I am reading about the Variational Method to solve Elliptic PDEs but I have been unable to find a precise definition of the concept Elliptic PDE.

Of course, I am aware of the definition given in general PDE courses for starters, and the one given in [The Wikipedia page for Elliptic PDEs][1], which only considers second order PDEs defined on $\Omega \subset \mathbb{R}^2$.

Haim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations gives what he calls the ellipticity condition:

For an open set $\Omega \subset \mathbb{R}^n$ we are given the functions $a_{ij} \in C^1(\bar{\Omega})$ satisfying the ellipticity condition: $$\sum_{i,j}^n a_{ij}(x) \xi_i \xi_j \geq \alpha |\xi|^2 \quad \forall x \in \Omega, \forall \xi \in \mathbb{R}^n \ with \ \alpha >0 $$ and a function $a_0 \in C(\bar{\Omega})$, we look for a function $u: \bar{\Omega} \to \mathbb{R}$ satisfying: $$-\sum_{i,j=1}^{n}\partial_{x_j}(a_{ij}\partial_{x_i}u)+a_0u=f $$

But this is not a "general" second order PDE. I understand a general second order n-dimensional PDE as:

$\sum_{i=1}^n \sum_{j=1}^n a_{ij}\partial_{x_ix_j}u + \sum_{i=1}^n a_i \partial_{x_i}u + a_0u=f$

So the question is: What is the general definition for an elliptic n-dimensional Partial Differential Equation?


In general, if $\Omega \subset \mathbb{R}^n$ is bounded open, an elliptic operators in divergence form and the second order is of the type

$\displaystyle Lu:= - \sum_{i,j=1}^n (a_{ij} u_{x_i})_{x_j} + \sum_{i=1}^n (b_i u)_{x_i} +c u$

where $a_{ij}, b_i, c \in L^\infty(\Omega)$, and it assumes that $L$ is uniformly elliptic, i.e. $\sum_{i,j=1}^n a_{ij} \xi_i \xi_j \geq \theta |\xi|^2$ $\forall x \in \Omega$ and $\forall \xi \in \mathbb{R}^n$. Now considers the problem:

($\star$) $Lu=f$ in $\Omega$ and $u=0$ in $\partial \Omega$

where $f \in L^2(\Omega)$. The weak formulation is introduced by considering test functions, and assuming temporarily $u \in C^2(\Omega) \cap C^1(\overline{\Omega})$. In such cases, it is not difficult to verify that $\forall v \in C_{c}^\infty(\Omega)$, we have the $L^2$ scalar product

$\displaystyle \int_{\Omega}Lu v dx = \int_{\Omega} f v dx$

and assumptions made, with integration by part, we have

$\displaystyle B(u,v):=\int_{\Omega} \left( \sum_{ij=1}^n a_{ij} u_{x_i} v_{x_j} - \sum_{i=1}^n b_i u v_{x_i} + cuv \right) dx = \int_{\Omega} f v dx$

since $v \in C^{\infty}_c(\Omega)$. Now, since $H_{0}^1(\Omega):=\overline {C_{c}^\infty (\Omega)}^{|| \cdot ||_{H^1}}$, the previous identity is true also $\forall v \in H_{0}^1(\Omega)$ by approssimation. All this allows us to justify the weak solution of the problem ($\star$). Then $u \in H_{0}^1(\Omega)$ is a weak solution of the ($\star$) problem if

$\displaystyle B(u,v)=(f,v)_{L^2}$ $\forall v \in H_{0}^1(\Omega)$

For example, if $b_i=c=0$, condition of uniform ellipticity allows us to try that $B(u,v)$ is a continuous coercive bilinear form, and Lax-Milgram theorem applies, proving the uniqueness for the problem ($\star$). In the case where $b_i, c \neq 0$ essentially using Fredholm alternative theorem's. In the end, if $b_i=c=0$ and $a_{ij}=\delta_{ij}$, $(\star)$ is a Dirichlet problem for the Poisson equations, and for the uniqueness you can also use the Riesz representation theorem.

Similarly, you can also define a weak formulation for elliptic operators of the second order does not in divergence form. For example, let

$\displaystyle Lu:= - \sum_{i,j=1}^n a^{ij} u_{x_i x_j} + \sum_{i=1}^n b_i u_{x_i} + cu$

can be rewritten as

$\displaystyle Lu=-\sum_{i,j=1}^n (a^{ij} u_{x_i})_{x_j} + \sum_{i=1}^n (b^i + \sum_{j=1}^n a^{ij}_{x_j})u_{x_i} + c u$

assuming $a^{ij}, a^{ij}_{x_j}, b^i, c \in L^{\infty}(\Omega)$, a weak solution of the corresponding problem ($\star$) can again be obteined by solving $B(u,v)=(f,v)_{L^2}$ where in this case

$\displaystyle B(u,v)=\int_{\Omega} \left( \sum_{i,j=1}^n a^{ij} u_{x_i})v_{x_j} + \sum_{i=1}^n (b^i + \sum_{j=1}^n a^{ij}_{x_j})u_{x_i} v + c u v \right) dx$

  • $\begingroup$ Thanks for the answer. Could you provide me some reference in which I can find the information given? $\endgroup$ – D1X May 6 '16 at 20:39
  • 1
    $\begingroup$ Sure, "Lecture Notes on Functional Analysis: With Applications to Linear Partial Differential Equations" by A. Bressan, but also "PDE" by Evans. $\endgroup$ – Andrew May 6 '16 at 20:42

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