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Hardt, Robert M. Meng, Zheng 2009-06-04T08:21:29Z 2009-06-04T08:21:29Z 2005 Meng, Zheng. "Geometric variational problems with cross-sectional constraints." (2005) Diss., Rice University. <a href="https://hdl.handle.net/1911/18846">https://hdl.handle.net/1911/18846</a>. https://hdl.handle.net/1911/18846 THESIS MATH. 2005 MENG We study here some geometric variational problems motivated by the modeling of plants' growth. In Chapter 2, we conclude the general existence of an area minimizing surface with given boundary on two parallel hyperplanes and the constraint that the intersection of the surface with each hyperplane parallel to these of the boundary encloses the same area. In Chapter 3, we study the area minimizing surface in R 3 whose intersection with each of the hyperplanes R 2 x {h}, h ∈ [0, 1] encloses a prescribed area. We conclude that, up to a translation, the minimizer exists and is invariant under revolution. In Chapter 4, as a specific case of the problem in Chapter 2, the minimizing surface bounded by two parallel circles of the same size is studied carefully. We conclude that such an area minimizing surface is the skewed cylinder determined by the two circles. In Chapter 5, we study an analogous energy minimizing problem in PDE with a boundary constraint and a cross-sectional constraint on the L1 norm over a rectangular region. The even terms and the conditions for the odd terms in the Fourier expansion of the energy minimizer are given. 57 p. application/pdf eng Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder. Mathematics Geometric variational problems with cross-sectional constraints Thesis Text Mathematics Natural Sciences Rice University Doctoral Doctor of Philosophy open.access