×

Singular solutions to some non linear oblique derivative problems. (English) Zbl 0661.35033

The purpose of this paper, which is a sequel to a previous paper by the author [Am. J. Math. 107, 591-615 (1985; Zbl 0582.35022)], is to study non-coercive oblique derivative problems obtained by combining a nonlinear elliptic second order partial differential equation with a nonlinear first order boundary condition. Using a refinement of E. Zehnder’s generalized implicit function theorem [Commun. Pure Appl. Math. 28, 91-140 (1975; Zbl 0309.58006)], the author constructs local solutions which are not smooth up to the boundary and extends the results of the linear theory to the nonlinear case.
Reviewer: Simeon Reich

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI

References:

[1] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math., 12, 623-727 (1959) · Zbl 0093.10401
[2] Bony, J.-M, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. Ecole Norm. Sup. 4e sér., 14, 209-246 (1981) · Zbl 0495.35024
[3] Egorov, Yu. V.; Kontrat’ev, V. A., The oblique derivative problem, Math. U.S.S.R. Sb., 7, 139-169 (1969) · Zbl 0186.43202
[4] Gilbarg, S.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (1977), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0361.35003
[5] Godin, P., Subelliptic non linear oblique derivative problems, Amer. J. Math., 107, 591-615 (1985) · Zbl 0582.35022
[6] Godin, P., Non linear oblique derivative problems with non smooth solutions, Contemp. Math., 27, 97-106 (1984) · Zbl 0545.35036
[7] Hartman, P., Ordinary Differential Equations (1964), Wiley: Wiley New York · Zbl 0125.32102
[8] Hörmander, L., Pseudo-differential operators and non-elliptic boundary value problems, Ann. Math., 83, 129-209 (1966) · Zbl 0132.07402
[9] Hörmander, L., The boundary problems of physical geodesy, Arch. Rational Mech. Anal., 62, 1-52 (1976) · Zbl 0331.35020
[10] Melin, A.; Sjöstrand, J., A calculus for Fourier integral operators in domains with boundary and applications to the oblique derivative problem, Comm. Partial Differential Equations, 2, 857-935 (1977) · Zbl 0392.35055
[11] Winzell, B., The oblique derivative problem I, Math. Ann., 229, 267-278 (1977) · Zbl 0362.35025
[12] Winzell, B., The oblique derivative problem II, Ark. Mat., 17, 107-122 (1979) · Zbl 0414.35025
[13] Zehnder, E., Generalized implicit function theorems with applications to some small divisor problems I, Comm. Pure Appl. Math., 28, 91-140 (1975) · Zbl 0309.58006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.