Elliptic boundary value problem

Sobolev spaces

The analysis of elliptic boundary value problems requires some fairly sophisticated tools of functional analysis. We require the space ${\displaystyle H^1(\mathrm{\Omega })}$, the Sobolev space of "once-differentiable" functions on ${\displaystyle \mathrm{\Omega }}$, such that both the function ${\displaystyle u}$ and its partial derivatives ${\displaystyle u_{x_i}}$, ${\displaystyle i=1,\dots ,n}$ are all square integrable. That is: $${\displaystyle H^1(\mathrm{\Omega })=\left\{u\in L^2(\mathrm{\Omega }),u_{x_j}\in L^2(\mathrm{\Omega }),1\le i\le n\right\}.}$$ There is a subtlety here in that the partial derivatives must be defined "in the weak sense" (see the article on Sobolev spaces for details.) The space ${\displaystyle H^1}$ is a Hilbert space, which accounts for much of the ease with which these problems are analyzed.

Unless otherwise noted, all derivatives in this article are to be interpreted in the weak, Sobolev sense. We use the term "strong derivative" to refer to the classical derivative of calculus. We also specify that the spaces ${\displaystyle C^k}$, ${\displaystyle k=0,1,\dots }$ consist of functions that are ${\displaystyle k}$ times strongly differentiable, and that the ${\displaystyle k}$th derivative is continuous.

Weak or variational formulation

The first step to cast the boundary value problem as in the language of Sobolev spaces is to rephrase it in its weak form. Consider the Laplace problem ${\displaystyle \mathrm{\Delta }u=f}$. Multiply each side of the equation by a "test function" ${\displaystyle \phi }$ and integrate by parts using Green's theorem to obtain

${\displaystyle -{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\cdot \mathrm{\nabla }\phi +{\int }_{\mathrm{\partial }\mathrm{\Omega }}{u}_{\nu }\phi ={\int }_{\mathrm{\Omega }}f\phi }$.

We will be solving the Dirichlet problem, so that ${\displaystyle u=0\text{ on }\mathrm{\partial }\mathrm{\Omega }}$. For technical reasons, it is useful to assume that ${\displaystyle \phi }$ is taken from the same space of functions as ${\displaystyle u}$ is so we also assume that ${\displaystyle \phi =0\text{ on }\mathrm{\partial }\mathrm{\Omega }}$. This gets rid of the ${\displaystyle {\int }_{\mathrm{\partial }\mathrm{\Omega }}}$ term, yielding

${\displaystyle A(u,\phi )=F(\phi )}$ (*)

where

${\displaystyle A(u,\phi )={\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\cdot \mathrm{\nabla }\phi }$ and

${\displaystyle F(\phi )=-{\int }_{\mathrm{\Omega }}f\phi }$.

If ${\displaystyle L}$ is a general elliptic operator, the same reasoning leads to the bilinear form

${\displaystyle A(u,\phi )={\int }_{\mathrm{\Omega }}\mathrm{\nabla }{u}^{T}a\mathrm{\nabla }\phi -{\int }_{\mathrm{\Omega }}{b}^T\mathrm{\nabla }u\phi -{\int }_{\mathrm{\Omega }}cu\phi }$.

We do not discuss the Neumann problem but note that it is analyzed in a similar way.

Continuous and coercive bilinear forms

The map ${\displaystyle A(u,\phi )}$ is defined on the Sobolev space ${\displaystyle H_0^1\subset H^1}$ of functions which are once differentiable and zero on the boundary ${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$, provided we impose some conditions on ${\displaystyle a,b,c}$ and ${\displaystyle \mathrm{\Omega }}$. There are many possible choices, but for the purpose of this article, we will assume that

1. ${\displaystyle a_{ij}(x)}$ is continuously differentiable on ${\displaystyle \overline{\mathrm{\Omega }}}$ for ${\displaystyle i,j=1,\dots ,n,}$
2. ${\displaystyle b_i(x)}$ is continuous on ${\displaystyle \overline{\mathrm{\Omega }}}$ for ${\displaystyle i=1,\dots ,n,}$
3. ${\displaystyle c(x)}$ is continuous on ${\displaystyle \overline{\mathrm{\Omega }}}$ and
4. ${\displaystyle \mathrm{\Omega }}$ is bounded.

The reader may verify that the map ${\displaystyle A(u,\phi )}$ is furthermore bilinear and continuous, and that the map ${\displaystyle F(\phi )}$ is linear in ${\displaystyle \phi }$, and continuous if (for instance) ${\displaystyle f}$ is square integrable.

