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Convexity Properties of Solutions to Elliptic P.D.E.’S

In: Variational Methods for Free Surface Interfaces

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  • Nicholas J. Korevaar

Abstract

How do the data of an elliptic boundary value problem (domain, boundary values, elliptic operator) affect the shape of the solution v? Estimates for v, Dv, or even D 2 v may be necessary to prove existence and regularity theorems, and they often also characterize fundamental geometric behavior of v. In this note we shall study some particular estimates involving v, Dv, D 2 v: ones that are related to convexity properties of v. The results do not usually lead to existence theorems (with some exceptions, e.g. [3]), but are surprising and have independent beauty.

Suggested Citation

  • Nicholas J. Korevaar, 1987. "Convexity Properties of Solutions to Elliptic P.D.E.’S," Springer Books, in: Paul Concus & Robert Finn (ed.), Variational Methods for Free Surface Interfaces, pages 115-121, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-4656-5_13
    DOI: 10.1007/978-1-4612-4656-5_13
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