4.8 Das Maximumprinzip

Inhalt:
Maximumprinzipien gestatten mit relativ geringem technischen Aufwand den Beweis manch interessanten und sehr anschaulichen Resultats über die Gestalt von Lösungen vor allem elliptischer (“Laplace”) und parabolischer (”Wärmeleitung”) Differentialgleichungen.

Literatur:

1. D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, 2nd edition, Berlin etc.: Springer-Verlag, 1983.
2. J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43, 304–318 (1971).
3. H. Weinberger, Remark on the preceding paper by Serrin, Arch. Rational Mech. Anal. 43, 319–320 (1971).
4. R. Finn, Equilibrium Capillary Surfaces, New York etc.: Springer-Verlag, 1986.
5. B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics 1150, Berlin etc.: Springer-Verlag, 1985.
6. B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys. 68, 209–243 (1979).
7. A. Bennett, Symmetry in an overdetermined fourth order elliptic boundary value problem, SIAM J. Math. Anal. 17, 1354–1358 (1986).
8. Chen, Wen Xiong; Li, Congming, Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63 (1991), no. 3, 615–622.
9. F. Gazzola, H.-Ch. Grunau, Critical dimensions and higher order Sobolev inequalities with remainder terms, Nonl. Differ. Equ. Appl. NoDEA 8, 35–44 (2001).