We say that the map ${\displaystyle A}$ is coercive if there is an ${\displaystyle \alpha >0}$ for all ${\displaystyle u,\phi \in H_0^1(\mathrm{\Omega })}$,

${\displaystyle A(u,\phi )\ge \alpha {\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\cdot \mathrm{\nabla }\phi .}$

This is trivially true for the Laplacian (with ${\displaystyle \alpha =1}$) and is also true for an elliptic operator if we assume ${\displaystyle b=0}$ and ${\displaystyle c\le 0}$. (Recall that ${\displaystyle u^{T}au>\alpha u^{T}u}$ when ${\displaystyle L}$ is elliptic.)

Existence and uniqueness of the weak solution

One may show, via the Lax–Milgram lemma, that whenever ${\displaystyle A(u,\phi )}$ is coercive and ${\displaystyle F(\phi )}$ is continuous, then there exists a unique solution ${\displaystyle u\in H_0^1(\mathrm{\Omega })}$ to the weak problem (*).

If further ${\displaystyle A(u,\phi )}$ is symmetric (i.e., ${\displaystyle b=0}$), one can show the same result using the Riesz representation theorem instead.

This relies on the fact that ${\displaystyle A(u,\phi )}$ forms an inner product on ${\displaystyle H_0^1(\mathrm{\Omega })}$, which itself depends on Poincaré's inequality.

Strong solutions

We have shown that there is a ${\displaystyle u\in H_0^1(\mathrm{\Omega })}$ which solves the weak system, but we do not know if this ${\displaystyle u}$ solves the strong system

${\displaystyle Lu=f\text{ in }\mathrm{\Omega },}$

${\displaystyle u=0\text{ on }\mathrm{\partial }\mathrm{\Omega },}$

Even more vexing is that we are not even sure that ${\displaystyle u}$ is twice differentiable, rendering the expressions ${\displaystyle u_{x_i{x}_j}}$ in ${\displaystyle Lu}$ apparently meaningless. There are many ways to remedy the situation, the main one being regularity.

Regularity

A regularity theorem for a linear elliptic boundary value problem of the second order takes the form

Theorem If (some condition), then the solution ${\displaystyle u}$ is in ${\displaystyle H^2(\mathrm{\Omega })}$, the space of "twice differentiable" functions whose second derivatives are square integrable.

There is no known simple condition necessary and sufficient for the conclusion of the theorem to hold, but the following conditions are known to be sufficient:

1. The boundary of ${\displaystyle \mathrm{\Omega }}$ is ${\displaystyle C^2}$, or
2. ${\displaystyle \mathrm{\Omega }}$ is convex.

It may be tempting to infer that if ${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$ is piecewise ${\displaystyle C^2}$ then ${\displaystyle u}$ is indeed in ${\displaystyle H^2}$, but that is unfortunately false.

Almost everywhere solutions

In the case that ${\displaystyle u\in H^2(\mathrm{\Omega })}$ then the second derivatives of ${\displaystyle u}$ are defined almost everywhere, and in that case ${\displaystyle Lu=f}$ almost everywhere.

Strong solutions

One may further prove that if the boundary of ${\displaystyle \mathrm{\Omega }\subset {\mathbb{R}}^n}$ is a smooth manifold and ${\displaystyle f}$ is infinitely differentiable in the strong sense, then ${\displaystyle u}$ is also infinitely differentiable in the strong sense. In this case, ${\displaystyle Lu=f}$ with the strong definition of the derivative.

The proof of this relies upon an improved regularity theorem that says that if ${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$ is ${\displaystyle C^k}$ and ${\displaystyle f\in H^{k-2}(\mathrm{\Omega })}$, ${\displaystyle k\ge 2}$, then ${\displaystyle u\in H^k(\mathrm{\Omega })}$, together with a Sobolev imbedding theorem saying that functions in ${\displaystyle H^k(\mathrm{\Omega })}$ are also in ${\displaystyle C^m(\overline{\mathrm{\Omega }})}$ whenever ${\displaystyle 0\le m<k-n/2}$.

Numerical solutions

While in exceptional circumstances, it is possible to solve elliptic problems explicitly, in general it is an impossible task. The natural solution is to approximate the elliptic problem with a simpler one and to solve this simpler problem on a computer.

Because of the good properties we have enumerated (as well as many we have not), there are extremely efficient numerical solvers for linear elliptic boundary value problems (see finite element method, finite difference method and spectral method for examples.)

