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Volumn 1670, Issue , 2010, Pages 1-289

Sobolev gradients and differential equations

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EID: 76349102779     PISSN: 00758434     EISSN: 16179692     Source Type: Book Series    
DOI: None     Document Type: Chapter
Times cited : (93)

References (229)
  • 3
    • 34548350707 scopus 로고
    • Dual variational methods in critical point theory and applications
    • A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Anal. 14 (1973), 349–381.
    • (1973) J. Functional Anal. , vol.14 , pp. 349-381
    • Ambrosetti, A.1    Rabinowitz, P.2
  • 4
    • 0001061872 scopus 로고
    • Numerical Differentiation for Non-exact Data
    • 74
    • R. S. Andressen, P. Bloomfield, Numerical Differentiation for Non-exact Data, Numer. Math. 22, (1973/74), 157–182.
    • (1973) Numer. Math. , vol.22 , pp. 157-182
    • Andressen, R.S.1    Bloomfield, P.2
  • 6
    • 0036391681 scopus 로고    scopus 로고
    • Sobolev space preconditioning for Newton’s method using domain decomposition
    • O. Axelsson, I. Faragó, J. Karátson, Sobolev space preconditioning for Newton’s method using domain decomposition, Numer. Lin. Alg. Appl., 9 (2002), 585–598.
    • (2002) Numer. Lin. Alg. Appl. , vol.9 , pp. 585-598
    • Axelsson, O.1    Faragó, I.2    Karátson, J.3
  • 8
    • 1242264760 scopus 로고    scopus 로고
    • Conditioning analysis of separate displacement pre-conditioners for some nonlinear elasticity systems
    • ), No
    • O. Axelsson, J. Karátson Conditioning analysis of separate displacement pre-conditioners for some nonlinear elasticity systems, Math. Comput. Simul. 64 (2004), No.6, 649–668.
    • (2004) Math. Comput. Simul. , vol.64 , Issue.6 , pp. 649-668
    • Axelsson, O.1    Karátson, J.2
  • 10
    • 0024105981 scopus 로고
    • On the updating and assembly of the Hessian matrix in finite element methods
    • O. Axelsson, J. Maubach, On the updating and assembly of the Hessian matrix in finite element methods, Comp. Meth. Appl. Mech. Engrg., 71 (1988), 41–67.
    • (1988) Comp. Meth. Appl. Mech. Engrg. , vol.71 , pp. 41-67
    • Axelsson, O.1    Maubach, J.2
  • 11
    • 0002057574 scopus 로고
    • On the Asymptotic Behavior of Nonexpansive Mappings and Semigroups in Banach Spaces
    • J. Baillon, R. Bruck, S. Reich, On the Asymptotic Behavior of Nonexpansive Mappings and Semigroups in Banach Spaces, Houston J. Math. 4 (1978), 1–9.
    • (1978) Houston J. Math. , vol.4 , pp. 1-9
    • Baillon, J.1    Bruck, R.2    Reich, S.3
  • 13
    • 12444314281 scopus 로고    scopus 로고
    • The Laplacian matrix of a graph
    • R. Bapat, The Laplacian matrix of a graph, Math. Student 65 (1996), 214–223.
    • (1996) Math. Student , vol.65 , pp. 214-223
    • Bapat, R.1
  • 16
    • 24644475107 scopus 로고
    • Finite Element Solution to Nonlinear Partial Differential Equations
    • C. Beasley, Finite Element Solution to Nonlinear Partial Differential Equations, Dissertation, North Texas State University (1981).
    • (1981) Dissertation, North Texas State University
    • Beasley, C.1
  • 19
    • 0000498337 scopus 로고
    • Bifurcation Analysis of Minimizing Harmonic Maps Describing the Equilibrium of Nematic Phases
    • F. Béthuel, H.Brezis, B. D. Coleman, F. Hélein, Bifurcation Analysis of Minimizing Harmonic Maps Describing the Equilibrium of Nematic Phases, Arch. Rat. Mech. 118 (1992), 149–168.
    • (1992) Arch. Rat. Mech. , vol.118 , pp. 149-168
    • Béthuel, F.1    Brezis, H.2    Coleman, B.D.3    Hélein, F.4
  • 22
    • 85072903085 scopus 로고
    • , Vol. , L. Carleson, P. Mallivan, J. Neuberger, J. Wermer, Editors, Birkhaüser
    • A. Beurling, Collected Works, Vol. 2, L. Carleson, P. Mallivan, J. Neuberger, J. Wermer, Editors, Birkhaüser, 1989.
    • (1989) Collected Works , vol.2
    • Beurling, A.1
  • 24
    • 11744273384 scopus 로고
    • The surface evolver
    • K. Brakke, The surface evolver, Experimental Math. 1 (1992), 141–165.
    • (1992) Experimental Math , vol.1 , pp. 141-165
    • Brakke, K.1
  • 25
    • 0004291558 scopus 로고
    • Opérateurs maximaux monotones et semi-groupes nonlinéares
    • North Holland
    • H. Brezis, Opérateurs maximaux monotones et semi-groupes nonlinéares, Math Studies 5, North Holland, 1973.
    • (1973) Math Studies 5
    • Brezis, H.1
  • 26
    • 84966221683 scopus 로고
    • A Fixed-point Theorem for certain Operator Valued Maps
    • D. R. Brown and M. J. O’Malley, A Fixed-point Theorem for certain Operator Valued Maps, Proc. Amer. Math. Soc. 74 (1979), 254–258.
    • (1979) Proc. Amer. Math. Soc. , vol.74 , pp. 254-258
    • Brown, D.R.1    O’Malley, M.J.2
  • 27
    • 84990255933 scopus 로고
    • A General Convergence Principle in Nonlinear Functional Analysis
    • R. E. Bruck and S. Reich, A General Convergence Principle in Nonlinear Functional Analysis, Nonlinear Analysis 4 (1980), 939–950.
    • (1980) Nonlinear Analysis , vol.4 , pp. 939-950
    • Bruck, R.E.1    Reich, S.2
  • 28
    • 33845773025 scopus 로고    scopus 로고
    • The Monge-Ampre equation and optimal transportation, an elementary review
    • L. Caffarelli, The Monge-Ampre equation and optimal transportation, an elementary review. Lecture Notes in Math., Springer 2003, 1–10.
    • (2003) Lecture Notes in Math., Springer , pp. 1-10
    • Caffarelli, L.1
  • 30
    • 0023344944 scopus 로고
    • Nonlinear preconditioned conjugate gradient and leastsquares finite elements
    • G. F. Carey, B.-N Jiang, Nonlinear preconditioned conjugate gradient and leastsquares finite elements, Comp. Meth. Appl. Mech. Engrg., 62 (1987), 145–154.
    • (1987) Comp. Meth. Appl. Mech. Engrg. , vol.62 , pp. 145-154
    • Carey, G.F.1    Jiang, B.-N.2
  • 33
    • 85072890469 scopus 로고    scopus 로고
    • Multiscale and hysteresis effects in vortex pattern simulations for Ginzburg-Landau Problems
    • to appear
    • G. F. Carey and D. J. Knezevic, Multiscale and hysteresis effects in vortex pattern simulations for Ginzburg-Landau Problems, Int. J. Numer. Math. Engng. (2009) (to appear).
    • (2009) Int. J. Numer. Math. Engng.
    • Carey, G.F.1    Knezevic, D.J.2
  • 34
    • 0031372470 scopus 로고    scopus 로고
    • A sign-changing solution for a superlinear Dirichlet problem
    • A. Castro, J. Cossio, John. M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem Rocky Mountain J. Math. 27 (1997), 1041–1053.
