On theory and numerical methods for finding multiple critical points with applications to semilinear PDE /
Almost all minimax theorems in critical point theory focus only on the existence issue. They require a two-level global optimization and therefore it is almost impossible to implement them as numerical methods. In this thesis, motivated by the Mountain Pass Lemma and its variants, we propose a new m...
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| Format: | Thesis Book |
| Language: | English |
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[Place of publication not identified] :
[publisher not identified] ;
1999.
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| Subjects: | |
| Online Access: | http://proxy.library.tamu.edu/login?url=http://proquest.umi.com/pqdweb?did=733668201&sid=1&Fmt=2&clientId=2945&RQT=309&VName=PQD |
| Summary: | Almost all minimax theorems in critical point theory focus only on the existence issue. They require a two-level global optimization and therefore it is almost impossible to implement them as numerical methods. In this thesis, motivated by the Mountain Pass Lemma and its variants, we propose a new minimax approach for finding multiple critical points. The approach is particularly designed to be easily implemented into a numerical minimax method for finding multiple critical points with high energy, or high Morse index. Based on the new approach, critical points are characterized as local minimax points, which are solutions to a two-level local optimization of minimax type. Mathematical justification of our approach and existence theorems are established. The minimax method is devised as a minimax algorithm with a step-size rule. Based on the step-size rule, some convergence results of the algorithm are proved. Results on estimate of the Morse index of a solution based on the minimax method are also established. The new minimax approach is then applied to solve a class of semi-linear elliptic partial differential equations(PDE) with Dirichlet boundary condition for multiple solutions. Numerical experiments for semi-linear partial differential equations on various domains have confirmed our theory and shown that the new minimax algorithm is stable in finding those unstable solutions. Numerical solutions are illustrated by their graphics for visualization. At the end of this thesis, other potential applications are discussed. |
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| Item Description: | Vita. "Major Subject: Mathematics". |
| Physical Description: | vi, 88 leaves : illustrations ; 28 cm. Issued also on microfiche from University Microfilm Inc. |
| Bibliography: | Includes bibliographical references (leaves 84-87). |