Abstract
In this paper a rearrangement minimization problem corresponding to solutions of thep-Laplacian equation is considered. The solution of the minimization problem determines the optimal way of exerting external forces on a membrane in order to have a minimum displacement. Geometrical and topological properties of the optimizer is derived and the analytical solution of the problem is obtained for circular and annular membranes. Then, we find nearly optimal solutions which are shown to be good approximations to the minimizer for specific ranges of the parameter values in the optimization problem. A robust and efficient numerical algorithm is developed based upon rearrangement techniques to derive the solution of the minimization problem for domains with different geometries in ℝ2and ℝ3.
Highlights
Finding an analytical solution for even geometrically simple domains such as rectangles and ellipses remains as an open problem
This is the reason that we turned our attention to determine nearly optimal solutions which are approximations to the minimizer for specific ranges of the parameter values in the optimization problem
We studied the nearly optimal solution in the low contrast regime by using rearrangement techniques
Summary
Rearrangement (shape) optimization problems arise in many different fields of applications such as fluid and structural mechanics, population biology, photonic crystals and nano structures, see [1, 4, 5, 9,10,11,12, 14, 24,25,26, 30, 31, 36, 40, 41, 43, 44], to name just a few In such problems, a functional corresponding to solutions of a given differential equation should be optimized over a rearrangement class of functions. Optimization problem (3) corresponding to nonlinear differential equation (1) has been studied by several authors, see [18, 19, 38] They have addressed existence and uniqueness of solutions of (3) using continuity, differentiability and convexity of the functional F. We will develop a robust and efficient numerical algorithm which is capable to determine the solution of (3) for different domains in R2 and R3
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