    • (1997) Rocky Mountain J. Math. , vol.27 , pp. 1041-1053
    • Castro, A.1    Cossio, J.2    Neuberger, J.M.3
  • 36
    • 3342925295 scopus 로고    scopus 로고
    • , A minimax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems
    • A. Castro, J. Cossio and John M. Neuberger, A minimax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems, Electronic J. Differential Equations 2 (1998).
    • (1998) Electronic J. Differential Equations , vol.2
    • Castro, A.1    Cossio, J.2    Neuberger, J.M.3
  • 37
    • 84864703738 scopus 로고    scopus 로고
    • An Inverse Function Theorem by Means of Continuous Newton’s Method
    • with A. Castro
    • A. Castro and J. W. Neuberger. An Inverse Function Theorem by Means of Continuous Newton’s Method, J. Nonlinear Analysis 47 (2001), 3259–3270 (with A. Castro).
    • (2001) J. Nonlinear Analysis , vol.47 , pp. 3259-3270
    • Castro, A.1    Neuberger, J.W.2
  • 38
    • 0000478038 scopus 로고    scopus 로고
    • Méthode générale pour la reśolution des systemes d’équations simultanées
    • P. L. Cauchy Méthode générale pour la reśolution des systemes d’équations simultanées, C.R. Acad. Sci. Paris, 25 (1847), 536–538.
    • C.R. Acad. Sci. Paris , vol.25 , Issue.1847 , pp. 536-538
    • Cauchy, P.L.1
  • 39
    • 0038116267 scopus 로고    scopus 로고
    • On the ̷Lojasiewicz-Simon gradient inequality
    • R. Chill, On the ̷Lojasiewicz-Simon gradient inequality, J. Funct. Anal. 201 (2003), 572–601.
    • (2003) J. Funct. Anal. , vol.201 , pp. 572-601
    • Chill, R.1
  • 40
    • 33749673101 scopus 로고    scopus 로고
    • Convergence to steady states of solutions of the Chan-Hilliard and Caginalp equations with dynamic boundary conditions
    • R. Chill, E. Fašangová andJ.Prüss, Convergence to steady states of solutions of the Chan-Hilliard and Caginalp equations with dynamic boundary conditions. Math. Nachr. 279 (2006), 1448–1462.
    • (2006) Math. Nachr. , vol.279 , pp. 1448-1462
    • Chill, R.1    Fašangová, E.2    Prüss, J.3
  • 41
  • 42
    • 0037410017 scopus 로고    scopus 로고
    • Convergence to steady states in asymptotically autonomous semilinear evolution equations
    • R. Chill and M. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlin. Anal.: Theory, Methods and Applications 3A (2003), 1017–1039.
    • (2003) Nonlin. Anal.: Theory, Methods and Applications , vol.3A , pp. 1017-1039
    • Chill, R.1    Jendoubi, M.2
  • 47
    • 0035419534 scopus 로고    scopus 로고
    • Neuberger, A reduction algorithm for sublinear Dirichlet problems
    • J. Cossio, S. Lee, John M. Neuberger, A reduction algorithm for sublinear Dirichlet problems, Nonlinear Analysis 47 (2001), 3379–3390.
    • (2001) Nonlinear Analysis , vol.47 , pp. 3379-3390
    • Cossio, J.1    Lee, S.2    John, M.3
  • 48
    • 0035361440 scopus 로고    scopus 로고
    • Neuberger, A numerical investigation of sign-changing solutions to superlinear elliptic equations on symmetric domains
    • D. Costa, Z. Ding, John M. Neuberger, A numerical investigation of sign-changing solutions to superlinear elliptic equations on symmetric domains, J. Comp. Appl. Math. 131 (2001), 299–319.
    • (2001) J. Comp. Appl. Math. , vol.131 , pp. 299-319
    • Costa, D.1    Ding, Z.2    John, M.3
  • 49
    • 84917794708 scopus 로고
    • Conformal Mapping, and Minimal Surfaces, Interscience
    • R. Courant, Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces, Interscience, 1950.
    • (1950) Dirichlet’s Principle
    • Courant, R.1
  • 50
    • 0001914191 scopus 로고
    • The Method of Steepest Descent for Nonlinear Minimization Problems
    • H. B. Curry, The Method of Steepest Descent for Nonlinear Minimization Problems, Quart. Appl Math 2 (1944), 258–261.
    • (1944) Quart. Appl Math , vol.2 , pp. 258-261
    • Curry, H.B.1
  • 51
    • 0025568645 scopus 로고
    • On Finding Equilibria for Isotropic Hyperelastic Materials
    • J. G. Dix and T. W. McCabe, On Finding Equilibria for Isotropic Hyperelastic Materials, Nonlinear Analysis 15 (1990), 437–444.
    • (1990) Nonlinear Analysis , vol.15 , pp. 437-444
    • Dix, J.G.1    McCabe, T.W.2
  • 52
    • 0030600694 scopus 로고    scopus 로고
    • A Theory of Strongly Continuous Semigroups in Terms of Lie Generators
    • J. R. Dorroh and J. W. Neuberger, A Theory of Strongly Continuous Semigroups in Terms of Lie Generators, J. Functional Analysis 136 (1996), 114–126.
    • (1996) J. Functional Analysis , vol.136 , pp. 114-126
    • Dorroh, J.R.1    Neuberger, J.W.2
  • 54
    • 0026838281 scopus 로고
    • Analysis and Approximation of the Ginzburg-Landau Model of Superconductivity
    • Q. Du, M. Gunzburger, and J. S. Peterson, Analysis and Approximation of the Ginzburg-Landau Model of Superconductivity, SIAM Review 34 (1992), 54–81.
    • (1992) SIAM Review , vol.34 , pp. 54-81
    • Du, Q.1    Gunzburger, M.2    Peterson, J.S.3
  • 55
    • 35949005560 scopus 로고
    • Computational simulation of type-II superconductivity including pinning phenomena
    • Q. Du, M. Gunzburger, and J.S. Peterson, Computational simulation of type-II superconductivity including pinning phenomena, Physical Review B 51, 22 (1995), 16194–16203.
    • (1995) Physical Review B , vol.51 , Issue.22 , pp. 16194-16203
    • Du, Q.1    Gunzburger, M.2    Peterson, J.S.3
  • 57
    • 0013499061 scopus 로고
    • Unrestricted Iteration of Nonexpansive Mappings in Hilbert Space
    • J. M. Dye and S. Reich, Unrestricted Iteration of Nonexpansive Mappings in Hilbert Space, Trans. Am. Math. Soc. 325 (1991), 87–99.
    • (1991) Trans. Am. Math. Soc. , vol.325 , pp. 87-99
    • Dye, J.M.1    Reich, S.2
  • 59
    • 0003285448 scopus 로고
    • Weak Convergence Methods for Nonlinear Partial Differential Equations
    • L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Series in Mathematics 74, 1990.
    • (1990) CBMS Series in Mathematics , vol.74
    • Evans, L.C.1
  • 60
    • 45149136816 scopus 로고
    • On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differenti al equations
    • V. Faber, T. Manteuffel, S. V. Parter, On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differenti al equations, Adv. in Appl. Math., 11 (1990), 109–163.
    • (1990) Adv. in Appl. Math. , vol.11 , pp. 109-163
    • Faber, V.1    Manteuffel, T.2    Parter, S.V.3
  • 62
    • 3042639237 scopus 로고    scopus 로고
    • Numerical solution of nonlinear elliptic problems via preconditioning operators: Theory and applications
    • , Volume , NOVA Science Publishers, New York
    • Faragó, I., Karátson, J., Numerical solution of nonlinear elliptic problems via preconditioning operators: Theory and applications. Advances in Computation, Volume 11, NOVA Science Publishers, New York, 2002.
