The inverse function theorem of Nash and Moser

The Continuous Newton's Method, Inverse Functions, and Nash-Moser

The conventional Newton’s method for finding a zero of a function F : R → R, assuming that (F ′(y))−1 exists for at least some y in R, is the familiar iteration: pick z0 in R n and define zk+1 = zk − (F ′(zk))−1F (zk) (k = 0, 1, 2, . . . ), hoping that z1, z2, . . . converges to a zero of F . What can stop this process from finding a zero of F ? For one thing, there might not be a zero of F . F...

A Nash-moser Theorem with Near-minimal Hypothesis

A proof of a Nash-Moser type inverse function theorem is given under substantially weaker hypothesis than previously known. Our method is associated with continuous Newton’s method rather than the more conventional Newton’s method.

On the Inverse Function Theorem

The De Giorgi-Nash-Moser Estimates

with a positive constant λ. The equation Lu = 0 is then a second-degree elliptic equation. We also require the aij to be bounded and measurable, satisfying ‖aij‖L∞(B4) ≤ Λ with another constant Λ > 0. (In case you wonder, the radius ’4’ of the ball is to avoid fractions. Most of our estimates will be of the kind “some expression on B1 ≤ another expression on B4 ” and it would be unconvenient to...

Nash–Moser iteration and singular perturbations

We present a simple and easy-to-use Nash–Moser iteration theorem tailored for singular perturbation problems admitting a formal asymptotic expansion or other family of approximate solutions depending on a parameter ε → 0. The novel feature is to allow loss of powers of ε as well as the usual loss of derivatives in the solution operator for the associated linearized problem. We indicate the util...