Regularity theory for Hamilton-Jacobi equations

REGULARITYTHEORYFORHAMILTON-JACOBIEQUATIONSDIOGOAGUIARGOMESAbstract.TheobjectiveofthispaperistodiscusstheregularityofviscositysolutionsoftimeindependentHamilton-JacobiEqua-tions.WeproveanalogsoftheKAMtheorem,showstabilityoftheviscositysolutionsandMathersetsundersmallperturbationsoftheHamiltonian.Contents1.Introduction12.Mathermeasuresandviscositysolutions33.L2-Perturbationtheory84.UniformContinuity13References161.IntroductionTheobjectiveofthispaperistostudytheregularityandstabil-ityundersmallperturbationsofviscositysolutionsofHamilton-Jacobiequations(1)H(P+Dxu;x)=H(P);usinganewsetofideasthatcombinesdynamicalsystemstechniqueswithcontroltheoryandviscositysolutionsmethods.In(1),H(p;x):R2n!RisasmoothHamiltonian,strictlyconvex,andcoerciveinp(limjpj!1H(p;x)jpj=1),andZnperiodicinx(H(p;x+k)=H(p;x)fork2Zn).SinceRnistheuniversalcoveringofthen-dimensionaltorus,weidentifyHwithitsprojectionprH:TnRn!R.BychangingconvenientlytheHamiltonianwemaytakeP=0andH(P)=H,whichwewilldothroughoutthepapertosimplifythenotation.12DIOGOAGUIARGOMESIngeneral,(1)doesnotadmitglobalsmoothsolutions.TheKAMtheoremdealswiththecaseinwhich(2)H(p;x)=H0(p)+H1(p;x):UndergenericconditionsitispossibletoprovethatformostvaluesofPandsucientlysmall(1)admitsasmoothsolutionthatcanbeapproximatedbyapowerseriesin[Arn89].Inthispaperwewillproveanalogousresultsforviscositysolutionsof(1).Theoutlineofthispaperisthefollowing:insection2wereviewbasicfactsconcerningtheconnectionsbetweenMathermeasuresandviscositysolutions.Ageneralreferenceoncontroltheoryandviscositysolutionsis[FS93].Thespecialresultsconcerningviscositysolutionsof(1)canbefoundin[LPV88],[Con95],and[Con97].Themainref-erencesonMather'stheoryare[Mat91],[Mat89a],[Mat89b],[Mn92],and[Mn96].TheuseofviscositysolutionstostudyHamiltoniansys-tems,andinparticularMather'stheoryisdiscussedbyFathi[Fat97a],[Fat97b],[Fat98a],[Fat98b],E[E99],andJauslin,KreissandMoser[JKM99](forconservationlawsinonedimension).Furtherdevelop-mentsandapplicationswereconsideredin[EG99],[Gom00],[Gom01b].Insection??wediscussrepresentationformulasforHandstudythebehaviorofHasafunctionof.WeprovethatHisLipschitzin,anddependingonlyonpropertiesoftheunperturbedproblem,weshowthatHisdierentiablewithrespectto.Insection3weobtainL2estimates(withrespecttoMathermea-sures)onthedierencesDxuDxu(uanduaresolutionsof(1)for=0;,respectively),aswellassomeperturbativeresultsfortheexpansionofuisapowerseriesin.SuchresultsareananalogoftheKAMtheoremforviscositysolutions.Inparticulartheyshow(L2)stabilityoftheMathersets.Theseestimatesarefairlygeneral,andtoprovenerresults,insec-tions4weassumetheadditionalhypothesisthattheMathermeasureisuniquelyergodic.Themainideaisthat,likeinKAMtheory,anon-resonancetypeconditionshouldbeimposedtoprovestrongerstabilityresults.ThisroleisplayedbyuniqueergodicityoftheMathermeasure.Weshow,insection4,thatuisuniformlycontinuousin.REGULARITYTHEORYFORHAMILTON-JACOBIEQUATIONS32.MathermeasuresandviscositysolutionsThepurposeofthissectionistoreviewsomeresultsconcerningviscositysolutionsandMathermeasures.Theorem1(Lions,Papanicolaou,Varadhan).ForeachP2RnthereexistsanumberH(P)andaperiodicviscositysolutionuof(1).ThesolutionuisLipschitz,semiconcave,andHisaconvexfunctionofP.BothHandtheviscositysolutionsof(1)encodethedynamicscertaintrajectories(globalminimizers,see[EG99])oftheHamiltonequations(3)_x=DpH(p;x)_p=DxH(p;x):LetL,theLagrangian,betheLegendretransformofHL(x;v)=supvp
vH(p;x):ThisLagrangianisdenedonthetangentspaceofthetorus(orwhenconvenientoneconsidersitsliftingtotheuniversalcoveringRnRn).Theorem2(Mather).ForeachPthereexistsapositiveprobabilitymeasure(Mathermeasure)onTnRninvariantunderthedynamics(3).ThismeasureminimizesZL(x;v)+Pvdoverallsuchmeasures.SeveralimportantpropertiesofMathermeasurescanbedescribedintermsofviscositysolutions.Mathermeasures,asdenedintheprevi-oustheorem,aresupportedinthetangentspaceofthetorus-howeveritisconvenienttoconsideranothermeasureonthecotangentspaceofthetorusinducedbyusingthedieomorphismv=DpH(p;x).ByabuseoflanguagewewillcallagainMathermeasuretosuchmeasure.Theorem3(Fathi).SupposeisaMathermeasureandletuanysolutionof(1).Thenissupportedonthegraph(x;P+Dxu).Fur-thermoreDxuisLipschitzonthesupportof.ThefactthatthesupportofaMathermeasureisaLipschitzgraphwasprovenbyMather[Mat89b].Thereforeonceitisknownthatissupportedonthegraph(x;P+Dxu)thelastpartofthetheoremfollowstrivially.Similarstatementscanalsobefoundin[E99]or,usingentropysolutionsforconservationlawsinsteadofviscositysolutionsofHamilton-Jacobiequations,in[JKM99].Thenextpropositiongives4DIOGOAGUIARGOMESmorepreciseLipschitzestimatesonDxu.andshowsthatevenoutsidetheMathersetDxuisLipschitz.Proposition1.Suppose(x;p)isapointinthegraphG=f(x;Dxu(x)):uisdierentiableatxg:Thenforallt0thesolution(x(t);p(t))of(3)withinitialconditions(x;p)belongsG.IfforsomeT0,(x(T);p(T))2GthenforanyysuchthatDxu(y)existsjDxu(x)Dxu(y)jCjxyj;withaconstantdependingonT.Proof.Therstpartofthetheorem(invarianceofthegraphfort0)isaconsequenceoftheoptimalcontrolinterpretationofviscos-ity