinearelliptic functionalsintheformofafinite sumofelementary 1(http://www.icts.res.in/program/ccpd2012) viii Adimurthi, K. Sandeep, I. Schindler and C. Tintarev concentrations, that is, sequences of the form 𝑔 𝑤, where (𝑔 ) is a sequence of 𝑘 𝑘 rescalings(actionsofdilationsandtranslationsoftheform𝑢(cid:2)→𝑢(𝜆(⋅−𝑦)),𝜆>0, 𝑦 ∈ ℝ𝑁, normalized in the corresponding Sobolev norm) and 𝑤, often called the blowup profile, satisfies an asymptotic scale-invariant equation, and is typically the famous “standard bubble”, also known as instanton or Talenti–Bliss solution. Struwe’s global compactness, as it was shown later by Sergio Solimini (Ann. Inst. Henri Poincar´e 1995), is a structure common to all sequences bounded in the ho- mogeneous Sobolev space 𝒟1,𝑝(ℝ𝑁), 𝑁 > 𝑝, once one allows a countable number ofelementaryconcentrationsandaremaindervanishingin𝐿𝑁𝑝−𝑁𝑝.Solimini’sprofile decompositionwasindependentlyreproduced,threeyearslater(andwithaweaker remainder)thepapersofPatrickG´erard(𝑝=2)andStephaneJaffard(general𝑝), who used the wavelet basis Γ𝜓 generated by actions of a suitable subset Γ of the rescaling group on the “mother wavelet” 𝜓 to form the concentration profiles by considering suitable finite-dimensional spans of the wavelets. The rediscovery of profile decomposition by the “wavelet school” gained substantial circulation and stimulated several important applications to evolution equations (Keraani, Kenig &Merle,Gallagher).A2003paperofSchindlerandTintarevgeneralizedtheSoli- mini’s result to the abstract Hilbert space 𝐻, allowing elementary concentrations 𝑔 𝑤 to be formed by a general set 𝐷 of unitary operators on 𝐻. This allowedthe 𝑘 inclusionofgeneralgaugesets intotheframeworkofconcentrationanalysissuchas actions of the conformal groups of Riemannian manifolds, and most recently, the non-standarddilationsthatformconcentrationsinthelimitingSobolevdimension (i.e.,Moser–Trudinger)imbeddings.Themeetingincludedtwocommunicationson profile decompositions for the Moser–Trudinger case. We refer the reader to a popular description of the topic in a blog entry by TerenceTaowrittenonprofiledecompositions2.Toputitconcisely,profiledecom- positions, given by the abstract concentration analysis, involve a 𝐷-weakly van- ishing remainder sequence 𝑟 , that is a sequence convergentweakly to zero under 𝑘 actions of an arbitrarygauge sequence, ∀{𝑔 }⊂𝐷, 𝑔 𝑟 ⇀0 in a givenBanach 𝑘 𝑘 𝑘 space 𝑋. For meaningful applications, a local metrization of such convergence is required,or,inotherwords,oneneedstofindauseful Banachspace𝑌 ←(cid:19)𝑋 (e.g., 𝑌 =𝐿𝑝)inwhich𝐷-weaklyvanishingsequencesvanishinnorm.Thisgivesriseto thenotionof𝐷-cocompactimbeddings,thatgivetheanalyticgroundsforabstract profile decompositions. Different proofs of cocompactness of Sobolev imbeddings are found in papers of Lieb (for subcritical Sobolev imbeddings and translations only) Lions, Solimini and Gerard, and in a monograph of Fieseler and Tintarev. Cocompactness of the Moser–Trudinger imbedding 𝑊1,2(Ω)(cid:21)→exp𝐿2(Ω) is veri- 0 fied in the recent work of Adimurthi–Tintarev as well as in the subsequent work of Bahouri–Majdoub–Masmoudi. Cocompactness, relative to standard rescalings (under the name Assumption 1, which describes a formally stronger, but in many casesequivalent,property),foralargescopeofSobolev-typeimbeddinginvolving, 2http://terrytao.wordpress.com/2008/11/05/ concentration-compactness-and-the-profile-decomposition/ Introduction ix in particular,Triebel–LizorkinandBesovspaces, hasbeen recently establishedby Bahouri,CohenandKoch.CocompactnessinStrichartzimbeddingsfordispersive equations,thatregulateslossofcompactnessinthemass-criticalcase(theenergy- critical case is extension of concentration analysis in the Sobolev space) requires engagingalargergroupoftransformations.Inparticular,fortheSchr¨odingerequa- tion,thisgroupisaproductgroupofspaceshifts,dilationsandshiftsintheFourier variable. Profile decompositions of Solimini’s type were used initially to establish the compactnessofsequencesconstructedtoapproximatesolutionsofnonlinearelliptic problems.Ifthestructureoftheequationpreventselementaryconcentrationsfrom occurring,oneconcludescompactness.Ontheotherhand,inellipticproblemswith asmallparameter𝜖,aprofiledecompositionsequenceofStruwe–Soliminitype,us- ing 𝜖 as a concentration parameter, is often a very efficient initial approximation for a solution. Furthermore, it is possible, for positive solutions of semilinear el- liptic equations, to associate location of the concentrationpoint with geometry of the problem, In many such problems concentration occurs at the critical points of the Robin function of the domain. For construction of sign-changing solutions, profiledecompositionsofStruwe–Soliminitype,involvingprofiles(typically“stan- dard bubbles”) of alternating signs, provide natural initial approximations,which allow usage of the Lyapunov–Schmidt method to reduce finding the solution to a finite-dimensional problem. As this takes the study beyond the usual energy spaces,thisapproachallowstheconstructionofsolutionsforproblemswithsuper- critical nonlinearities. Another, similar, use of concentrating approximations is in variational problems of Bahri–Coron type, where topology of the domain affects the geometry of the energy functional, and non-contractive subsets of level sets can be constructed out of families of concentrating profiles, typically involving the standard bubbles. Separation of distinct terms in the profile decompositions is not always spatial, by different concentration cores, but may also occur at the same (or convergent to the same) points, whenever the rates of concentration re- main incomparably different. Such “bubble towers” have been also used as initial approximations of solutions. Blowup analysis of partial differential equations is not limited to solutions of finite energy, that is, to analysis of convergence for sequences bounded in some particular norm. In many cases rescaled limits of sequences of solutions or of ap- proximate solutions satisfy an equation for which a suitable Liouville theorem guarantees that only a trivial solution exists, which in turn provides grounds for existence results. Existence and asymptotics results based on the blowup analy- sis of PDE at points of maximum has been done in recent years by Adimurthi, Druet, Hebey, YanyanLi,Robert, StruweandJunchengWei. The conference,fur- thermore,includedsomecommunicationsonvariousaspectsofnon-compactPDE problemsthatprovidebasisforfutureconcentrationanalysis,suchasellipticprob- lems on the hyperbolic space and inequalities of Caffarelli–Kohn–Nirenberg type. Lyapunov–Schmidtmethod or the finite-dimensional reduction is closely adjacent to concentration analysis. The abstract problem is to find the critical points of a
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