04321nam 22006255 450 991029978770332120200702052346.03-319-12829-910.1007/978-3-319-12829-0(CKB)3710000000306160(EBL)1967017(SSID)ssj0001386561(PQKBManifestationID)11755226(PQKBTitleCode)TC0001386561(PQKBWorkID)11374353(PQKB)10427939(DE-He213)978-3-319-12829-0(MiAaPQ)EBC1967017(PPN)183091361(EXLCZ)99371000000030616020141126d2015 u| 0engur|n|---|||||txtccrAn Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L∞[electronic resource] /by Nikos Katzourakis1st ed. 2015.Cham :Springer International Publishing :Imprint: Springer,2015.1 online resource (125 p.)SpringerBriefs in Mathematics,2191-8198Description based upon print version of record.3-319-12828-0 Includes bibliographical references.Preface; Acknowledgments; Contents; 1 History, Examples, Motivation and First Definitions; References; 2 Second Definitions and Basic Analytic Properties of the Notions; References; 3 Stability Properties of the Notions and Existence via Approximation; References; 4 Mollification of Viscosity Solutions and Semiconvexity; References; 5 Existence of Solution to the Dirichlet Problem via Perron's Method; References; 6 Comparison Results and Uniqueness of Solution to the Dirichlet Problem; References7 Minimisers of Convex Functionals and Existence of Viscosity Solutions to the Euler-Lagrange PDEReferences; 8 Existence of Viscosity Solutions to the Dirichlet Problem for the infty-Laplacian; References; 9 Miscellaneous Topics and Some Extensions of the Theory; 9.1 Fundamental Solutions of the infty-Laplacian; 9.1.1 The infty-Laplacian and Tug-of-War Differential Games; 9.1.2 Discontinuous Coefficients, Discontinuous Solutions; 9.1.3 Barles-Perthame Relaxed Limits (1-Sided Uniform Convergence) and Generalised 1-Sided Stability; 9.1.4 Boundary Jets and Jets Relative to Non-open Sets9.1.5 Nonlinear Boundary Conditions9.1.6 Comparison Principle for Viscosity Solutions Without Decoupling in the x-variable; ReferencesThe purpose of this book is to give a quick and elementary, yet rigorous, presentation of the rudiments of the so-called theory of Viscosity Solutions which applies to fully nonlinear 1st and 2nd order Partial Differential Equations (PDE). For such equations, particularly for 2nd order ones, solutions generally are non-smooth and standard approaches in order to define a "weak solution" do not apply: classical, strong almost everywhere, weak, measure-valued and distributional solutions either do not exist or may not even be defined. The main reason for the latter failure is that, the standard idea of using "integration-by-parts" in order to pass derivatives to smooth test functions by duality, is not available for non-divergence structure PDE.SpringerBriefs in Mathematics,2191-8198Partial differential equationsCalculus of variationsPartial Differential Equationshttps://scigraph.springernature.com/ontologies/product-market-codes/M12155Calculus of Variations and Optimal Control; Optimizationhttps://scigraph.springernature.com/ontologies/product-market-codes/M26016Partial differential equations.Calculus of variations.Partial Differential Equations.Calculus of Variations and Optimal Control; Optimization.510515.353515.64Katzourakis Nikosauthttp://id.loc.gov/vocabulary/relators/aut768293BOOK9910299787703321Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L∞1564836UNINA