10.3. Functionals, extremums and variations (multidimensional)

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10.3. Functionals, extremums and variations (multidimensional)

  1. Functionals: definitions
  2. Variations of functionals
  3. Stationary points of functionals
  4. Extremums of functionals

Now we consider functionals, defined on functions of several variables.

Variations of functionals

Let us consider functional \begin{equation} \Phi[u]= \iiint_\Omega L(x, u,\nabla u)\,dx \label{eq-10.3.4} \end{equation} where $\Omega$ is $n$-dimensional domain and $L$ is some function of $n+2$ variables. Let us consider $u+\delta u$ where $\delta u $ is a small function. We do not formalize this notion, just $\varepsilon \phi$ with fixed $\phi$ and $\varepsilon\to 0$ is considered to be small. We call $\delta u$ variation of $u$ and important is that we change a function as a whole object. Let us consider \begin{multline} \Phi[u+\delta u]-\Phi[u]= \iiint_\Omega \Bigl(L(x,u+\delta u,\nabla u +\nabla \delta u)-L(x, u,\nabla u) \Bigr)\,dx\\ \approx \iiint_\Omega \Bigl(\frac{\partial L}{\partial u}\delta u +\sum_{1\le j\le n} \frac{\partial L}{\partial u_{x_j}}\delta u_{x_j} \Bigr)\,dx\qquad \label{eq-10.3.5} \end{multline} where we calculated the linear part of expression in the parenthesis; if $\delta u=\varepsilon \phi$ and all functions are sufficiently smooth then $\approx$ would mean equal modulo $o(\varepsilon)$ as $\varepsilon\to 0$.

Definition 1.

  1. Function $L$ we call Lagrangian.
  2. The right-hand expression of (\ref{eq-10.3.5}) which is a linear functional with respect to $\delta u$ we call variation of functional $\Phi$ and denote by $\delta \Phi$.

Assumption 1. All functions are sufficiently smooth.

Under this assumption, we can integrate the right-hand expression of (\ref{eq-10.3.5}) by parts: \begin{multline} \delta \Phi:= \iiint_\Omega \Bigl(\frac{\partial L}{\partial u}\delta u +\sum_{1\le j\le n} \frac{\partial L}{\partial u_{x_j}}\delta u_{x_j} \Bigr)\,dx\\ = \iiint_\Omega\Bigl(\frac{\partial L}{\partial u} - \sum_{1\le j\le n} \frac{\partial\ }{\partial x_j} \frac{\partial L}{\partial u_{x_j}} u\Bigr)\delta u \,dx - \iint_{\partial \Omega} \Bigl(\sum_{1\le j\le n} \frac{\partial L}{\partial u_{x_j}}\nu_j \Bigr)\delta u \,d\sigma\qquad \label{eq-10.3.6} \end{multline} where $d\sigma$ is an area element and $\nu$ is a unit interior normal to $\partial \Omega$.

Stationary points of functionals

Definition 2. If $\delta \Phi=0$ for all admissible variations $\delta u$ we call $u$ a stationary point or extremal of functional $\Phi$.

Remark 1.

  1. We consider $u$ as a point in the functional space.
  2. In this definition we did not specify which variations are admissible. Let us consider as admissible all variations which are $0$ at the boundary: \begin{equation} \delta u |_{\partial\Omega}=0. \label{eq-10.3.7} \end{equation} We will consider different admissible variations later.

In this framework \begin{equation} \delta \Phi= \iiint_\Omega\Bigl(\frac{\partial L}{\partial u} - \sum_{1\le j\le n} \frac{\partial\ }{\partial x_j} \frac{\partial L}{\partial u_{x_j}} \Bigr)\delta u \,dx . \label{eq-10.3.8} \end{equation}

Lemma 1. Let $f$ be a continuos function in $\Omega$. If $\iiint_\Omega f(x)\phi(x)\,dx=0$ for all $\phi$ such that $\phi|_{\partial \Omega}=0$ then $f=0$ in $\Omega$.

Proof. Indeed, let us assume that $f(\bar{x})> 0$ at some point $\bar{x}\in \Omega$ (case $f(\bar{x})< 0$ is analyzed in the same way). Then $f(x)>0$ in some vicinity $\mathcal{V}$ of $\bar{x}$. Consider function $\phi(x)$ which is $0$ outside of $\mathcal{V}$, $\phi\ge 0$ in $\mathcal{V}$ and $\phi(\bar{x})>0$. Then $f(x)\phi(x)$ has the same properties and $\iiint_{\Omega} f(x)\phi(x)\, dx>0$. Contradiction!

