Boundary value problems for variational integrals involving surface curvatures

Nitsche, Johannes C. C.

doi:10.1090/qam/1218374

Author: Johannes C. C. Nitsche
Journal: Quart. Appl. Math. 51 (1993), 363-387
MSC: Primary 58E12; Secondary 35J55, 53A10, 76B45
DOI: https://doi.org/10.1090/qam/1218374
MathSciNet review: MR1218374
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Abstract: The following investigation deals with surfaces governed by and extremal for a free energy functional which is quadratic in the principal curvatures. The associated Euler-Lagrange differential equations are derived, as are the corresponding intricate natural boundary conditions. Pertinent boundary value problems—without and with volume constraints—are formulated and discussed$^{1}$ and existence proofs are provided for certain situations. The discussion opens the view onto an arena of rich mathematical problems which will also be of interest in engineering applications where the surfaces in question are utilized frequently as idealized models for the interfaces separating phases in real materials.

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