    • (2002) Advances in Computation , vol.11
    • Faragó, I.1    Karátson, J.2
  • 64
    • 0000615669 scopus 로고
    • Function Minimisation by Conjugate Gradients
    • R. Fletcher and C. M. Reeves, Function Minimisation by Conjugate Gradients, Computer J. 7 (1964), 149–154.
    • (1964) Computer J , vol.7 , pp. 149-154
    • Fletcher, R.1    Reeves, C.M.2
  • 66
    • 0004227964 scopus 로고
    • Cambridge University Press, Cambridge
    • T. M. Flett, Differential Analysis, Cambridge University Press, Cambridge, 1980.
    • (1980) Differential Analysis
    • Flett, T.M.1
  • 73
    • 85072902922 scopus 로고
    • Semigroups of Linear Operators and Applications
    • J. Goldstein, Semigroups of Linear Operators and Applications, Oxford Math. Studies, (1985).
    • (1985) Oxford Math. Studies
    • Goldstein, J.1
  • 74
    • 85072884984 scopus 로고    scopus 로고
    • The GAP Group, GAP-Groups, Algorithums, and Programming, Version 4.4.9
    • The GAP Group, GAP-Groups, Algorithums, and Programming, Version 4.4.9, (2006).
    • (2006)
  • 75
    • 0003270066 scopus 로고    scopus 로고
    • Singularites and Groups in Bifurcation Theory
    • Springer-Verlag, New York
    • M. Golubitsky, I. Stewart and D. Schaeffer, Singularites and Groups in Bifurcation Theory, Appl. Math. Series, Springer-Verlag, New York, 1998.
    • (1998) Appl. Math. Series
    • Golubitsky, M.1    Stewart, I.2    Schaeffer, D.3
  • 81
    • 33750963973 scopus 로고    scopus 로고
    • Neuberger, GNGA for general regions: Semilinear elliptic PDE and crossing eigenvalues
    • J. Hineman and John M. Neuberger, GNGA for general regions: semilinear elliptic PDE and crossing eigenvalues, Commun. Nonlinear Sci. Numer. Simul. 12 (2007), 447–464.
    • (2007) Commun. Nonlinear Sci. Numer. Simul. , vol.12 , pp. 447-464
    • Hineman, J.1    John, M.2
  • 86
    • 84968468698 scopus 로고    scopus 로고
    • Weak Convergence and Reflexivity
    • R. C. James, Weak Convergence and Reflexivity, Trans.Amer.Math.Soc.113 (1964), 129–140.
    • Trans.Amer.Math.Soc. , vol.113 , Issue.1964 , pp. 129-140
    • James, R.C.1
  • 87
    • 34548500804 scopus 로고    scopus 로고
    • As flat as possible
    • J. Jacobsen, As flat as possible, SIAM Review 49 (2007), 491–507.
    • (2007) SIAM Review , vol.49 , pp. 491-507
    • Jacobsen, J.1
  • 89
    • 0001518119 scopus 로고
    • On the regularity of the solution of the Poisson problem on a domain with boundary locally similar to the boundary of a convex open set
    • J. Kadlec, On the regularity of the solution of the Poisson problem on a domain with boundary locally similar to the boundary of a convex open set, Czechosl. Math. J., 14 (89), (1964), 386–393.
    • (1964) Czechosl. Math. J. , vol.14 , Issue.89 , pp. 386-393
    • Kadlec, J.1
  • 90
    • 83055172340 scopus 로고    scopus 로고
    • Gradient method in Sobolev space for nonlocal boundary value problems
    • No
    • J. Karátson, Gradient method in Sobolev space for nonlocal boundary value problems, Electron. J. Diff. Eqns., Vol. 2000(2000), No. 51, 1–17.
    • (2000) Electron. J. Diff. Eqns , vol.2000 , Issue.51 , pp. 1-17
    • Karátson, J.1
  • 91
    • 3042546519 scopus 로고    scopus 로고
    • I.,Variable preconditioning via quasi-Newton methods for nonlinear problems in Hilbert space, SIAM
    • , No
    • J. Karátson, I. Faragó I.,Variable preconditioning via quasi-Newton methods for nonlinear problems in Hilbert space, SIAM J. Numer. Anal. 41 (2003), No. 4, 1242–1262.
    • (2003) J. Numer. Anal , vol.41 , Issue.4 , pp. 1242-1262
    • Karátson, J.1    Faragó, I.2
  • 92
    • 26444618152 scopus 로고    scopus 로고
    • Preconditioning operators and Sobolev gradients for nonlinear elliptic problems
    • J. Karátson, I. Faragó, Preconditioning operators and Sobolev gradients for nonlinear elliptic problems, Comput. Math. Appl. 50 (2005), 1077–1092.
    • (2005) Comput. Math. Appl. , vol.50 , pp. 1077-1092
    • Karátson, J.1    Faragó, I.2
  • 93
    • 14544271374 scopus 로고    scopus 로고
    • Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions
    • J. Karátson, S. Korotov, Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions, Numer. Math. 99 (2005), 669–698.
    • (2005) Numer. Math. , vol.99 , pp. 669-698
    • Karátson, J.1    Korotov, S.2
  • 94
    • 34948860120 scopus 로고    scopus 로고
    • Newton’s method in the context of gradients
    • ), No., pp
    • J. Karátson, J. W. Neuberger, Newton’s method in the context of gradients, Electron. J. Diff. Eqns. Vol. 2007 (2007), No. 124, pp. 1–13.
    • (2007) Electron. J. Diff. Eqns , vol.2007 , Issue.124 , pp. 1-13
    • Karátson, J.1    Neuberger, J.W.2
  • 95
    • 84868028151 scopus 로고    scopus 로고
    • The gradient method for non-differentiable operators in product Hilbert spaces and applications to elliptic systems of quasilinear differential equations
    • ) No
    • J. Karátson, The gradient method for non-differentiable operators in product Hilbert spaces and applications to elliptic systems of quasilinear differential equations, J. Appl. Anal., 3 (1997) No. 2., 205–217.
    • (1997) J. Appl. Anal. , vol.3 , Issue.2 , pp. 205-217
    • Karátson, J.1
  • 96
    • 51249181779 scopus 로고
    • A New Polynomial Time Algorithm for Linear Programming
    • N. A. Karmarkar, A New Polynomial Time Algorithm for Linear Programming, Combinatorica 4 (1984), 373–395.
    • (1984) Combinatorica , vol.4 , pp. 373-395
    • Karmarkar, N.A.1
  • 97
    • 45049087076 scopus 로고    scopus 로고
    • Potential theory and applications in a contructive method for finding critical points of Ginzburg-Landau type
    • P. Kazemi and J. W. Neuberger, Potential theory and applications in a contructive method for finding critical points of Ginzburg-Landau type, Nonlinear Analysis 69 (2008), 925–930.
    • (2008) Nonlinear Analysis , vol.69 , pp. 925-930
    • Kazemi, P.1    Neuberger, J.W.2
  • 98
    • 85072879703 scopus 로고    scopus 로고
    • Using projections in a direct minimization scheme for a wide class of energy functionals
    • to appear
    • P. Kazemi, Using projections in a direct minimization scheme for a wide class of energy functionals, Comm. Applied Nonlinear Analysis (to appear).
    • Comm. Applied Nonlinear Analysis
    • Kazemi, P.1
  • 100
    • 0035361531 scopus 로고    scopus 로고
    • Parameter Identification for elliptic problems
    • I. Knowles, Parameter Identification for elliptic problems, J.Comp.Appl.Math., 131, (2001), 175–194.
