Ngô Quốc Anh

March 1, 2011

The implicit function theorem: A PDE example

Filed under: Giải Tích 3, PDEs — Tags: — Ngô Quốc Anh @ 23:29

This entry devotes an existence result for the following semilinear elliptic equation

-\Delta u + u = u^p+f(x)

in the whole space \mathbb R^n where 0<u \in H^1(\mathbb R^n).

Our aim is to apply the implicit function theorem. It is known in the literature that

Theorem (implicit function theorem). Let X, Y, Z be Banach spaces. Let the mapping f:X\times Y\to Z be continuously Fréchet differentiable.

If

(x_0,y_0)\in X\times Y, \quad F(x_0,y_0) = 0,

and

y\mapsto DF(x_0,y_0)(0,y)

is a Banach space isomorphism from Y onto Z, then there exist neighborhoods U of x_0 and V of y_0 and a Frechet differentiable function g:U\to V such that

F(x,g(x)) = 0

and F(x,y) = 0 if and only if y = g(x), for all (x,y)\in U\times V.

Let us now consider

X=L^2(\mathbb R^n), \quad Y=H_+^2(\mathbb R^n), \quad Z=L^2(\mathbb R^n).

Let us define

F(f,u)=-\Delta u + u - u^p-f(x), \quad f \in X, \quad u \in Y, \quad x \in \mathbb R^n.

It is not hard to see that Fréchet derivative of F at (f,u) with respect to u in the direction v is given by

{D_u}F(f,u)v = - \Delta v + v - p{u^{p - 1}}v.

Since -\Delta +I defines an isomorphism from Y to Z, it is clear to see that our PDE is solvable for f small enough in the X-norm.

6 Comments »

  1. Hi,
    Why, in the end, f should enough small?

    Comment by R.T — February 5, 2014 @ 13:30

    • Hi there, the IFT basically says that as long as F(x_0,y_0)=0, there exists a mapping g between neighborhoods of x_0 and y_0 such that F\big( x,g(x) \big)=0. In the example above, the point (x_0, y_0) is nothing but (0, u_0) where 0 \in X is a function and u_0 is a solution to -\Delta u+u=u^p. As such, by a small neighborhood of 0 we mean functions f such that \|f\|_X \ll 1. I hope this is clear for you now.

      Comment by Ngô Quốc Anh — February 5, 2014 @ 22:51

  2. […] Please see this page: The implicit function theorem: A PDE example. […]

    Pingback by Implicit function theorem and PDE; do we get uniqueness? - MathHub — April 4, 2016 @ 15:24

  3. Hi, I have a couple of questions.
    1. For what values of p can this method be applied to your example?
    2. Is there anything special about using R^n as the domain. That is, can we still establish the existence for any subset of R^n?

    Comment by Melusi — August 10, 2016 @ 17:43

  4. Hi, what if f is large in the X-norm. How can we say that there is a solution? iteration?

    Comment by boytaehun@naver.com — October 20, 2016 @ 13:26

    • Thanks for your comment, it is likely that such a result cannot hold in view of a perturbation result.

      Comment by Ngô Quốc Anh — October 21, 2016 @ 10:12


RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Create a free website or blog at WordPress.com.

%d bloggers like this: