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An Introduction to Partial Differential Equations by Jacob Rubinstein, Yehuda Pinchover

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10

Variational methods

The PDEs we have considered so far were derived by modeling a variety of phenomena in physics, engineering, etc. In this chapter we shall derive PDEs from a new perspective. We shall show that many PDEs are related to optimization problems. The theory that associates optimization with PDEs is called the calculus of variations [20]. It is an extremely useful theory. On the one hand, we shall be able to solve many optimization problems by solving the corresponding PDEs. On the other hand, sometimes it is simpler to study (and solve) certain optimization problems than to study (and solve) the related PDE. In such cases, the calculus of variations is an indispensable theoretical and practical tool in the study of PDEs. The ...

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