    • (2001) J.Comp.Appl.Math. , vol.131 , pp. 175-194
    • Knowles, I.1
  • 102
    • 21844510167 scopus 로고
    • A variational method for numerical differentiation
    • I. Knowles and R. Wallace, A variational method for numerical differentiation, Numerische Mathematik 70 (1995), 91–110.
    • (1995) Numerische Mathematik , vol.70 , pp. 91-110
    • Knowles, I.1    Wallace, R.2
  • 104
    • 84966254369 scopus 로고
    • The Nonlinear Geometry of Linear Programming III. Projective Legendre Transform Co-ordinates and Hilbert Geometry
    • J. C. Lagarias, The Nonlinear Geometry of Linear Programming III. Projective Legendre Transform Co-ordinates and Hilbert Geometry, Trans. Amer. Math. Soc 320 (1990), 193–225.
    • (1990) Trans. Amer. Math. Soc , vol.320 , pp. 193-225
    • Lagarias, J.C.1
  • 106
    • 85072893176 scopus 로고    scopus 로고
    • Neuberger Asymptotic Solutions
    • J. Lee and John M. Neuberger Asymptotic Solutions, Preprint.
    • Preprint
    • Lee, J.1    John, M.2
  • 108
    • 85072897557 scopus 로고
    • The Steepest Descent Method Using Finite Elements for Systems of Nonlinear Partial Differential Equations
    • M. Liaw, The Steepest Descent Method Using Finite Elements for Systems of Nonlinear Partial Differential Equations, Dissertation, North Texas State University, (1983).
    • (1983) Dissertation, North Texas State University
    • Liaw, M.1
  • 109
    • 0000298073 scopus 로고
    • Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities
    • E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math. 118 (1983), 349–374.
    • (1983) Ann. Math. , vol.118 , pp. 349-374
    • Lieb, E.1
  • 110
    • 27844475132 scopus 로고    scopus 로고
    • Discrete periodic geodesics in a surface
    • , No
    • A. LinnérandR.J.Renka,Discrete periodic geodesics in a surface, Experimental Math. 14, No. 2 (2005), 145–152.
    • (2005) Experimental Math , vol.14 , Issue.2 , pp. 145-152
    • Linnérandr, A.1    Renka, J.2
  • 113
    • 0003366108 scopus 로고
    • Colloques internationaux du C.N.R.S.: Les equations aux derivées partielles, Paris (1962), Editions du C.N.R.S., Paris
    • S. Lojasiewicz, Une propriété topologique des sousenembles analytiques réeles partielles, Colloques internationaux du C.N.R.S.: Les equations aux derivées partielles, Paris (1962), Editions du C.N.R.S., Paris, 1963, 87–89.
    • (1963) Une propriété Topologique Des Sousenembles Analytiques réeles Partielles , pp. 87-89
    • Lojasiewicz, S.1
  • 114
    • 23044523349 scopus 로고    scopus 로고
    • On the maximum principle for generalized solutions to the Tricomi problem
    • D.LupoandK.R.Payne,On the maximum principle for generalized solutions to the Tricomi problem, Commun. Contemp. Math. 2 (2000), 535–557.
    • (2000) Commun. Contemp. Math. , vol.2 , pp. 535-557
    • Lupoandk, D.1    Payne, R.2
  • 115
    • 0142013458 scopus 로고    scopus 로고
    • A convergence result for discrete steepest descent in weighted Sobolev spaces
    • ), no
    • W. T. Mahavier, A convergence result for discrete steepest descent in weighted Sobolev spaces, Abstr. Appl. Anal., 2 (1997), no. 1–2, 67–72.
    • (1997) Abstr. Appl. Anal. , vol.2 , Issue.1-2 , pp. 67-72
    • Mahavier, W.T.1
  • 116
    • 85072887739 scopus 로고    scopus 로고
    • Weighted descent for singular first-order differential equations
    • W. T. Mahavier, Weighted descent for singular first-order differential equations, Southwest J. Pure Appl. Math. 5 (1999), 41–51.
    • (1999) Southwest J. Pure Appl. Math. , vol.5 , pp. 41-51
    • Mahavier, W.T.1
  • 117
    • 33745167617 scopus 로고    scopus 로고
    • Solving boundary value problems numerically using steepest descent in Sobolev spaces, Missouri
    • W. T. Mahavier, Solving boundary value problems numerically using steepest descent in Sobolev spaces, Missouri J. Math. Sci. 11 (1999), 19–32.
    • (1999) J. Math. Sci. , vol.11 , pp. 19-32
    • Mahavier, W.T.1
  • 118
    • 33745156752 scopus 로고    scopus 로고
    • A numerical method utilizing weighted Sobolev descent to solve singular differential equations
    • W. T. Mahavier, A numerical method utilizing weighted Sobolev descent to solve singular differential equations, Nonlinear World 4 (1997), 435–455.
    • (1997) Nonlinear World , vol.4 , pp. 435-455
    • Mahavier, W.T.1
  • 119
    • 0025447701 scopus 로고
    • Preconditioning and boundary conditions
    • ), no
    • T. Manteuffel, S. V. Parter, Preconditioning and boundary conditions, SIAM J. Numer. Anal. 27 (1990), no. 3, 656–694.
    • (1990) SIAM J. Numer. Anal. , vol.27 , Issue.3 , pp. 656-694
    • Manteuffel, T.1    Parter, S.V.2
  • 121
    • 85072886175 scopus 로고
    • A Solution-Giving Transformation for Systems of Differential Equations
    • L. C. May, A Solution-Giving Transformation for Systems of Differential Equations, Dissertation, Univ. of North Texas (1990).
    • (1990) Dissertation, Univ. of North Texas
    • May, L.C.1
  • 122
    • 0000388191 scopus 로고
    • A Rapidly Convergent Iteration Method and Non-linear Partial Differential Equations
    • J. Moser, A Rapidly Convergent Iteration Method and Non-linear Partial Differential Equations, Annali Scuola Norm. Sup. Pisa 20 (1966), 265–315.
    • (1966) Annali Scuola Norm. Sup. Pisa , vol.20 , pp. 265-315
    • Moser, J.1
  • 123
    • 54049090077 scopus 로고    scopus 로고
    • Recursive form of Sobolev gradient for ODEs on long intervals
    • D. Mujeeb, J. W. Neuberger, S. Sial, Recursive form of Sobolev gradient for ODEs on long intervals, Int. J. Computer Math. 85 (2008), 1727–1740.
    • (2008) Int. J. Computer Math. , vol.85 , pp. 1727-1740
    • Mujeeb, D.1    Neuberger, J.W.2    Sial, S.3
  • 124
    • 0002702476 scopus 로고
    • The embedding problem for Riemannian manifolds
    • J. Nash, The embedding problem for Riemannian manifolds, Annals of Mathematics 63 (1956), 20–63.
    • (1956) Annals of Mathematics , vol.63 , pp. 20-63
    • Nash, J.1
  • 125
    • 57649156763 scopus 로고
    • The convergence of the method of steepest descents for nonlinear equations with variational or quasi-variational operators
    • M. Z. Nashed, The convergence of the method of steepest descents for nonlinear equations with variational or quasi-variational operators, J.Math.Mech.13 (1964), 765–794.
    • (1964) J.Math.Mech. , vol.13 , pp. 765-794
    • Nashed, M.Z.1
  • 127
    • 0001831970 scopus 로고    scopus 로고
    • , A numerical method for finding sign-changing solutions of superlinear Dirichlet problems
    • John M. Neuberger, A numerical method for finding sign-changing solutions of superlinear Dirichlet problems, Nonlinear World 4 (1997), 73–83.
