[Solutions visqueuses du problème avec une dérivée oblique dégénérée pour une classe d'équations complètement non linéaires]
On démontre dans cette Note un principe de comparaison entre les sous et supersolutions visqueuses semi-continues du problème avec une dérivée oblique tangentielle et aussi le problème mixte du type de Dirichlet–Neumann pour une classe d'équations elliptiques complètement non-linéaires. En appliquant ce principe de comparaison on démontre l'existence d'une solution visqueuse unique.
In this paper we prove a comparison principle between the semicontinuous viscosity sub- and supersolutions of the tangential oblique derivative problem and the mixed Dirichlet–Neumann problem for fully nonlinear elliptic equations. By means of the comparison principle, the existence of a unique viscosity solution is obtained.
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Publié le :
Petar Popivanov 1 ; Nickolai Kutev 1
@article{CRMATH_2002__334_8_661_0,
author = {Petar Popivanov and Nickolai Kutev},
title = {Viscosity solutions to the degenerate oblique derivative problem for fully nonlinear elliptic equations},
journal = {Comptes Rendus. Math\'ematique},
pages = {661--666},
publisher = {Elsevier},
volume = {334},
number = {8},
year = {2002},
doi = {10.1016/S1631-073X(02)02321-X},
language = {en},
}
TY - JOUR AU - Petar Popivanov AU - Nickolai Kutev TI - Viscosity solutions to the degenerate oblique derivative problem for fully nonlinear elliptic equations JO - Comptes Rendus. Mathématique PY - 2002 SP - 661 EP - 666 VL - 334 IS - 8 PB - Elsevier DO - 10.1016/S1631-073X(02)02321-X LA - en ID - CRMATH_2002__334_8_661_0 ER -
%0 Journal Article %A Petar Popivanov %A Nickolai Kutev %T Viscosity solutions to the degenerate oblique derivative problem for fully nonlinear elliptic equations %J Comptes Rendus. Mathématique %D 2002 %P 661-666 %V 334 %N 8 %I Elsevier %R 10.1016/S1631-073X(02)02321-X %G en %F CRMATH_2002__334_8_661_0
Petar Popivanov; Nickolai Kutev. Viscosity solutions to the degenerate oblique derivative problem for fully nonlinear elliptic equations. Comptes Rendus. Mathématique, Volume 334 (2002) no. 8, pp. 661-666. doi : 10.1016/S1631-073X(02)02321-X. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02321-X/
[1] Viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc., Volume 277 (1983), pp. 1-42
[2] User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., Volume 27 (1992), pp. 1-67
[3] On the oblique derivative problem, Math. Sb., Volume 78 (1969), pp. 148-176
[4] Regularity estimates for the oblique derivative problem, Ann. Math., Volume 137 (1993), pp. 1-70
[5] On the existence and regularity of solutions of linear pseudodifferential equations, Enseign. Math., Volume 17 (1971), pp. 99-163
[6] , The Analysis of Linear Differential Operators, IV, Springer-Verlag, Berlin, 1985
[7] Perron's method for Hamilton–Jacobi equations, Duke Math. J., Volume 55 (1987), pp. 369-384
[8] On uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic PDE's, Comm. Pure Appl. Math., Volume 42 (1989), pp. 14-45
[9] Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, Volume 83 (1990), pp. 26-78
[10] Neumann type boundary conditions for Hamilton–Jacobi equations, Duke Math. J., Volume 52 (1985), pp. 793-820
[11] The Oblique Derivative Problem, Ser. Math. Topics, 17, Wiley–VCH, Berlin, 2000
[12] The tangential oblique derivative problem for nonlinear elliptic equations, Comm. Partial Differential Equations, Volume 16 (1989), pp. 413-428
[13] The Degenerate Oblique Derivative Problem for Elliptic and Parabolic Equations, Academie-Verlag–VCH, 1997
[14] A boundary value problem with an oblique derivative, Comm. Partial Differential Equations, Volume 6 (1981), pp. 305-328
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