Eigenvalues and eigensolutions

Another Sobolev imbedding theorem states that the inclusion ${\displaystyle H^1\subset L^2}$ is a compact linear map. Equipped with the spectral theorem for compact linear operators, one obtains the following result.

Theorem Assume that ${\displaystyle A(u,\phi )}$ is coercive, continuous and symmetric. The map ${\displaystyle S:f\to u}$ from ${\displaystyle L^2(\mathrm{\Omega })}$ to ${\displaystyle L^2(\mathrm{\Omega })}$ is a compact linear map. It has a basis of eigenvectors ${\displaystyle u_1,u_2,\cdots \in H^1(\mathrm{\Omega })}$ and matching eigenvalues ${\displaystyle {\lambda }_1,{\lambda }_2,\cdots \in \mathbb{R}}$ such that

1. ${\displaystyle S{u}_k={\lambda }_k{u}_k,k=1,2,\dots ,}$
2. ${\displaystyle {\lambda }_k\to 0}$ as ${\displaystyle k\to \mathrm{\infty }}$,
3. ${\displaystyle {\lambda }_k\gneqq 0\mathrm{\forall }k}$,
4. ${\displaystyle {\int }_{\mathrm{\Omega }}{u}_j{u}_k=0}$ whenever ${\displaystyle j\ne k}$ and
5. ${\displaystyle {\int }_{\mathrm{\Omega }}{u}_j{u}_j=1}$ for all ${\displaystyle j=1,2,\dots .}$

Series solutions and the importance of eigensolutions

If one has computed the eigenvalues and eigenvectors, then one may find the "explicit" solution of ${\displaystyle Lu=f}$,

${\displaystyle u=\sum _{k=1}^{\mathrm{\infty }}\hat{u}(k)u_k}$

via the formula

${\displaystyle \hat{u}(k)={\lambda }_k\hat{f}(k),k=1,2,\dots }$

where

${\displaystyle \hat{f}(k)={\int }_{\mathrm{\Omega }}f(x)u_k(x)dx.}$

(See Fourier series.)

The series converges in ${\displaystyle L^2}$. Implemented on a computer using numerical approximations, this is known as the spectral method.

An example

Consider the problem

${\displaystyle u-u_{xx}-u_{yy}=f(x,y)=xy}$ on ${\displaystyle (0,1)\times (0,1),}$

${\displaystyle u(x,0)=u(x,1)=u(0,y)=u(1,y)=0\mathrm{\forall }(x,y)\in (0,1)\times (0,1)}$ (Dirichlet conditions).

The reader may verify that the eigenvectors are exactly

${\displaystyle u_{jk}(x,y)=\sin (\pi jx)\sin (\pi ky)}$, ${\displaystyle j,k\in \mathbb{N}}$

with eigenvalues

${\displaystyle {\lambda }_{jk}=\frac{1}{1+{\pi }^2{j}^2+{\pi }^2{k}^2}.}$

The Fourier coefficients of ${\displaystyle g(x)=x}$ can be looked up in a table, getting ${\displaystyle \hat{g}(n)=\frac{(-1{)}^{n+1}}{\pi n}}$. Therefore,

${\displaystyle \hat{f}(j,k)=\frac{(-1{)}^{j+k+1}}{{\pi }^{2}jk}}$

yielding the solution

${\displaystyle u(x,y)=\sum _{j,k=1}^{\mathrm{\infty }}\frac{(-1{)}^{j+k+1}}{{\pi }^{2}jk(1+{\pi }^2{j}^2+{\pi }^2{k}^2)}\sin (\pi jx)\sin (\pi ky).}$

Maximum principle

There are many variants of the maximum principle. We give a simple one.

Theorem. (Weak maximum principle.) Let ${\displaystyle u\in C^2(\mathrm{\Omega })\cap C^1(\overline{\mathrm{\Omega }})}$, and assume that ${\displaystyle c(x)=0\mathrm{\forall }x\in \mathrm{\Omega }}$. Say that ${\displaystyle Lu\le 0}$ in ${\displaystyle \mathrm{\Omega }}$. Then ${\displaystyle \underset{x\in \overline{\mathrm{\Omega }}}{max}u(x)=\underset{x\in \mathrm{\partial }\mathrm{\Omega }}{max}u(x)}$. In other words, the maximum is attained on the boundary.

A strong maximum principle would conclude that ${\displaystyle u(x)\lneqq \underset{y\in \mathrm{\partial }\mathrm{\Omega }}{max}u(y)}$ for all ${\displaystyle x\in \mathrm{\Omega }}$ unless ${\displaystyle u}$ is constant.