    • (1997) Nonlinear World , vol.4 , pp. 73-83
    • Neuberger, J.M.1
  • 128
    • 33745676148 scopus 로고    scopus 로고
    • Nonlinear elliptic partial differential equations on graphs
    • John M. Neuberger, Nonlinear elliptic partial differential equations on graphs, Experimental Mathematics 15 (2006), 91–107.
    • (2006) Experimental Mathematics , vol.15 , pp. 91-107
    • Neuberger, J.M.1
  • 129
    • 0035606761 scopus 로고    scopus 로고
    • Newton’s method and Morse index for semilinear elliptic PDEs
    • John M. Neuberger and J. Swift, Newton’s method and Morse index for semilinear elliptic PDEs, Int. J. Bifur, Chaos Appl. Sci. Engrg. 11 (2001), 801–820.
    • (2001) Int. J. Bifur, Chaos Appl. Sci. Engrg. , vol.11 , pp. 801-820
    • Neuberger, J.M.1    Swift, J.2
  • 130
    • 33644858405 scopus 로고    scopus 로고
    • Computing eigenfunctions on the Koch snowflake: A new grid and symmetry
    • John M. Neuberger, N. Sieben and J. Swift, Computing eigenfunctions on the Koch snowflake: a new grid and symmetry, J. Comp. Appl. Math. 191 (2006), 126–142.
    • (2006) J. Comp. Appl. Math. , vol.191 , pp. 126-142
    • Neuberger, J.M.1    Sieben, N.2    Swift, J.3
  • 131
    • 57249111985 scopus 로고    scopus 로고
    • Symmetry and automated branch following for a semilinear elliptic PDE on a fractal region
    • John M. Neuberger, N. Sieben and J. Swift, Symmetry and automated branch following for a semilinear elliptic PDE on a fractal region, SIAMJ.Appl.Dyn. Syst., 5 (2006), 476–507.
    • (2006) Siamj.Appl.Dyn. Syst. , vol.5 , pp. 476-507
    • Neuberger, J.M.1    Sieben, N.2    Swift, J.3
  • 133
    • 0001831970 scopus 로고    scopus 로고
    • A numerical method for finding sign-changing solutions of superlinear Dirichlet problems
    • ), no
    • John M. Neuberger, A numerical method for finding sign-changing solutions of superlinear Dirichlet problems, Nonlinear World 4 (1997), no. 1, 73–83.
    • (1997) Nonlinear World , vol.4 , Issue.1 , pp. 73-83
    • Neuberger, J.M.1
  • 134
    • 84972500750 scopus 로고
    • An exponential formula for one-parameter semi-groups of nonlinear transformations
    • J. W. Neuberger, An exponential formula for one-parameter semi-groups of nonlinear transformations, J. Math. Soc. Japan 18 (1963), 154–157.
    • (1963) J. Math. Soc. Japan , vol.18 , pp. 154-157
    • Neuberger, J.W.1
  • 135
    • 84972503072 scopus 로고
    • Concerning Boundary Value Problems
    • J. W. Neuberger, Concerning Boundary Value Problems, Pac.J.of Math.,10 (1960), 1385–1392.
    • (1960) Pac.J.Of Math. , vol.10 , pp. 1385-1392
    • Neuberger, J.W.1
  • 137
    • 85072882146 scopus 로고    scopus 로고
    • Some global steepest descent results for nonlinear systems
    • (Arlington, Tex., 1982),, Lecture Notes in Pure and Appl. Math., 90, Dekker, New York
    • J. W. Neuberger, Some global steepest descent results for nonlinear systems. Trends in theory and practice of nonlinear differential equations (Arlington, Tex., 1982), 413–418, Lecture Notes in Pure and Appl. Math., 90, Dekker, New York, 1984.
    • (2013) Trends in Theory and Practice of Nonlinear Differential Equations
    • Neuberger, J.W.1
  • 138
    • 0039507037 scopus 로고
    • R. J., Minimal surfaces and Sobolev gradients, SIAM
    • , no
    • J. W. Neuberger, R. J. Renka, R. J., Minimal surfaces and Sobolev gradients, SIAM J. Sci. Comput. 16 (1995), no. 6, 1412–1427.
    • (1995) J. Sci. Comput , vol.16 , Issue.6 , pp. 1412-1427
    • Neuberger, J.W.1    Renka, R.J.2
  • 140
    • 0003413877 scopus 로고    scopus 로고
    • Sobolev gradients and differential equations
    • No. 1670, Springer, (first edition)
    • J. W. Neuberger, Sobolev gradients and differential equations, Lecture Notes in Math., No. 1670, Springer, 1997 (first edition).
    • (1997) Lecture Notes in Math
    • Neuberger, J.W.1
  • 141
    • 51649190232 scopus 로고
    • Tensor Products and Successive Approximations for Partial Differential Equations
    • J. W. Neuberger, Tensor Products and Successive Approximations for Partial Differential Equations, Isr. J. Math., 6 (1968), 121–132.
    • (1968) Isr. J. Math. , vol.6 , pp. 121-132
    • Neuberger, J.W.1
  • 142
    • 0040334091 scopus 로고
    • Norm of Symmetric Product Compared with Norm of Tensor Product
    • J. W. Neuberger,, Norm of Symmetric Product Compared with Norm of Tensor Product, J. Linear Multi-Linear Algebra, 2 (1974), 115–121.
    • (1974) J. Linear Multi-Linear Algebra , vol.2 , pp. 115-121
    • Neuberger, J.W.1
  • 143
    • 28044468671 scopus 로고
    • An Iterative Method for Solving Nonlinear Partial Differential Equations
    • J. W. Neuberger, An Iterative Method for Solving Nonlinear Partial Differential Equations, Advances in Math., 19 (1976), 245–265.
    • (1976) Advances in Math , vol.19 , pp. 245-265
    • Neuberger, J.W.1
  • 144
    • 84968516157 scopus 로고
    • A Resolvent for an Iteration Method for Nonlinear Partial Differential Equations
    • J. W. Neuberger, A Resolvent for an Iteration Method for Nonlinear Partial Differential Equations, Trans. Amer. Math. Soc., 226 (1977), 231–343.
    • (1977) Trans. Amer. Math. Soc. , vol.226 , pp. 231-343
    • Neuberger, J.W.1
  • 145
    • 85003279776 scopus 로고
    • Projection Methods for Linear and Nonlinear Systems of Partial Differential Equations
    • Proceedings 1976 Dundee Conference on Differential Equations, Springer-Verlag Lecture
    • J. W. Neuberger, Projection Methods for Linear and Nonlinear Systems of Partial Differential Equations, Proceedings 1976 Dundee Conference on Differential Equations, Springer-Verlag Lecture Notes 564 (1976), 341–349.
    • (1976) Notes , vol.564 , pp. 341-349
    • Neuberger, J.W.1
  • 146
    • 49349136544 scopus 로고
    • Square Integrable Solutions to Linear Inhomogenous Systems
    • J. W. Neuberger, Square Integrable Solutions to Linear Inhomogenous Systems, J. Differential Equations, 27 (1978), 144–152.
    • (1978) J. Differential Equations , vol.27 , pp. 144-152
    • Neuberger, J.W.1
  • 148
    • 84976151644 scopus 로고
    • Boundary Value Problems for Linear Systems
    • J. W. Neuberger, Boundary Value Problems for Linear Systems, Proc.Royal Soc., Edinburgh, 83A (1979), 297–302.
    • (1979) Proc.Royal Soc., Edinburgh , vol.83A , pp. 297-302
    • Neuberger, J.W.1
  • 149
    • 49149139134 scopus 로고
    • Finite Dimensional Potential Theory Applied to Numerical Analysis
    • J. W. Neuberger, Finite Dimensional Potential Theory Applied to Numerical Analysis, Linear Alg. Appl., 35 (1981), 193–202.
    • (1981) Linear Alg. Appl. , vol.35 , pp. 193-202
    • Neuberger, J.W.1
  • 150
    • 0019240744 scopus 로고
    • A Type-Independent Iterative Method for Systems of Non-Linear Partial Differential Equations: Application to the Problem of Transonic Flow
    • J. W. Neuberger, A Type-Independent Iterative Method for Systems of Non-Linear Partial Differential Equations: Application to the Problem of Transonic Flow, Computers and Mathematics With Applications, 7 (1980), 67–78.
    • (1980) Computers and Mathematics with Applications , vol.7 , pp. 67-78
    • Neuberger, J.W.1
  • 151
    • 85072881309 scopus 로고
    • A Solvability Condition for Systems of Functional Differential Equations
    • Marcel Dekker, Inc
    • J. W. Neuberger, A Solvability Condition for Systems of Functional Differential Equations, Proceedings Conf. Functional and Integral Equations, Marcel Dekker, Inc., (1981).
    • (1981) Proceedings Conf. Functional and Integral Equations
    • Neuberger, J.W.1
  • 153
    • 85072882000 scopus 로고
    • A Type-Independent Numerical Method with Applications to Navier-Stokes Equations
    • Northwestern University
    • J. W. Neuberger, A Type-Independent Numerical Method with Applications to Navier-Stokes Equations, Proc. Ninth Conference on Numerical Simulation of Plasmas, Northwestern University, 1980.
    • (1980) Proc. Ninth Conference on Numerical Simulation of Plasmas
    • Neuberger, J.W.1
  • 157
    • 0038914414 scopus 로고
    • Steepest Descent for General Systems of Linear Differential Equations in Hilbert Space, Springer-Verlag
    • J. W. Neuberger, Steepest Descent for General Systems of Linear Differential Equations in Hilbert Space, Springer-Verlag Lecture Notes in Mathematics 1032 (1982), 390–406.
    • (1982) Lecture Notes in Mathematics , vol.1032 , pp. 390-406
    • Neuberger, J.W.1
  • 159
    • 84972562507 scopus 로고
    • Steepest Descent and Differential Equations
    • J. W. Neuberger, Steepest Descent and Differential Equations, J. Math. Soc. Japan 37 (1985), 187–195.
    • (1985) J. Math. Soc. Japan , vol.37 , pp. 187-195
    • Neuberger, J.W.1
  • 160
    • 69549100129 scopus 로고
    • Calculation of Sharp Shocks Using Sobolev Gradients
    • American Mathematical Society
    • J. W. Neuberger, Calculation of Sharp Shocks Using Sobolev Gradients, Contemporary Mathematics, American Mathematical Society 108 (1990), 111–118.
    • (1990) Contemporary Mathematics , vol.108 , pp. 111-118
    • Neuberger, J.W.1
  • 161
    • 85037192992 scopus 로고
    • Constructive Variational Methods for Differential Equations, Nonlinear Analysis
    • J. W. Neuberger, Constructive Variational Methods for Differential Equations, Nonlinear Analysis, Theory, Meth. & Appl. 13 (1988), 413–428.
    • (1988) Theory, Meth. & Appl. , vol.13 , pp. 413-428
    • Neuberger, J.W.1
  • 162
    • 85072898170 scopus 로고
    • Prevalance of Chaotic Differences for Unpredictable Functions
    • J. W. Neuberger, Prevalance of Chaotic Differences for Unpredictable Functions, Acta Scientiarum Math., 61 (1995), 181–196.
    • (1995) Acta Scientiarum Math , vol.61 , pp. 181-196
    • Neuberger, J.W.1
  • 163
    • 84996131867 scopus 로고    scopus 로고
    • Sobolev Gradients and Boundary Conditions for Partial Differential Equations
    • J. W. Neuberger, Sobolev Gradients and Boundary Conditions for Partial Differential Equations, Contemporary Mathematics 204 (1997), 171–181.
    • (1997) Contemporary Mathematics , vol.204 , pp. 171-181
    • Neuberger, J.W.1
  • 164
    • 0041544029 scopus 로고    scopus 로고
    • Continuous Steepest Descent in Higher Order Sobolev Spaces
    • J. W. Neuberger, Continuous Steepest Descent in Higher Order Sobolev Spaces, Nonlinear Analysis 30 (1997), 3517–3519.
    • (1997) Nonlinear Analysis , vol.30 , pp. 3517-3519
    • Neuberger, J.W.1
  • 165
    • 33646973624 scopus 로고    scopus 로고
    • Laplacians and Sobolev Gradients
    • J. W. Neuberger, Laplacians and Sobolev Gradients, Proc. Am. Math. Soc. 126 (1998), 2053–2060.
    • (1998) Proc. Am. Math. Soc. , vol.126 , pp. 2053-2060
    • Neuberger, J.W.1
  • 166
    • 34948885876 scopus 로고    scopus 로고
    • A near minimal hypothesis Nash-Moser Theorem
    • J. W. Neuberger A near minimal hypothesis Nash-Moser Theorem, Int.J.Pure. Appl. Math.. 4 (2003), 269–280.
    • (2003) Int.J.Pure. Appl. Math. , vol.4 , pp. 269-280
    • Neuberger, J.W.1
  • 168
    • 84919876995 scopus 로고    scopus 로고
    • Least squares and chaotic behavior in initial value problems
    • J. W. Neuberger and R. J. Renka, Least squares and chaotic behavior in initial value problems, J. Nonlinnear Analysis Convexity 6 (2005), 65–70.
    • (2005) J. Nonlinnear Analysis Convexity , vol.6 , pp. 65-70
    • Neuberger, J.W.1    Renka, R.J.2
  • 169
    • 0039254818 scopus 로고    scopus 로고
    • Continuous Newton’s method for polynomials
    • J. W. Neuberger, Continuous Newton’s method for polynomials, Math. Intelligencer 21 (1999), 18–23.
    • (1999) Math. Intelligencer , vol.21 , pp. 18-23
    • Neuberger, J.W.1
  • 171
    • 0037992554 scopus 로고
    • Non-uniqueness for Plateau’s Problem. A Bifurcation Process
    • J.C.C. Nitsche, Non-uniqueness for Plateau’s Problem. A Bifurcation Process, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 361–373.
    • (1976) Ann. Acad. Sci. Fenn. Ser. a I Math. , vol.2 , pp. 361-373
    • Nitsche, J.C.C.1
  • 172
    • 41849128900 scopus 로고    scopus 로고
    • Sobolev Gradients for Differential Algebraic Equations
    • R. Nittka and M. Sauter, Sobolev Gradients for Differential Algebraic Equations, Elec. J. Differential Equations, Vol. 2008 (2008), 1–31.
    • (2008) Elec. J. Differential Equations , vol.2008 , pp. 1-31
    • Nittka, R.1    Sauter, M.2
  • 176
    • 0040122523 scopus 로고
    • +-ion fluxes and bioelectric phenomena in mesophyll cells of Atriplex spongiosa
    • +-ion fluxes and bioelectric phenomena in mesophyll cells of Atriplex spongiosa, Zeit.für Pflanz 62 (1970), 417–425.
