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Global existence of strong solutions to incompressible MHD
Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature
1. | Research group of the project PN-II-ID-PCE-2012-4-0021, "Simion Stoilow" Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania |
References:
[1] |
A. D. Alexandrov, A characteristic property of the sphere,, \emph{Ann. Mat. Pura Appl.}, 58 (1962), 303.
|
[2] |
L. Barbu and C. Enache, A maximum principle for some fully nonlinear elliptic equations with applications to Weingarten surfaces,, \emph{Complex Var. Elliptic Equ.}, 58 (2013), 1725. Google Scholar |
[3] |
B. Brandolini, C. Nitsch, P. Salani and C. Trombetti, Serrin type overdetermined problems : An alternative proof,, \emph{Arch. Ration. Mech. Anal.}, 190 (2008), 267.
doi: 10.1007/s00205-008-0119-3. |
[4] |
L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet Problem for Nonlinear Second-Order Elliptic Equations I. Monge-Ampère Equation,, \emph{Comm. Pure App. Math.}, 37 (1984), 369.
doi: 10.1002/cpa.3160370306. |
[5] |
C. Enache, Maximum principles and symmetry results for a class of fully nonlinear elliptic PDEs,, \emph{NODEA - Nonlinear Differ. Equ. Appl.}, 17 (2010), 591.
doi: 10.1007/s00030-010-0070-5. |
[6] |
C. Enache, Necessary conditions of solvability and isoperimetric estimates for some Monge-Ampère problems in the plane,, \emph{Proc. Amer. Math. Soc.}, (). Google Scholar |
[7] |
E. Hopf, Elementare Bemerkung über die Lösung partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus,, \emph{Berlin Sber. Preuss. Akad. Wiss}, 19 (1927), 147. Google Scholar |
[8] |
E. Hopf, A remark on linear elliptic differential equations of the second order,, \emph{Proc. Amer. Math. Soc.}, 36 (1952), 791.
|
[9] |
H. Hopf, Differential geometry in the large: seminar lectures New York University 1946 and Stanford University 1956,, vol. 1000 of Lecture Notes in Mathematics, (1000).
|
[10] |
N. M. Ivochkina, Solution of the Dirichlet problem for equations of mth order curvature, (Russian), \emph{Mat. Sb.}, 180 (1989), 867.
|
[11] |
P. L. Lions, N. S. Trudinger and J. Urbas, The Neumann problem for equations of Monge-Ampère type,, \emph{Comm. Pure Appl. Math.}, 39 (1986), 539.
doi: 10.1002/cpa.3160390405. |
[12] |
X.-N. Ma, A necessary condition of solvability for the capillarity boundary of Monge-Ampère equations in two dimensions,, \emph{Proc. Amer. Math. Soc.}, 127 (1999), 763.
doi: 10.1090/S0002-9939-99-04750-4. |
[13] |
X.-N. Ma, Sharp size estimates for capillary free surfaces without gravity,, \emph{Pacific J. Math.}, 192 (2000), 121.
doi: 10.2140/pjm.2000.192.121. |
[14] |
L. E. Payne and G. A. Philippin, Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature,, \emph{Nonlinear Anal.}, 3 (1979), 193.
doi: 10.1016/0362-546X(79)90076-2. |
[15] |
G. A. Philippin, A minimum principle for the problem of torsional creep,, \emph{J. Math. Anal. Appl.}, 68 (1979), 526.
doi: 10.1016/0022-247X(79)90133-1. |
[16] |
G. A. Philippin and A. Safoui, Some maximum principles and symmetry results for a class of boundary value problems involving the Monge-Ampère equation,, \emph{Math. Models Meth. Appl. Sci.}, 11 (2001), 1073.
doi: 10.1142/S0218202501001240. |
[17] |
G. A. Philippin and A. Safoui, Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE's,, \emph{Z. angew. Math. Phys.}, 54 (2003), 739.
doi: 10.1007/s00033-003-3200-7. |
[18] |
G. A. Philippin and A. Safoui, Some minimum principles for a class of elliptic boundary value problems,, \emph{Appl. Anal.}, 83 (2004), 231.
doi: 10.1080/00036810310001632754. |
[19] |
H. Rosenberg, Hypersurfaces of constant curvature in space forms,, \emph{Bull. Sci. Math.}, 117 (1993), 211.
|
[20] |
J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Ration. Mech. Anal.}, 43 (1971), 304.
|
[21] |
R. P. Sperb, Maximum Principles and Their Applications,, Academic Press, (1981).
|
[22] |
J. Urbas, Nonlinear oblique boundary value problem for Hessian equations in two dimensions,, \emph{Ann. Inst. Henri Poincare}, 12 (1995), 507.
|
[23] |
J. Urbas, A note on the contact angle boundary condition for Monge-Ampère equations,, \emph{Proc. Amer. Math. Soc.}, 128 (2000), 853.
