×

Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature. (English) Zbl 0408.35015


MSC:

35B50 Maximum principles in context of PDEs
35J60 Nonlinear elliptic equations
Full Text: DOI

References:

[1] Protter, M. H.; Weinberger, H. F., Maximum principles in differential equations (1967), Prentice-Hall · Zbl 0153.13602
[2] Hopf, E., Elementare Bemerkung über die Lösung partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Berlin. Sber. Preuss. Akad. Wiss., 19, 147-152 (1927) · JFM 53.0454.02
[3] Hopf, E., A remark on elliptic differential equations of the second order, Proc. Am. math. Soc., 3, 791-793 (1952) · Zbl 0048.07802
[4] Hill, G. N.; Protter, M. H., Maximum principles for a class of first-order elliptical systems, J. diff. Eqns, 24, 136-151 (1977) · Zbl 0351.35007
[5] Ladyzenskaja, O. A.; Ural’ceva, N. N., Local estimates for gradients of solutions of nonuniformly elliptic and parabolic equations, Communs pure appl. Math., 33, 677-703 (1970) · Zbl 0193.07202
[6] Payne, L. E., Bounds for the maximum stress in the Saint-Venant torsion problem, Ind. J. Mech. Math., 51-59 (1968), Special issue
[7] Payne, L. E., Some isoperimetric inequalities for harmonic functions, SIAM J. math. Analysis, 1, 354-359 (1970) · Zbl 0199.16902
[8] Payne, L. E., Some remarks on maximum principles, J. Analyse math., 30, 421-433 (1976) · Zbl 0334.35029
[9] Payne, L. E.; Philippin, G. A., Some applications of the maximum principle in the problem of torsional creep, SIAM J. appl. Math., 33, 446-455 (1977) · Zbl 0378.73028
[10] Payne, L. E.; Philippin, G. A., Some remarks on the problems of elastic torsion and of torsional creep, (Some Aspects of Mechanics of Continua (1977), Jadavpur University), 32-40, Part I
[11] Payne, L. E.; Philippin, G. A., On some maximum principles involving harmonic functions and their derivatives, SIAM J. math. Analysis, 10, 1 (1979) · Zbl 0402.35014
[12] Payne, L. E.; Sperb, R.; Stakgold, I., On Hopf type maximum principles for convex domains, J. Nonlinear Analysis, 1, 547-559 (1977) · Zbl 0367.35023
[13] Payne, L. E.; Stakgold, I., On the mean value of the fundamental mode in the fixed membrane problem, Appl. Analysis, 3, 295-303 (1973) · Zbl 0323.35057
[14] Philippin, G. A., On the first eigenfunction of the fixed membrane: Some extensions of results of Payne and Stakgold, ZAMP, 28, 151-159 (1977) · Zbl 0354.35036
[15] Philippin, G. A., Some remarks on the elastically supported membrane, ZAMP, 29, 306-314 (1978) · Zbl 0378.73069
[16] PHILIPPINJ. math. Analysis Applic.; PHILIPPINJ. math. Analysis Applic.
[17] Protter, M. H.; Weinberger, H. F., A maximum principle and gradient bounds for linear elliptic equations, Ind. Univ. math. J., 23, 239-249 (1973-74) · Zbl 0276.35030
[18] SHAEFER; SHAEFER
[19] Shaefer, P. W., Some maximum principles for nonlinear elliptic boundary value problems, Q. appl. Math., 35, 517-523 (1977) · Zbl 0379.35026
[20] Shaefer, P. W.; Sperb, R. P., Maximum principles and bounds in some inhomogeneous elliptic boundary value problems, SIAM J. math. Analysis, 8, 871-878 (1977) · Zbl 0372.35032
[21] Shaefer, P. W.; Sperb, R. P., Maximum principles for some functionals associated with the solution of elliptic boundary value problems, Archs ration. mech. Analysis, 61, 65-76 (1976) · Zbl 0328.35004
[22] Shaefer, P. W.; Sperb, R. P., A maximum principle for a class of functionals in nonlinear Dirichlet problems, Ordinary and partial differential equations, (Lecture Notes in Mathematics, 564 (1976), Springer), 400-406, Dundee 1976 · Zbl 0338.35041
[23] Serrin, J. B., Gradient estimates for solutions of nonlinear elliptic and parabolic equations, (Contributions to Nonlinear functional analysis, Univ. of Wisconsin (1971), Academic Press), 565-601 · Zbl 0271.35004
