# 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.

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.

## February 11, 2011

### The implicit function theorem: An ODE example

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 23:35

We want to continue the series of notes involving some applications of  the implicit function theorem. As in the previous note, here we consider the solvability of the following ODE

$x''+\mu x+f(x)=0, \quad J= [0,1]$

with the following boundary conditions

$x(0)=0=x(1)$.

We assume that $f \in C^1(\mathbb R)$ and $f(0)=0$. For the sake of convenience, let us recall the standard implicit function theorem

Theorem (implicit function theorem). Let $X, Y, Z$ be Banachspaces, $U\subset X$ and $V\subset Y$ neighbourhoods of $x_0$ and $y_0$ respectively, $F: U\times V\to Z$ continuous and continuously differentiable with respect to $y$. Suppose also that

$F(x_o, y_o) = 0,\quad F_y^{-1}(x_o,y_o) \in L(Z, Y).$

Then  there exist balls $\overline B_r(x_o) \subset U$, $\overline B_r (y_o) \subset V$ and exactly one map $T: B_r(x_o) \to B_r (y_o)$ such that

$Tx_o = y_o$ and $F(x, Tx) = 0$ on $B_r(x_o)$.

This map $T$ is continuous.

## July 27, 2010

### The implicit function theorem: How to prove a continuously dependence on parameters for solutions of ODEs

Filed under: Giải Tích 3, PDEs — Tags: — Ngô Quốc Anh @ 0:25

It is clear that the implicit function theorem plays an important role in analysis. From now on, I am going to demonstrate this significant matter from the theory of differential equations, both ODE and PDE, point of view.

$-u''-\alpha^2 u^{-q-1}+\beta^2u^{q-1}=0$

on some domain $\Omega \subset \mathbb R^n$ with $\alpha \not\equiv 0$ and $\beta \not\equiv 0$. We assume the existence result on $W_+^{2,p}$ is proved for some $p>1$. We prove the following

Theorem. The solution $u \in W_+^{2,p}$ depends continuously on $(\alpha, \beta) \in L^\infty \times L^\infty$.

Proof. Consider the map

$\mathcal N : W_+^{2,p} \times (L^\infty \times L^\infty) \to L^p$

taking

$(u,\alpha,\beta) \mapsto -u''-\alpha^2 u^{-q-1}+\beta^2u^{q-1}$.

This map is evidently continuous (since $W_+^{2,p}$ is an algebra). One readily shows that its Fréchet derivative at $(u, \alpha, \beta)$ with respect to $u$ in the direction $h$ is

$\mathcal N'[u,\alpha ,\beta ]h = - h'' + \left[ {(q + 1){\alpha ^2}{u^{ - q - 2}} + (q - 1){\beta ^2}{u^{q - 2}}} \right]h$.

The continuity of the map

$(u,\alpha,\beta) \mapsto \mathcal N'[u,\alpha ,\beta ]$

follows from the fact that $W_+^{2,p}$ is an algebra continuously embedded in $C^0(\Omega)$.

Since $\alpha \not\equiv 0$ and $\beta \not\equiv 0$, the potential

$V={(q + 1){\alpha ^2}{u^{ - q - 2}} + (q - 1){\beta ^2}{u^{q - 2}}}$

is not identically zero. Thus it is well-known that the map

$-\Delta +V : W^{2,p} \to L^p$

is an isomorphism.

The implicit function theorem then implies that if $u_0$ is a solution for data $(\alpha_0, \beta_0)$, there is a continuous map defined near $(\alpha_0, \beta_0)$ taking $(\alpha, \beta)$ to the corresponding solution of the ODE. This establishes the conclusion.

For the more details, we prefer the reader to this preprint.