    • (1970) Zeit.für Pflanz , vol.62 , pp. 417-425
    • Pallaghy, C.K.1    Lüttge, U.2
  • 177
    • 49549131043 scopus 로고
    • Lower Bounds for the Norm of the Symmetric Product
    • T. Pate, Lower Bounds for the Norm of the Symmetric Product, Linear Alg. App. 14 (1976), 285–292.
    • (1976) Linear Alg. App. , vol.14 , pp. 285-292
    • Pate, T.1
  • 178
    • 48749124681 scopus 로고
    • Some Improvements on Neuberger’s Iteration Procedure for Solving Partial Differential Equations
    • T. Pate, Some Improvements on Neuberger’s Iteration Procedure for Solving Partial Differential Equations, J. Diff. Eqns. 34 (1979), 261–272.
    • (1979) J. Diff. Eqns. , vol.34 , pp. 261-272
    • Pate, T.1
  • 179
    • 0018737335 scopus 로고
    • An Application of the Neuberger Iteration Procedure to the Constant Coefficient Linear Partial Differential Equation
    • T. Pate, An Application of the Neuberger Iteration Procedure to the Constant Coefficient Linear Partial Differential Equation, J. Math. Anal. Appl. 72 (1979), 771–782.
    • (1979) J. Math. Anal. Appl. , vol.72 , pp. 771-782
    • Pate, T.1
  • 180
    • 84972546436 scopus 로고
    • A Characterization of a Neuberger type Iteration Procedure that Leads to Solutions of Classical Boundary Value Problems
    • T. Pate, A Characterization of a Neuberger type Iteration Procedure that Leads to Solutions of Classical Boundary Value Problems, Pac. J. Math. 104 (1983), 191–204.
    • (1983) Pac. J. Math. , vol.104 , pp. 191-204
    • Pate, T.1
  • 181
    • 85072897612 scopus 로고    scopus 로고
    • A Unifying Context for the Classical Cauchy Initial Value Problem and the Type of Problem Solvable Using-an Iteration Procedure Suggested by Neuberger, Houston J
    • T. Pate, A Unifying Context for the Classical Cauchy Initial Value Problem and the Type of Problem Solvable Using-an Iteration Procedure Suggested by Neuberger, Houston J. Mcath. 8 (1982), 273–284.
    • Mcath , vol.8 , Issue.1982 , pp. 273-284
    • Pate, T.1
  • 183
    • 44149107770 scopus 로고    scopus 로고
    • Adjoint-based optimization of PDE systems with alternative gradients
    • B. Protas Adjoint-based optimization of PDE systems with alternative gradients, J. Comp. Physics 227 (2008), 6490–510.
    • (2008) J. Comp. Physics , vol.227 , pp. 6490-6510
    • Protas, B.1
  • 184
    • 1542636290 scopus 로고    scopus 로고
    • A comprehensive framework for the regularization of adjoint analysis in multiscale PDE systems
    • B. Protas, T. Bewley and G. Hagen, A comprehensive framework for the regularization of adjoint analysis in multiscale PDE systems, J. Comp. Physics 195 (1) (2004), 48–89.
    • (2004) J. Comp. Physics , vol.195 , Issue.1 , pp. 48-89
    • Protas, B.1    Bewley, T.2    Hagen, G.3
  • 185
    • 60449100399 scopus 로고    scopus 로고
    • Energy minimization related to the nonlinear Schrödinger equation
    • N. Raza, S. Sial, S. Siddiqi, T. Lookman, Energy minimization related to the nonlinear Schrödinger equation, J. Comp. Physics 228 (2009), 2572–2577.
    • (2009) J. Comp. Physics , vol.228 , pp. 2572-2577
    • Raza, N.1    Sial, S.2    Siddiqi, S.3    Lookman, T.4
  • 186
    • 60349102326 scopus 로고    scopus 로고
    • Sobolev gradient approach for the time evolution related to energy minimization of Ginzburg-Landau functionals
    • N. Raza, S. Sial, S. Siddiqui, Sobolev gradient approach for the time evolution related to energy minimization of Ginzburg-Landau functionals, J. Comp. Physics 228 (2009), 2566–2571.
    • (2009) J. Comp. Physics , vol.228 , pp. 2566-2571
    • Raza, N.1    Sial, S.2    Siddiqui, S.3
  • 187
    • 65149087622 scopus 로고    scopus 로고
    • Simulation study of propagation of pulses in optical fiber communicaton systems using Sobolev gradient and split-step Fourier Methods
    • N. Raza, S. Sial, S. Siddiqui, Simulation study of propagation of pulses in optical fiber communicaton systems using Sobolev gradient and split-step Fourier Methods, Int. J. Comp. Methods 6 (2009), 1–12.
    • (2009) Int. J. Comp. Methods , vol.6 , pp. 1-12
    • Raza, N.1    Sial, S.2    Siddiqui, S.3
  • 188
    • 0002079211 scopus 로고
    • A Nonlinear Hille-Yosida Theorem in Banach Spaces
    • S. Reich, A Nonlinear Hille-Yosida Theorem in Banach Spaces, J. Math. Analy. Appl. 84 (1981), 1–5.
    • (1981) J. Math. Analy. Appl. , vol.84 , pp. 1-5
    • Reich, S.1
  • 190
    • 0039507037 scopus 로고
    • Minimal Surfaces and Sobolev Gradients
    • R. J. Renka and J. W. Neuberger Minimal Surfaces and Sobolev Gradients, Siam J. Sci. Comp 16 (1995), 1412–1427.
    • (1995) Siam J. Sci. Comp , vol.16 , pp. 1412-1427
    • Renka, R.J.1    Neuberger, J.W.2
  • 191
    • 0032228725 scopus 로고    scopus 로고
    • Sobolev Gradients and the Ginzburg-Landau Equations
    • R. J. Renka and J. W. Neuberger, Sobolev Gradients and the Ginzburg-Landau Equations. Siam J. Sci. Comp. 20 (1998), 582–590.
    • (1998) Siam J. Sci. Comp. , vol.20 , pp. 582-590
    • Renka, R.J.1    Neuberger, J.W.2
  • 192
    • 2442486949 scopus 로고    scopus 로고
    • Algorithm 828. DNSPLIN1: Discrete nonlinear spline interpolation
    • , No
    • R. J. Renka, Algorithm 828. DNSPLIN1: Discrete nonlinear spline interpolation, ACM Trans. Math. Softw. 29, No. 4 (2003), 1–11.
    • (2003) ACM Trans. Math. Softw. , vol.29 , Issue.4 , pp. 1-11
    • Renka, R.J.1
  • 193
    • 0347899446 scopus 로고    scopus 로고
    • Constructing fair curves and surfaces with a Sobolev gradient method
    • R. J. Renka, Constructing fair curves and surfaces with a Sobolev gradient method, CAGD 21 (2004), 137–149.
    • (2004) CAGD , vol.21 , pp. 137-149
    • Renka, R.J.1
  • 194
    • 28044470759 scopus 로고    scopus 로고
    • A Sobolev gradient method for construction of elastic curves in regular surfaces
    • R. J. Renka, A Sobolev gradient method for construction of elastic curves in regular surfaces, J. Nonlinear Analysis: Theory, Methods and Applications 63 (2005), e1789–e1796.
    • (2005) J. Nonlinear Analysis: Theory, Methods and Applications , vol.63
    • Renka, R.J.1
  • 195
    • 25844443709 scopus 로고    scopus 로고
    • 3 space curves
    • , No
    • 3 space curves, CAGD 22, No. 8 (2005), 793–809.
