An application of Brouwer fixed point Theorem

calculuscalculus-of-variationseuclidean-domainfixed-point-theoremsreal-analysis

  1. Let $\zeta:\mathbb{R}\to [0,+\infty)$ be a continuous non-negative function such that $\zeta(0)=0$ and $\tau\mapsto \zeta(\tau)\tau$ is a non-decreasing differentiable function whose derivative is bounded on every compact subset of $\mathbb{R}$.

  2. Let $\{\phi_{k}, \lambda_{k}\}_{k \in \Bbb N}$ be the the Dirichlet eigenpairs of the n Laplace operator on an open bounded set $\Omega\subset \Bbb R^N$, i.e., $\phi_k\in H_0^1(\Omega)$ and $-\Delta \phi_k= \lambda_k\phi_k$. Recall $\{\phi_k\}_{k}$ is an orthonormal basis in $L^2(\Omega)$.

  3. Question: Let $\mathcal{V}_{k}= \operatorname{span}\{\phi_1,\dotsc, \phi_{k}\}$. According to page 5 Eq (3.3) of Starovoitov – Boundary value problem for a global in time parabolic equation, the Brouwer fixed-point theorem implies the existence of a vector $v_k\in \mathcal{V}_k$ such that
    \begin{equation}\label{Star-3.3}
    \int_\Omega \nabla v_{k}\cdot \nabla \phi dx + \int_\Omega\zeta(v_k)v_k\phi dx=\int_\Omega f\phi dx\quad\text{for all}\quad\phi\in \mathcal{V}_{k}.
    \end{equation}

How can one justify this claim?

In fact, that $\phi_k\in L^\infty (\Omega)$ is the only important property needed from $\phi_k$. So that by assumption the function $\phi(v_k)v_k$ is bounded.
Since we are in finite-dimensional space and $\int_\Omega \nabla \phi_{i}\cdot \nabla \phi_j dx=\lambda_i\delta_{ij}$, the above equation reduces into finding $v_k=(v_{k,1}\phi_1+ \dotsb+v_{k,k}\phi_k)$ satisfying
\begin{equation}\label{Star-3.v}
\sum_{i=1}^k v_{k,i}\lambda_i + \zeta(v_k)v_k = f_k\quad\text{in} \quad \mathcal{V}_{k},
\end{equation}
where $f_k=(f_{k,1}\phi_1+ \dotsb+f_{k,k}\phi_k)$ is the projection of $f$ on $\mathcal V_k$ and by abuse of notation, we write again $\zeta(v_k)v_k$ to represent its own projection on $\mathcal V_k$.

Recall Brouwer fixed-point theorem: Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.

Best Answer

First using Brouwer

Let $w\in \mathcal{V}_k$, necessarily $\zeta(w)$ is a bounded function since $\phi_k $'s are also bounded. The Lax-Milgram lemma implies there is a unique function $\widehat{w}\in \mathcal{V}_k$ such that \begin{align} \int_\Omega\nabla\widehat{w}\nabla\phi +\zeta(w)\widehat{w}\phi -f\psi dx=0\quad\text{for all}\quad\phi\in \mathcal{V}_{k}. \end{align}

The Poincar'{e}--Friedrichs inequality yields \begin{equation} \int_\Omega|\nabla\widehat{w}|^2dx + \int_\Omega \zeta(w )\widehat{w}^2\, d x\leq \|f\|_{L^{2}(\Omega)}\,\|\widehat{w}\|_{L^{2}(\Omega)}\leq C\|f\|_{L^{2}(\Omega)}\, \|\widehat{w}\|_{H_0^1(\Omega )} \end{equation}

Thus, letting $R=C\,\|f\|_{L^{2}(\Omega)}$, since $\varphi\geq0 $ we obtain the following estimates \begin{align}\label{eq:boundedmapT} &\|\widehat{w}\|_{H_0^1(\Omega )}\leq R \quad\text{ and }\quad \int_\Omega \zeta(w )\widehat{w}^2\,dx\leq R^2. \end{align} We let $ \mathcal{B}_R=\big\{ w\in \mathcal{V}_k: \|w\|_{H_0^1(\Omega)} \leq R\big\}$, be the closed ball in $\mathcal{V}_k$ of radius $R$ centered at the origin. Clearly, the mapping $T:\mathcal{V}_k\to \mathcal{B}_R$ with $Tw=\widehat{w}$ is well defined.

It remains to prove that $T$ is a continuous mapping. Indeed, let $\{w_n\}$ be a sequence in $\mathcal{V}_k$ with $w_n= \lambda_{1,n}\phi_1+\cdots+ \lambda_{k,n}\phi_k$ converging in $\mathcal{V}_k$ to a function $w= \lambda_1\phi_1+\cdots+ \lambda_k\phi_k$; i.e., $\lambda_{\ell,n}\xrightarrow{n\to\infty } \lambda_\ell$, $\ell=1,2,\cdots,k$. By continuity we have $ \varphi(w_n)\xrightarrow{n\to \infty}\varphi(w)$ almost everywhere. In addition, the convergence in $L^2(\Omega)$ also holds, since the continuity gives $\sup_{n\geq 0} \|\varphi(w_n) \|_{L^\infty(\Omega)} <\infty$ because $\sup_{n\geq 0}\|w_n\|_{L^\infty(\Omega) } <\infty$.

On the other side, in virtue of the first estimate above the sequence $\{Tw_n\}$ is bounded in finite dimensional space $\mathcal{V}_k$ and thus converges $\mathcal{V}_k$ up to a subsequence to some $w_*\in \mathcal{V}_k$. Altogether, it follows that, for all $\phi\in \mathcal{V}_k\subset L^\infty(\Omega)$ % \begin{align*} (f,\phi)= \lim_{n\to \infty} \int_\Omega\nabla\widehat{w}_n\nabla\phi +\zeta(w_n)\widehat{w}_n\phi dx= \int_\Omega\nabla\widehat{w}\nabla\phi +\zeta(w)w_* \phi. \end{align*}

The uniqueness of $\widehat{w}$ entails $w_*=\widehat{w}=Tw$ and hence the whole sequence $\{Tw_n\}$ converges in $Tw$ in $\mathcal V_k$, which gives the continuity of $T$.

Therefore, by the Brouwer fixed-point theorem, $T$ has a fixed point $v_k\in \mathcal V_k$, i.e., $v_k=Tv_k$ which clearly satisfies the announced relation.

An alternative.

The given problem is equivalent to the minimization problem \begin{align} \mathcal{J}(v_k)= \min_{v\in \mathcal{V}_k} \mathcal{J}(v)\quad \text{with}\quad \mathcal{J}(v):= \frac12 \int_\Omega|\nabla v|^2 dx + \int_\Omega G(v)d x + \int_\Omega fvd x \end{align} and we define the function $G(v)= \int_0^v\zeta(\tau)\tau d \tau $. Note that $G$ is non-negative since $\zeta(\tau)\geq 0$ and that $\mathcal J$ is continuous on $\mathcal V_k$. Using the Poincaré-Friedrichs inequality we find that $\mathcal{J}(v)\to \infty$, as $\|v\|_{L^2(\Omega)} \to \infty$ and $v\in \mathcal{V}_k$. Which implies the existence of a minimizer $v_k\in \mathcal{V}_k$ of \mathcal{J}, since we are in finite dimension space.

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