zbMATH — the first resource for mathematics

Isoperimetric inequalities for the length of level lines of solutions of quasilinear capacity problems in the plane. (English) Zbl 0712.35034
Estimates are given for the length of level curves of solutions of certain quasilinear elliptic PDE’s in plane annuli. The equations are of the type $$div(a(| \nabla u|)\nabla u)=0$$, and they include the p- Laplace and the nonparametric minimal surface equation. The annuli are of general type with no convexity or star shapedness requirements.
Reviewer: F.Schulz

MSC:
 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
Full Text:
References:
 [1] G. Alessandrini,On the identification of the leading coefficient of an elliptic equation. Boll. Un. Mat. Ital. C6, 87-111 (1985). · Zbl 0598.35129 [2] G. Alessandrini,Critical points of solutions of elliptic equations in two variables. Ann. Scuola Norm. Sup. Pisa Cl. Sci.4 XIV, 229-256 (1987). · Zbl 0649.35026 [3] G. Alessandrini,Critical points of solutions to the p-Laplace equation in dimension two. Boll. Un. Mat. Ital. A7, 239-246 (1987). · Zbl 0634.35027 [4] G. Alessandrini,The length of level lines of solutions of elliptic equations in the plane. Arch. Rat. Mech. Anal.102, 183-191 (1988). · Zbl 0664.35016 · doi:10.1007/BF00251498 [5] D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer Verlag, New York 1983. · Zbl 0562.35001 [6] P. Hartman and A. Wintner,On the local behaviour of solutions of non-parabolic partial differential equations. Amer. J. Math.76, 449-476 (1953). · Zbl 0052.32201 · doi:10.2307/2372496 [7] P. Laurence,On the convexity of geometric functionals of level for solutions of certain elliptic partial differential equations. J. Appl. Math. Phys.40, 258-284 (1989). · Zbl 0679.35026 · doi:10.1007/BF00945002 [8] M. Longinetti,Some isoperimetric inequalities for the level curves of capacity and Green’s functions on convex plane domains. SIAM J. Math. Anal.19, 377-389 (1988). · Zbl 0647.31001 · doi:10.1137/0519028 [9] M. Longinetti,On minimal surfaces bounded by two convex curves in parallel planes. J. Differential Equations67, 344-358 (1987). · Zbl 0626.53002 · doi:10.1016/0022-0396(87)90131-8 [10] W. F. Osgood, Lehrbuch der Funktionentheorie, vol. 1, Berlin 1907. · JFM 38.0412.01 [11] C. E. Weatherburn,On families of surfaces, Math. Ann.99, 473-478 (1928). · JFM 54.0747.02 · doi:10.1007/BF01459108
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.