    • (2005) CAGD , vol.22 , Issue.8 , pp. 793-809
    • Renka, R.J.1
  • 197
    • 0142013456 scopus 로고    scopus 로고
    • Sobolev Gradient Preconditioning for PDE Applications
    • (Kincaid, D.R., Elster, A.C., eds.), pp., IMACS, New Jersey
    • W. B. Richardson, JR., Sobolev Gradient Preconditioning for PDE Applications, in: Iterative Methods in Scientific Computation IV. (Kincaid, D.R., Elster, A.C., eds.), pp. 223–234, IMACS, New Jersey, 1999.
    • (1999) Iterative Methods in Scientific Computation IV , pp. 223-234
    • Richardson, W.B.1
  • 198
    • 0039507039 scopus 로고
    • Steepest Descent and the Least C for Sobolev’s Inequality
    • W. B. Richardson, Jr., Steepest Descent and the Least C for Sobolev’s Inequality, Bull. London Math. Soc. 18 (1986), 478–484.
    • (1986) Bull. London Math. Soc. , vol.18 , pp. 478-484
    • Richardson, W.B.1
  • 200
    • 0038243581 scopus 로고    scopus 로고
    • Steeptest descent using smoothed gradients
    • W. B. Richardson, Jr., Steeptest descent using smoothed gradients, Applied Math. Comp. 117 (2000), 241–254.
    • (2000) Applied Math. Comp , vol.117 , pp. 241-254
    • Richardson, W.B.1
  • 201
    • 0033640001 scopus 로고    scopus 로고
    • Sobolev preconditioning for the Poisson-Boltzman equation
    • W. B. Richardson, Jr., Sobolev preconditioning for the Poisson-Boltzman equation, Comput. Methods. Appl. Mech. Engrg. 181 (2000), 425–436.
    • (2000) Comput. Methods. Appl. Mech. Engrg. , vol.181 , pp. 425-436
    • Richardson, W.B.1
  • 202
    • 28044457668 scopus 로고    scopus 로고
    • High-order Sobolev Preconditioning
    • W. B. Richardson, Jr., High-order Sobolev Preconditioning, Nonlinear Analysis 63 (2005), 1779–1787.
    • (2005) Nonlinear Analysis , vol.63 , pp. 1779-1787
    • Richardson, W.B.1
  • 203
    • 48949117329 scopus 로고    scopus 로고
    • Sobolev gradient preconditioning for image-processing PDEs, Comm
    • W. B. Richardson, Jr., Sobolev gradient preconditioning for image-processing PDEs, Comm. Numerical Methods in Engineering 24 (2008), 493–504.
    • (2008) Numerical Methods in Engineering , vol.24 , pp. 493-504
    • Richardson, W.B.1
  • 205
    • 0036344565 scopus 로고    scopus 로고
    • Optimizing Schrödinger Functionals using Sobolev gradients: Applications to Quantum mechanics aqnd nonlinear optics
    • J.J. García-Ripoll and V. M. Pêrez-García, Optimizing Schrödinger Functionals using Sobolev gradients: applications to Quantum mechanics aqnd nonlinear optics, Siam J. Sci. Comp. 23 (2001), 1315–1333.
    • (2001) Siam J. Sci. Comp. , vol.23 , pp. 1315-1333
    • García-Ripoll, J.J.1    Pêrez-García, V.M.2
  • 207
    • 84972571552 scopus 로고
    • ′ (T) =F (t, y(g(t)))
    • ′ (t) =F (t, y(g(t))), Pacific J. Math. 43 (1972), 483–492.
    • (1972) Pacific J. Math. , vol.43 , pp. 483-492
    • Robertson, M.L.1
  • 211
    • 0018929621 scopus 로고
    • Two-Dimensional Magnetohydrodynamic Calculation
    • D. Schnack and J. Killeen, Two-Dimensional Magnetohydrodynamic Calculation, J. Comp. Physics 35 (1980), 110–145.
    • (1980) J. Comp. Physics , vol.35 , pp. 110-145
    • Schnack, D.1    Killeen, J.2
  • 212
    • 24644442200 scopus 로고    scopus 로고
    • Sobolev gradient algorithm for minimum energy states of S-wave superconductors-finite element setting
    • S. Sial, Sobolev gradient algorithm for minimum energy states of S-wave superconductors-finite element setting, Supercond. Sci. Technol 18 (2005), 675–677.
    • (2005) Supercond. Sci. Technol , vol.18 , pp. 675-677
    • Sial, S.1
  • 213
    • 0037777657 scopus 로고    scopus 로고
    • Energy minimization using Sobolev gradients: Application to phase separation and ordering
    • S. Sial, J. W. Neuberger, T. Lookman, A. Saxena, Energy minimization using Sobolev gradients: application to phase separation and ordering, J. Comp. Physics 189 (2003), 88–97.
    • (2003) J. Comp. Physics , vol.189 , pp. 88-97
    • Sial, S.1    Neuberger, J.W.2    Lookman, T.3    Saxena, A.4
  • 215
    • 0000137102 scopus 로고
    • Asymptotics for a class of nonlinear evolution equations with applications to geometric problems
    • L. Simon, Asymptotics for a class of nonlinear evolution equations with applications to geometric problems, Ann. of Math. 118 (1983), 525571.
    • (1983) Ann. of Math. , vol.118
    • Simon, L.1
  • 217
    • 34250392866 scopus 로고
    • Best constant in Sobolev inequality
    • G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura. Appl. (4) (1976), 352–372.
    • (1976) Ann. Mat. Pura. Appl. , Issue.4 , pp. 352-372
    • Talenti, G.1
  • 220
    • 0039269495 scopus 로고
    • On the Convergence of the Method of Steepest Descent for Nonlinear Equations
    • M. M. Vainberg, On the Convergence of the Method of Steepest Descent for Nonlinear Equations, Sibirsk. Math. A. 2 (1961), 201–220.
    • (1961) Sibirsk. Math. A. , vol.2 , pp. 201-220
    • Vainberg, M.M.1
  • 222
    • 68649116471 scopus 로고    scopus 로고
    • An inverse model for a free-boundary problem with a contact line: Steady case
    • to appear
    • O. Volkov and B. Protas, An inverse model for a free-boundary problem with a contact line: steady case. J. Comp. Physics (2009) (to appear).
    • (2009) J. Comp. Physics
    • Volkov, O.1    Protas, B.2
  • 224
  • 225
    • 84967491274 scopus 로고
    • The method of orthogonal projections in potential theory
    • H. Weyl, The method of orthogonal projections in potential theory, Duke Math. J. 7 (1940), 411–444.
    • (1940) Duke Math. J. , vol.7 , pp. 411-444
    • Weyl, H.1
  • 227
    • 2442492200 scopus 로고    scopus 로고
    • Steepest Descent on a Uniformly Convex Space
    • M. Zahran, Steepest Descent on a Uniformly Convex Space, RockyMountain J. Math. 33 (2003), 1539–1555.
    • (2003) Rockymountain J. Math. , vol.33 , pp. 1539-1555
    • Zahran, M.1
  • 228
    • 85072899306 scopus 로고    scopus 로고
    • Sobolev gradients in uniformly convex spaces
    • M. Zahran Sobolev gradients in uniformly convex spaces, Rocky Mountain J. Math. 33 (2009), 1–16.
    • (2009) Rocky Mountain J. Math. , vol.33 , pp. 1-16
    • Zahran, M.1
  • 229
    • 33845895199 scopus 로고    scopus 로고
    • Sturm-Liouville Theory, Mathematical Surveys
    • A. Zettl, Sturm-Liouville Theory, Mathematical Surveys 121 (2005), American Math. Soc.
    • (2005) American Math. Soc. , vol.121
    • Zettl, A.1


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