LEADER 03645nam 2200637 450 001 9910146292203321 005 20220420154900.0 010 $a3-540-69594-X 024 7 $a10.1007/BFb0092831 035 $a(CKB)1000000000437343 035 $a(SSID)ssj0000326692 035 $a(PQKBManifestationID)12069599 035 $a(PQKBTitleCode)TC0000326692 035 $a(PQKBWorkID)10296698 035 $a(PQKB)10963532 035 $a(DE-He213)978-3-540-69594-3 035 $a(MiAaPQ)EBC5577146 035 $a(MiAaPQ)EBC6691771 035 $a(Au-PeEL)EBL5577146 035 $a(OCoLC)1066180869 035 $a(Au-PeEL)EBL6691771 035 $a(PPN)155195549 035 $a(EXLCZ)991000000000437343 100 $a20220420d1997 uy 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aSobolev gradients and differential equations /$fJ. W. Neuberger 205 $a1st ed. 1997. 210 1$aBerlin, Germany ;$aNew York, New York :$cSpringer,$d[1997] 210 4$dİ1997 215 $a1 online resource (VIII, 152 p.) 225 1 $aLecture Notes in Mathematics,$x0075-8434 ;$v1670 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-540-63537-8 320 $aIncludes bibliographical references (pages [145]-149) and index. 327 $aSeveral gradients -- Comparison of two gradients -- Continuous steepest descent in Hilbert space: Linear case -- Continuous steepest descent in Hilbert space: Nonlinear case -- Orthogonal projections, Adjoints and Laplacians -- Introducing boundary conditions -- Newton's method in the context of Sobolev gradients -- Finite difference setting: the inner product case -- Sobolev gradients for weak solutions: Function space case -- Sobolev gradients in non-inner product spaces: Introduction -- The superconductivity equations of Ginzburg-Landau -- Minimal surfaces -- Flow problems and non-inner product Sobolev spaces -- Foliations as a guide to boundary conditions -- Some related iterative methods for differential equations -- A related analytic iteration method -- Steepest descent for conservation equations -- A sample computer code with notes. 330 $aA Sobolev gradient of a real-valued functional is a gradient of that functional taken relative to the underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. Equal emphasis is placed on numerical and theoretical matters. Several concrete applications are made to illustrate the method. These applications include (1) Ginzburg-Landau functionals of superconductivity, (2) problems of transonic flow in which type depends locally on nonlinearities, and (3) minimal surface problems. Sobolev gradient constructions rely on a study of orthogonal projections onto graphs of closed densely defined linear transformations from one Hilbert space to another. These developments use work of Weyl, von Neumann and Beurling. 410 0$aLecture Notes in Mathematics,$x0075-8434 ;$v1670 606 $aDifferential equations$xNumerical solutions 606 $aSobolev gradients 615 0$aDifferential equations$xNumerical solutions. 615 0$aSobolev gradients. 676 $a515/.353 686 $a65N30$2msc 686 $a35A15$2msc 700 $aNeuberger$b J. W$g(John W.),$f1934-$061864 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910146292203321 996 $aSobolev gradients and differential equations$9374787 997 $aUNINA