Viscosity subsolutions and supersolutions for non-uniformly and degenerate elliptic equations

Aris S. Tersenov

Archivum Mathematicum (2009)

  • Volume: 045, Issue: 1, page 19-35
  • ISSN: 0044-8753

Abstract

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In the present paper we study the Dirichlet boundary value problem for quasilinear elliptic equations including non-uniformly and degenerate ones. In particular, we consider mean curvature equation and pseudo p-Laplace equation as well. It is well-known that the proof of the existence of continuous viscosity solutions is based on Ishii’s implementation of Perron’s method. In order to use this method one has to produce suitable subsolution and supersolution. Here we introduce new methods to construct subsolutions and supersolutions for the above mentioned problems. Using these subsolutions and supersolutions one may prove the existence of unique continuous viscosity solution for a wide class of degenerate and non-uniformly elliptic equations.

How to cite

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Tersenov, Aris S.. "Viscosity subsolutions and supersolutions for non-uniformly and degenerate elliptic equations." Archivum Mathematicum 045.1 (2009): 19-35. <http://eudml.org/doc/250686>.

@article{Tersenov2009,
abstract = {In the present paper we study the Dirichlet boundary value problem for quasilinear elliptic equations including non-uniformly and degenerate ones. In particular, we consider mean curvature equation and pseudo p-Laplace equation as well. It is well-known that the proof of the existence of continuous viscosity solutions is based on Ishii’s implementation of Perron’s method. In order to use this method one has to produce suitable subsolution and supersolution. Here we introduce new methods to construct subsolutions and supersolutions for the above mentioned problems. Using these subsolutions and supersolutions one may prove the existence of unique continuous viscosity solution for a wide class of degenerate and non-uniformly elliptic equations.},
author = {Tersenov, Aris S.},
journal = {Archivum Mathematicum},
keywords = {viscosity subsolution; viscosity supersolution; mean curvature equation; pseudo $p$-Laplace equation; viscosity subsolution; viscosity supersolution; mean curvature equation; pseudo -Laplace equation},
language = {eng},
number = {1},
pages = {19-35},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Viscosity subsolutions and supersolutions for non-uniformly and degenerate elliptic equations},
url = {http://eudml.org/doc/250686},
volume = {045},
year = {2009},
}

TY - JOUR
AU - Tersenov, Aris S.
TI - Viscosity subsolutions and supersolutions for non-uniformly and degenerate elliptic equations
JO - Archivum Mathematicum
PY - 2009
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 045
IS - 1
SP - 19
EP - 35
AB - In the present paper we study the Dirichlet boundary value problem for quasilinear elliptic equations including non-uniformly and degenerate ones. In particular, we consider mean curvature equation and pseudo p-Laplace equation as well. It is well-known that the proof of the existence of continuous viscosity solutions is based on Ishii’s implementation of Perron’s method. In order to use this method one has to produce suitable subsolution and supersolution. Here we introduce new methods to construct subsolutions and supersolutions for the above mentioned problems. Using these subsolutions and supersolutions one may prove the existence of unique continuous viscosity solution for a wide class of degenerate and non-uniformly elliptic equations.
LA - eng
KW - viscosity subsolution; viscosity supersolution; mean curvature equation; pseudo $p$-Laplace equation; viscosity subsolution; viscosity supersolution; mean curvature equation; pseudo -Laplace equation
UR - http://eudml.org/doc/250686
ER -

References

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  2. Chen, Y. G., Giga, Y., Goto, S., Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), 749–786. (1991) Zbl0696.35087MR1100211
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  4. Crandall, M. G., Ishii, H., Lions, P. L., 10.1090/S0273-0979-1992-00266-5, Bull. Amer. Math. Soc. 27 (1992), 1–67. (1992) Zbl0755.35015MR1118699DOI10.1090/S0273-0979-1992-00266-5
  5. Crandall, M. G., Lions, P. L., 10.1090/S0002-9947-1983-0690039-8, Trans. Amer. Math. Soc. 277 (1983), 1–42. (1983) Zbl0599.35024MR0690039DOI10.1090/S0002-9947-1983-0690039-8
  6. Ishii, H., Lions, P. L., 10.1016/0022-0396(90)90068-Z, J. Differential Equations 83 (1990), 26–78. (1990) Zbl0708.35031MR1031377DOI10.1016/0022-0396(90)90068-Z
  7. Kawohl, B., Kutev, N., Comparison principle and Lipschitz regularity for viscosity solutions of some classes of nonlinear partial differential equations, Funkcial. Ekvac. 43 (2000), 241–253. (2000) Zbl1142.35315MR1795972
  8. Tersenov, Al., Tersenov, Ar., 10.1007/s00013-006-1873-9, Arch. Math. (Basel) 88 (3) (2007), 259–268. (2007) MR2305604DOI10.1007/s00013-006-1873-9
  9. Trudinger, N. S., Holder gradient estimates for fully nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), 57–65. (1988) MR0931007

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