Abstract
It follows from a result of Brascamp and Lieb that the level curves of the first eigenfunction in the clamped membrane problem for a convex regionΩ are convex. This paper gives an alternate proof of the above convexity theorem which makes use of the maximum principle — a method which also yields pointwise bounds for the curvature of the level curve through an arbitrary point inΩ. The convexity theorem is then used to establish the existence of convex solutions to a related free boundary problem.
Zusammenfassung
Aus einem Ergebnis von Brascamp und Lieb folgt die Konvexität der Niveaulinien der ersten Eigenfunktion einer eingespannten Membran auf einem konvexen ebenen GebietΩ. Hier wird ein anderer Beweis desselben Konvexitätssatzes gegeben, welcher auf dem Maximum-prinzip beruht; diese Methode liefert auch punktweise Schranken für die Krümmung der Niveaulinie durch einen beliebigen Punkt inΩ. Mit Hilfe des Konvexitätssatzes wird dann die Existenz von konvexen Lösungen einer verwandten freien Randwertaufgabe bewiesen.
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Acker, A., Payne, L.E. & Philippin, G. On the convexity of level lines of the fundamental mode in the clamped membrane problem, and the existence of convex solutions in a related free boundary problem. Z. angew. Math. Phys. 32, 683–694 (1981). https://doi.org/10.1007/BF00946979
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DOI: https://doi.org/10.1007/BF00946979