There is a strong connection between the implicit function theorem and the theory of differential equations. This is true even from the historical point of view, for Pi-card’s iterative proof of the existence theorem for ordinary differential equations inspired Goursat to give an iterative proof of the implicit function theorem (see Goursat [Go 03]). In the mid-twentieth century, John Nash pioneered the use of a sophisticated form of the implicit function theorem in the study of partial differential equations. We will discuss Nash’s work in Section 6.4. In this section, we limit our attention to ordinary (rather than partial) differential equations because the technical details are then so much simpler. Our plan is first to show how a theorem on the existence of solutions to ordinary differential equations can be used to prove the implicit function theorem. Then we will go the other way by using a form of the implicit function theorem to prove an existence theorem for differential equations.
KeywordsOrdinary Differential Equation Distance Function Implicit Function Theorem Directional Derivative Signed Distance Function
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- 1.This fundamental theorem is commonly known as Picard’s existence and uniqueness theorem. The classical proof uses a method that has come to be known as the Picard iteration technique. See [Pi 93].Google Scholar
- 2.Recall from Section 3.3 that we use the notation ‹, › to denote the application of the Jacobian matrix to a vector.Google Scholar
- 3.In fact, this is a nontrivial hypothesis. A surface with this property is called a “set of positive reach.” It is known, and we will prove below using the implicit function theorem, that a C2 surface is a set of positive reach.Google Scholar
- 4.On page 50 of Hdrmander [Ho 66] it is observed in passing that the implicit function theorem can be used to show the distance function is as smooth as the boundary.Google Scholar
- 5.In the second edition, Gilbarg and Trudinger [GT 83], the result appears as Lemma 14.16, page 355.Google Scholar
- 6.Recall from Section 3.3 that we use the notation ‹, › to denote the application of the Jacobian matrix to a vector.Google Scholar