[24] Stakgold, I., Global estimates for nonlinear reaction and diffusion, (Ordinary and partial differential equations. Ordinary and partial differential equations, Proc, of a conf. held in Dundee, Scotland, (1974). Ordinary and partial differential equations. Ordinary and partial differential equations, Proc, of a conf. held in Dundee, Scotland, (1974), Lecture in Mathematics Notes, 415 (1974), Springer), 252-266 · Zbl 0301.35033
[25] Stakgold, I.; Payne, L. E., Nonlinear problems in nuclear reactor analysis, (Proc, Conf. on Nonlin. Probl. in Phys. Sc. and Biology, 322 (1973), Springer), 298-307, Lecture Notes in Mathematics · Zbl 0259.35025
[26] Ladyzenskaja, O. A.; Ural’ceva, N. N., Equations aux Dérivées Partielles de Type Elliptique (1968), Dunod: Dunod Paris · Zbl 0164.13001
[27] Ural’ceva, N. N., The solvability of the capillarity problem, (Math. Meh. Astronom., 4 (1973), Vestnik Leningrad Univ), #19
[28] Spruck, J., On the existence of a capillary surface with prescribed contact angle, Communs pure appl. Math., 28, 189-200 (1975) · Zbl 0297.76018
[29] Simon, L.; Spruck, J., Existence and regularity of a capillary surface with prescribed contact angle, Archs ration. mech. Analysis, 61, 19-34 (1976) · Zbl 0361.35014
[30] Serrin, J. B., The problem of Dirichlet for quasilinear elliptic differential equations with many independent varaibles, Phil. Trans. R. Soc. Lond., 264, 413-496 (1969) · Zbl 0181.38003
[31] Bernstein, S., Über ein geometrisches Theorem und seine Anwendungen auf die partiellin Differentialgleichungen vom elliptischen Typus, Math. Z., 26, 551-588 (1927) · JFM 53.0670.01
[32] Emmer, M., Existenza, unicità e rigolarità delle superfici di equilibrio nei capillari, Ann. Univ. Ferrara, 18, 79-94 (1973) · Zbl 0275.49005
[33] Pepe, L., Analiticità delle superfici di equilibrio dei capillari in ogni dimensione, (Symposia Mathematica (1974), Academic Press) · Zbl 0305.35032
[34] Finn, R.; Gerhardt, C., The internal sphere and the capillary problem, Ann. Math., 62, 13-31 (1977) · Zbl 0349.49019
[35] Concus, P.; Finn, R., On the behavior of a capillary surface in a wedge, Appl. Math. sciences, Proc. natl. Acad. Sci. U.S.A., 292-299 (1969) · Zbl 0219.76104
[36] Concus, P.; Finn, R., On capillary free surfaces in the absence of gravity, Acta math., 132, 177-198 (1974) · Zbl 0382.76003
[37] Concus, P.; Finn, R., On capillary free surfaces in a gravitational field, Acta math., 132, 207-223 (1974) · Zbl 0382.76005
[38] Gerhardt, C., Existence and regularity of capillary surfaces, Boll. Un. mat. ital., 10, 317-335 (1974) · Zbl 0314.49019
[39] Gerhardt, C., Global regularity of the solutions to the capillary problem, Annali Scu. norm. sup. Pisa, 3, 157-175 (1976) · Zbl 0338.49008
[40] Giusti, E., On the regularity of the solutions to a mixed boundary value problem for the nonhomogeneous minimal surface equation, Boll. Un. math. ital., 11, 349-374 (1975) · Zbl 0316.35015
[41] Giusti, E., Boundary value problems for non-parametric surfaces of prescribed mean curvature, Annali Scu. norm. sup. Pisa, 3, 501-548 (1976) · Zbl 0344.35036
[42] Serrin, J. B., Nonlinear elliptic equations of second order, Lectures at Symposium on partial differential equations, 57 (1971), Mimeographed notes
[43] Miranda, C., Formule di maggiorazione e teorema di esistenza per le funzioni biarmoniche di due variabili, Giorn. Mat. Battaglini, 78, 97-118 (1948-1949) · Zbl 0037.07103
[44] Serrin, J. B., A symmetry problem in potential theory, Archs ration. mech. Analysis, 43, 304-318 (1971) · Zbl 0222.31007
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.