12.1 Notations

Definition 12.1.1 (Multi-Index).label For $\al=(\al_{1},...,\al_{n})$ with $\al_{i}\geq0$ integers then

• $\abs{\al}\define \sums{n }{i=1}\al_{i}$

• $\al!\define \al_{1}!\cds\al_{n}!$

• $x^{\al}\define x_{1}^{\al_1}\cds x_{n}^{\al_n}$

• $\pd^{\al}\define \prods{n}{i=1}\parens{\pder{}{x_i}}^{\al_i}=\frac{\pd^{\abs{\al}}}{\parens{\pd x_1}^{\al_1}\cds\parens{\pd x_n}^{\al_n}}$

We write $F(x,u,\pd_{1}u,...,\pd_{n}u,\pd_{1}^{2}u,...,\pd_{n}^{2}u,...,\pd_{n}^{k}u)=F(x,\parens{\pd^\al u}_{\abs{\al}\leq k})=0$ where $\pd_{i}=\frac{\pd}{\pd x_{i}}$ with $u=u(x_{1},...,x_{n})=y\in\R$ or $\com$.

Remark 12.1.2 (Examples).label

• $\la u=\sums{n }{i=1 }\pd_{i}^{2}u=b$ is the Poisson equation or Laplace equation for $b=0$ (Laplacian)

• $\square u=\pd_{1}^{2}u-\sums{n }{j=2}\pd_{j}^{2}u=0$ Wave equation (D’Alembertion)

• $\pd_{1}u=\sums{n }{j=2}\pd_{j}^{2}u=0$ Heat equation

We note that these equations behave very differently. Fix $n=2$ then

• $\la u=0,u\in C^{2}\implies u\in C^{\om}$. If $u$ is defined on $\R^{2}$ and $\SUP{x_1,x_2\in \R^2}\abs{u(x_1,x_2)}<\infty$ then $u$ is constant.

• $\square u=0,u=f(x_{1}+x_{2})+g(x_{1}-x_{2})$ viewing $x_{2}$ as a time variable and $u$ is discontinuous at some point $(x_{1}^{*},x_{2}^{*})$ then discontinuity propagates

• If initial condition $u(x_{1},0)=h(x_{1})$ is $C^{0}$ with $u$ solving heat equation then $u(x_{1},x_{2})\in C^{\infty}$

Definition 12.1.3 (Linear Function).label We say the function $F(x,\parens{\pd^\al u}_{\abs{\al}\leq k})$ is linear (in $u$) if it is equivalent to

For linear equations with coefficients $a_{\al}\in C^{\infty}(\R^{n})$ one can also talk about weak solutions. We write $\inner{f}{g}\define \integral{\R^n}{}f g dx$ if $f,g$ are integrable. $u$ is a weak solution if $\fall \p\in C^{\infty}_{c}(\R^{n})$ we have

Of course we observe that every solution is a weak solution since we can integrate only within the support of $\p$ which is compact so regularity of solution implies boundedness hence integrable.

Definition 12.1.4 (Quasi-Linear).label We say the function $F(x,\parens{\pd^\al u}_{\abs{\al}\leq k})$ is linear (in $u$) if it is equivalent to

i.e. $F$ is linear in $k$-th order derivatives. For example $\la u=\abs{\nabla u}^{2}$

Definition 12.1.5 (Characteristics).label For a linear differential operator $L=\sums{}{\abs{\al}\leq k}a_{\al}(x)\pd^{\al}$ we define the characteristic form of $L$ at $x$

where $\xi=(\xi_{1},...,\xi_{n})$ can be thought of as an element of the cotangent space. For example

The characteristics of $L$ is the zero locus of the characteristic form in the projective variety

We say that $L$ is elliptic at $x$ if $char_{x}(L)=\emp$. We observe that

• Suppose that at some $x\in\R^{n},\xi\in char_{x}(L)$ points in $x_{j}$ directions, i.e. $\xi_{i}= 0$ for $i\neq j$ then the coefficient of $\parens{\pd_j}^{k}$ at $x$ must vanish i.e. $L$ fail to be $k$-th order in the $\xi$ direction at $x$.

• One can show that $char_{x}(L)$ is defined invariantly, that is to say given a diffeomorphism $F:\OM\suf\R^{n}\to\OM'\suf\R^{n}$ with $x\mto y(x)$ then $\pd_{x}=J_{x}^{t}\pd_{y}=J^{t}_{F\inv(y)}\pd_{y}$ and $L$ gets transformed into $L'$ given by

  \begin{align*}L'=\sums{}{\abs{\al}\leq k}a_{\al}(F\inv(y))(J^{t}_{F\inv(y)}\pd_{y})^{\al}\end{align*}

  which has the property $char_{y}(L')$ is the image of $char_{x}(L)$. In particular, ellipticity is a coordinate-invariant property

Example 12.1.6 (Elliptic Operators).label

1. (1)

   $\la$ is elliptic everywhere in $\R^{n}$

2. (2)

   $\square$ is nowhere elliptic in $\R^{n}$

3. (3)

   $L=x_{1}\pd_{1}^{2}+\pd_{2}^{2}$ then $\chi_{L}(x,\xi)=x_{1}\xi_{1}^{2}+\xi_{2}^{2}$ so $L$ is elliptic if and only if $x_{1}>0$