Chapter 4 The heat equation on a general domain

Consider the Laplacian \(\Delta \) on \(\mR ^d\). In Cartesian coordinates (which we denote by \(\bf x\)) this means

\[ \Delta u:=\sum _{k=1}^d\partial _{{\bf x}_k{\bf x}_k}u. \]

For example when \(d=2\) (where we denote the Cartesian coordinates by \(x\) and \(y\)) this is

\[ \Delta u=\partial _{xx}u+\partial _{yy}u. \]

Let \(\Omega \subset \mR ^d\) be an open set with Lipschitz boundary \(\partial \Omega \) (this means that the boundary is locally the graph of a Lipschitz function; examples are rectangles (and more generally polyhedra), balls and cylinders). Such a set is called a Lipschitz domain. We can consider the heat equation on the spatial domain \(\Omega \) with Dirichlet boundary condition on the boundary \(\partial \Omega \)

\begin{equation} \label {eq:Dirichletheat} \partial _t u=\Delta u,\qquad u(t,\cdot )|_{\partial \Omega }=0,\qquad u(0,{\bf x})=u^0({\bf x}), \end{equation}

where the initial condition \(u^0:\Omega \to \mR \) is given and \(u:[0,\infty )\times \Omega \to \mR \) is to be found (the boundary condition must be understood in a suitable sense which we will not elaborate on).

The operator

\[ \dom (\Delta _D)=\left \{u\in L^2(\Omega ): \int _\Omega \sum _{k=1}^d\left |\partial _{{\bf x}_k}u\right |^2<\infty , \Delta u\in L^2(\Omega ), u|_{\partial \Omega }=0\right \}, \]

is called the Dirichlet Laplacian. Note that this is the generalization of the operator \(A\) considered in Section 2.2, which has \(\Omega =(0,L)\).

Theorem 4.1 allows us to write down the solution of the heat equation (4.1) as

\[ u(t,{\bf x})=\sum _{k=1}^\infty C_k\e ^{\lambda _kt}{\bf X}_k({\bf x}),\qquad C_k=\int _\Omega u^0({\bf x}){\bf X}_k({\bf x})\,d{\bf x}, \]

where \(\lambda _k\) are the eigenvalues and \(({\bf X}_k)\) is an orthonormal basis of eigenvectors of the Dirichlet Laplacian \(\Delta _D\). All that remains to be done is to explicitly calculate these eigenvalues and eigenvectors for specific domains \(\Omega \). For general domains this is impossible, but for certain domains such as rectangles, balls and cylinders it is possible.

The Dirichlet Laplacian on a bounded Lipschitz domain \(\Omega \) seen as an operator on \(L^2(\Omega )\) is self-adjoint and has compact resolvent (the latter follows from Rellich’s Theorem). From Theorem 2.6 we then obtain in particular Theorem 4.1.

4.1 The heat equation on a disc

For the heat equation on a disc \(\Omega =\{(x,y)\in \mR ^2: x^2+y^2<b^2\}\), it is convenient to calculate in polar coordinates (since we will obtain separated solutions in polar coordinates, but not in Cartesian coordinates).

The relation between polar and Cartesian coordinates is

\[ x=r\cos (\theta ),\qquad y=r\sin (\theta ). \]

The Laplacian in polar coordinates is (this follows from applying the chain rule several times):

\[ \Delta u=\partial _{rr}u+\frac {1}{r}\partial _ru+\frac {1}{r^2}\partial _{\theta \theta }u. \]

The heat equation with Dirichlet boundary condition then is (we have \(\partial \Omega =\{(x,y)\in \mR ^2: x^2+y^2=b^2\}\), which in polar coordinates is \(r=b\)):

\[ \partial _t u=\partial _{rr}u+\frac {1}{r}\partial _ru+\frac {1}{r^2}\partial _{\theta \theta }u,\qquad u(t,b,\theta )=0,\qquad u(0,r,\theta )=u^0(r,\theta ), \]

where \(r\in (0,b)\) and \(\theta \in (0,2\pi )\) and we have slightly abused notation in viewing \(u^0\) and \(u\) in terms of polar coordinates where we previously used the same symbols for these functions in Cartesian coordinates.

4.2 The inner-product

The original inner-product on the spatial domain \(\Omega \) considered is

\[ \ipd {g}{h}=\int _\Omega g({\bf x})h({\bf x})\,d{\bf x}. \]

For \(\Omega \) the disc with radius \(b\) in polar coordinates and with \(g\) and \(h\) separated, i.e.

\[ g=R_g(r)\Theta _g(\theta ),\qquad h=R_h(r)\Theta _h(\theta ), \]

this is (noting that the Jacobian determinant for polar coordinates is \(r\))

\[ \int _0^b \int _0^{2\pi } R_g(r)\Theta _g(\theta )R_h(r)\Theta _h(\theta )\,r\,d\theta \,dr, \]

which we can write as the product

\[ \int _0^{2\pi }\Theta _g(\theta )\Theta _h(\theta )\,d\theta ~~ \int _0^b R_g(r)R_h(r)\,r\,dr, \]

which means that in the variable \(r\) we do not have the standard \(L^2\) inner-product, but there is a “weight” in the integral (namely the \(r\) which comes from the Jacobian determinant). This will be abstracted to the weight \(w\) in Sturm–Liouville problems in Chapter 6.