LEADER 04787nam 22005655 450 001 9910309661003321 005 20250609112036.0 010 $a3-319-90572-4 024 7 $a10.1007/978-3-319-90572-3 035 $a(CKB)4100000007522479 035 $a(DE-He213)978-3-319-90572-3 035 $a(MiAaPQ)EBC5646080 035 $a(PPN)233799796 035 $a(MiAaPQ)EBC6236133 035 $a(EXLCZ)994100000007522479 100 $a20190121d2018 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aElliptic Systems of Phase Transition Type /$fby Nicholas D. Alikakos, Giorgio Fusco, Panayotis Smyrnelis 205 $a1st ed. 2018. 210 1$aCham :$cSpringer International Publishing :$cImprint: Birkhäuser,$d2018. 215 $a1 online resource (XII, 343 p. 59 illus., 10 illus. in color.) 225 1 $aProgress in Nonlinear Differential Equations and Their Applications,$x1421-1750 ;$v91 311 08$a3-319-90571-6 327 $aIntroduction -- Connections -- Basics for the PDE System -- The Cut-Off Lemma and a Maximum Principle -- Estimates -- Symmetry and the Vector Allen-Cahn Equation: the Point Group in Rn -- Symmetry and the Vector Allen-Cahn Equation: Crystalline and Other Complex Structures -- Hierarchical Structure - Stratification -- Vector Minimizers in R2 -- Radial Solutions of ?u = c2u. 330 $aThis book focuses on the vector Allen-Cahn equation, which models coexistence of three or more phases and is related to Plateau complexes ? non-orientable objects with a stratified structure. The minimal solutions of the vector equation exhibit an analogous structure not present in the scalar Allen-Cahn equation, which models coexistence of two phases and is related to minimal surfaces. The 1978 De Giorgi conjecture for the scalar problem was settled in a series of papers: Ghoussoub and Gui (2d), Ambrosio and Cabré (3d), Savin (up to 8d), and del Pino, Kowalczyk and Wei (counterexample for 9d and above). This book extends, in various ways, the Caffarelli-Córdoba density estimates that played a major role in Savin's proof. It also introduces an alternative method for obtaining pointwise estimates. Key features and topics of this self-contained, systematic exposition include: ? Resolution of the structure of minimal solutions in the equivariant class, (a) for general point groups, and (b) for general discrete reflection groups, thus establishing the existence of previously unknown lattice solutions. ? Preliminary material beginning with the stress-energy tensor, via which monotonicity formulas, and Hamiltonian and Pohozaev identities are developed, including a self-contained exposition of the existence of standing and traveling waves. ? Tools that allow the derivation of general properties of minimizers, without any assumptions of symmetry, such as a maximum principle or density and pointwise estimates. ? Application of the general tools to equivariant solutions rendering exponential estimates, rigidity theorems and stratification results. This monograph is addressed to readers, beginning from the graduate level, with an interest in any of the following: differential equations ? ordinary or partial; nonlinear analysis; the calculus of variations; the relationship of minimal surfaces to diffuse interfaces; or the applied mathematics of materials science. 410 0$aProgress in Nonlinear Differential Equations and Their Applications,$x1421-1750 ;$v91 606 $aDifferential equations, Partial 606 $aCalculus of variations 606 $aDifferential equations 606 $aPartial Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12155 606 $aCalculus of Variations and Optimal Control; Optimization$3https://scigraph.springernature.com/ontologies/product-market-codes/M26016 606 $aOrdinary Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12147 615 0$aDifferential equations, Partial. 615 0$aCalculus of variations. 615 0$aDifferential equations. 615 14$aPartial Differential Equations. 615 24$aCalculus of Variations and Optimal Control; Optimization. 615 24$aOrdinary Differential Equations. 676 $a515.353 700 $aAlikakos$b Nicholas D.$4aut$4http://id.loc.gov/vocabulary/relators/aut$0768235 702 $aFusco$b Giorgio$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aSmyrnelis$b Panayotis$4aut$4http://id.loc.gov/vocabulary/relators/aut 906 $aBOOK 912 $a9910309661003321 996 $aElliptic Systems of Phase Transition Type$92222640 997 $aUNINA