doi: 10.1090/S0002-9939-99-05222-3. |
[24] |
H. F. Weinberger, Remarks on the preceding paper of Serrin,, \emph{Arch. Ration. Mech. Anal.}, 43 (1971), 319.
|
show all references
References:
[1] |
A. D. Alexandrov, A characteristic property of the sphere,, \emph{Ann. Mat. Pura Appl.}, 58 (1962), 303.
|
[2] |
L. Barbu and C. Enache, A maximum principle for some fully nonlinear elliptic equations with applications to Weingarten surfaces,, \emph{Complex Var. Elliptic Equ.}, 58 (2013), 1725. Google Scholar |
[3] |
B. Brandolini, C. Nitsch, P. Salani and C. Trombetti, Serrin type overdetermined problems : An alternative proof,, \emph{Arch. Ration. Mech. Anal.}, 190 (2008), 267.
doi: 10.1007/s00205-008-0119-3. |
[4] |
L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet Problem for Nonlinear Second-Order Elliptic Equations I. Monge-Ampère Equation,, \emph{Comm. Pure App. Math.}, 37 (1984), 369.
doi: 10.1002/cpa.3160370306. |
[5] |
C. Enache, Maximum principles and symmetry results for a class of fully nonlinear elliptic PDEs,, \emph{NODEA - Nonlinear Differ. Equ. Appl.}, 17 (2010), 591.
doi: 10.1007/s00030-010-0070-5. |
[6] |
C. Enache, Necessary conditions of solvability and isoperimetric estimates for some Monge-Ampère problems in the plane,, \emph{Proc. Amer. Math. Soc.}, (). Google Scholar |
[7] |
E. Hopf, Elementare Bemerkung über die Lösung partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus,, \emph{Berlin Sber. Preuss. Akad. Wiss}, 19 (1927), 147. Google Scholar |
[8] |
E. Hopf, A remark on linear elliptic differential equations of the second order,, \emph{Proc. Amer. Math. Soc.}, 36 (1952), 791.
|
[9] |
H. Hopf, Differential geometry in the large: seminar lectures New York University 1946 and Stanford University 1956,, vol. 1000 of Lecture Notes in Mathematics, (1000).
|
[10] |
N. M. Ivochkina, Solution of the Dirichlet problem for equations of mth order curvature, (Russian), \emph{Mat. Sb.}, 180 (1989), 867.
|
[11] |
P. L. Lions, N. S. Trudinger and J. Urbas, The Neumann problem for equations of Monge-Ampère type,, \emph{Comm. Pure Appl. Math.}, 39 (1986), 539.
doi: 10.1002/cpa.3160390405. |
[12] |
X.-N. Ma, A necessary condition of solvability for the capillarity boundary of Monge-Ampère equations in two dimensions,, \emph{Proc. Amer. Math. Soc.}, 127 (1999), 763.
doi: 10.1090/S0002-9939-99-04750-4. |
[13] |
X.-N. Ma, Sharp size estimates for capillary free surfaces without gravity,, \emph{Pacific J. Math.}, 192 (2000), 121.
doi: 10.2140/pjm.2000.192.121. |
[14] |
L. E. Payne and G. A. Philippin, Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature,, \emph{Nonlinear Anal.}, 3 (1979), 193.
doi: 10.1016/0362-546X(79)90076-2. |
[15] |
G. A. Philippin, A minimum principle for the problem of torsional creep,, \emph{J. Math. Anal. Appl.}, 68 (1979), 526.
doi: 10.1016/0022-247X(79)90133-1. |
[16] |
G. A. Philippin and A. Safoui, Some maximum principles and symmetry results for a class of boundary value problems involving the Monge-Ampère equation,, \emph{Math. Models Meth. Appl. Sci.}, 11 (2001), 1073.
doi: 10.1142/S0218202501001240. |
[17] |
G. A. Philippin and A. Safoui, Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE's,, \emph{Z. angew. Math. Phys.}, 54 (2003), 739.
doi: 10.1007/s00033-003-3200-7. |
[18] |
G. A. Philippin and A. Safoui, Some minimum principles for a class of elliptic boundary value problems,, \emph{Appl. Anal.}, 83 (2004), 231.
doi: 10.1080/00036810310001632754. |
[19] |
H. Rosenberg, Hypersurfaces of constant curvature in space forms,, \emph{Bull. Sci. Math.}, 117 (1993), 211.
|
[20] |
J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Ration. Mech. Anal.}, 43 (1971), 304.
|
[21] |
R. P. Sperb, Maximum Principles and Their Applications,, Academic Press, (1981).
|
[22] |
J. Urbas, Nonlinear oblique boundary value problem for Hessian equations in two dimensions,, \emph{Ann. Inst. Henri Poincare}, 12 (1995), 507.
|
[23] |
J. Urbas, A note on the contact angle boundary condition for Monge-Ampère equations,, \emph{Proc. Amer. Math. Soc.}, 128 (2000), 853.
doi: 10.1090/S0002-9939-99-05222-3. |
[24] |
H. F. Weinberger, Remarks on the preceding paper of Serrin,, \emph{Arch. Ration. Mech. Anal.}, 43 (1971), 319.
|
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