Transition type solutions for some classes of quasilinear elliptic Allen-Cahn equations

Abstract

The goalofthisthesisistodevelopandstudythestructurerichofthesetof transition typesolutionsofsomeclassesofellipticPDEsoftheform −ΔΦu + A(x, y)V ′(u) =0 in R2, (PDE) where ΔΦ is aquasilinearoperatorindivergenceforminvolvingthe N-function Φ that doesnotincreasemorerapidlythanexponentialfunctions, A(x, y) is periodicinallits argumentsand V is adouble-wellpotentialwithminimaat t = ±α. Animportant prototypeof V is givenby V (t) =Φ(|t2−α2|), whichwasinspiredbytheclassicaldouble- wellGinzburg-Landaupotential.Oneofourmotivationsforlookingforsuchsolutions derivesfromaclassicAllen-Cahnmodelofphasetransitionsthatcanbeseenasavery specialcaseof (PDE). Inourinvestigations,suchsolutionsareobtainedbyvariational approachesusingminimizationmethodstolookforminimaofanactionfunctionalon a reasonableclassofadmissiblefunctionscontainedintheusualOrlicz-Sobolevspace W1,Φ loc (R2). Weprovideseveralqualitativeandquantitativepropertiesforthesesolutions and anumberofdi cultieshadtobeovercomeinourapproach.Forthisreason,it wasnecessarytodevelopnewestimatesbyusingforexampleHarnacktypeinequalities found in[91], C1,α regularitybyLieberman[67] andanewuniquenessresultforaclass of quasilinearODEsofthetype −(ϕ(|q′|)q′)′ + a(t)V ′(q) =0 in R, (ODE) where a(t) belongsto L∞(R) and ϕ(t) =Φ′(t)/t for t > 0. Among thetransitiontypesolutions,heteroclinicandsaddle-typesolutionsstand out inthiswork.Moreover,inthisthesis,itisalsoofparticularinteresttostudythe existence ofbasicheteroclinicsolutionsfortherelativelysimpleone-dimensionalequation (ODE), thatis,todeterminesolutionsthatnaturallyconnectthestationarypoints ±α and thatliebetween −α and α. Thedevelopmentofsuchsolutionsto (ODE) serves as supportfortheconstructionofmorecomplexsolutionsofspatialphase-transition problems. Inparticular,servestocharacterizetheasymptoticbehaviorofthesaddle-type solution for (PDE). Finally,wewilldiscusshowvariantsofwhatwasjustdescribedfor (PDE) hold equally wellforprescribedmeancurvatureequationofthetype −div

∇u p 1 + |∇u|2 ! + A(x, y)V ′(u) =0 in R2. Using thecuttingtechniquesforthedi erentialoperatorinvolvedwebuildauxiliary equations oftheform (PDE) to showthatsuchequationalsohasarichvarietyof transition typesolutionswheneverthedistancebetweentherootsofthesymmetric potential V is smalland V is similarto V (t) =(t2 − α2)2. Not least,wewillprovide su cientconditionsfortheexistenceofbasicheteroclinicsolutionsforthefollowing one-dimensional model −

q′ p 1 +(q′)2 !′ + a(t)V ′(q) =0 in R. Moreover,uniquenessresultsarealsoexploredunderappropriateconditionson a and V .