As a corollary we arrive to

Theorem 1. Let us consider a functional (\ref{eq-10.3.4}) and consider as admissible all $\delta u$ satisfying (\ref{eq-10.3.7}). Then $u$ is a stationary point of $\Phi$ if and only if it satisfies Euler-Lagrange equation \begin{equation} \frac{\delta \Phi}{\delta u}:= \frac{\partial L}{\partial u} - \sum_{1\le j\le n} \frac{\partial\ }{\partial x_j} \left(\frac{\partial L}{\partial u_{x_j}}\right) =0. \label{eq-10.3.9} \end{equation}

Remark 2.

  1. This is a second order PDE.
  2. We do not have statements similar to Remark 10.1.4 (b)-(d), see also Remark 10.2.2

Extremums of functionals

Definition 3. If $\Phi[u]\ge \Phi[u+\delta u]$ for all small admissible variations $\delta u$ we call $u$ a local maximum of functional $\Phi$. If $\Phi[u]\le \Phi[u+\delta u]$ for all small admissible variations $\delta u$ we call $u$ a local minimum of functional $\Phi$.

Here again we do not specify what is small admissible variation.

Theorem 2. If $u$ is a local extremum (that means either local minimum or maximum) of $\Phi$ and variation exits, then $u$ is a stationary point.

Proof. Consider case of minimum. Let $\delta u =\varepsilon \phi$. Then $\Phi [u+\delta u]- \Phi [u]=\varepsilon (\delta \Phi)(\phi) +o(\varepsilon)$. If $\pm \delta \Phi> 0$ then choosing $\mp \varepsilon <0$ we make $\varepsilon (\delta \Phi)(\phi)\le -2\epsilon\_0 \varepsilon$ with some $\epsilon\_0>0$. Meanwhile for sufficiently small $\varepsilon$ $o(\varepsilon)$ is much smaller and $\Phi [u+\delta u]- \Phi [u]\le -2\epsilon_0 \varepsilon<0$ and $u$ is not a local minimum.

Remark 3. We consider neither sufficient conditions of extremums nor second variations (similar to second differentials). In some cases they will be obvious.

Example 2.

  1. Consider a surface $\Sigma = \{(x,y,z):\, (x,y)\in \Omega , z=u(x,y)\}$ which has $(x,y)$-projection $\Omega$. Then the surface area of $\Sigma$ is \begin{equation} A(\Sigma)= \iint_{\Omega} \bigl(1+u_x^2+u_y^2\bigr)^{\frac{1}{2}}\,dxdy. \label{eq-10.3.10} \end{equation} We are interested in such surface of minimal area (aka minimal surface) under restriction $u=g$ at points $\partial \Omega$. It is a famous minimal surface problem (under the assumption that it projects nicely on $(x,y)$-plane (which is not necessarily the case). One can formulate it: find the shape of the soap film on the wire.

Then Euler-Lagrange equation is \begin{equation} -\frac{\partial\ }{\partial x} \Bigl(u_x\bigl(1+u_x^2+u_y^2\bigr)^{-\frac{1}{2}}\Bigr)- \frac{\partial\ }{\partial y} \Bigl(u_y\bigl(1+u_x^2+u_y^2\bigr)^{-\frac{1}{2}}\Bigr)=0. \label{eq-10.3.11} \end{equation} 2. Assuming that $u_x, u_y \ll 1$ one can approximate $A(\Sigma)-A(\Omega)$ by \begin{equation} \frac{1}{2}\iint_{\Omega} \bigl(u_x^2+u_y^2\bigr)\,dxdy \label{eq-10.3.12} \end{equation} and for this functional Euler-Lagrange equation is \begin{equation} -\Delta u=0. \label{eq-10.3.13} \end{equation} 3. Both (1) and (2) could be generalized to higher dimensions.

Remark 4. Both equations (\ref{eq-10.3.11}) and (\ref{eq-10.3.12}) come with the boundary condition $u|_{\partial\Omega}=g$. In the next section we analyse the case when such condition is done in the original variational problem only on the part of the boundary.

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