Initial value problems for Laplace equation

Question

Usually in the textbooks well-posedness of the boundary value problem is studied for Laplace equation $$-\Delta u=0.$$

Why dont we study the Initial value problems like

\begin{eqnarray} -\Delta u&=&0 \quad \text{ in } \mathbb{R}^2\\ u(x,0)=u_y(x,0)&=&0 \quad \text{ on } \mathbb{R} \end{eqnarray}

Clearly, in the one dimensional case, the IVP given by

\begin{eqnarray} -u_{xx}&=&0 \quad \text{ in } \mathbb{R}\\ u(0)=u'(0)&=&0 \end{eqnarray} is well-posed.

Thanks in advance

Answer 1

It is not correct to say that the Cauchy problem for the Laplacian or, more generally, for elliptic operators is not studied: simply stated, the analysis of such kind of problems requires a deeper care and attention, therefore it is not seen in every textbook on PDEs.
One of the basic problems of such kind of Cauchy problems is that as a rule they are not well posed i.e. the solution is not continuous respect to the given Cauchy (initial) data. But what is the precise meaning of this locution?
A problem is not well posed simply if, for a given choice of the topology of the function space to which the Cauchy (initial) data belongs, the solution operator is not continuous: nevertheless, there are other, more subtle, choices of the topology for which well posedness can be recovered. We'll follow Fichera's analysis of Hadamard's example given in [1], §2 pp. 164-165 to see how this can be done in that particular case.

Preliminary considerations.

Consider the general Cauchy problem for the Laplace equation in $\Bbb R^2$: $$\DeclareMathOperator{\Drv}{d} \begin{cases} \Delta u = 0 & \text{ in } \mathbb{R}^2\\ \\ u(x,0) = \varphi_0(x) \\ & \text{ on } \mathbb{R}\\ u_y(x,0) = \varphi_1(x) \end{cases}\label{1}\tag{1} $$ Each function in the Cauchy data $(\varphi_0, \varphi_1)$ is assumed to belong to the space of real analytic functions $\mathscr{A}([0,\pi])$. This choice implies that the Cauchy-Kovalevskaya theorem is applicable, thus there exists a $\Bbb R^2$-neighborhood of $[0,\pi]\subset\Bbb R$ where a solution $u(x,y)$ exists an is unique: for example (and without restriction to generality), let's say that this neighborhood has the form of the rectangle $[0,\pi]\times[-\sigma, \sigma]$, for some $\sigma>0$.
Hadamard's counterexample is constructed by explicitly solving \eqref{1} with the the particular choices $\varphi_0(x,n) = 0$ and $\varphi_1(x,n)=e^{-\sqrt{n}}\sin nx$. We have $$ u_n(x,y) = \frac{1}{n}e^{-\sqrt{n}}\sin nx\sinh ny\label{2}\tag{2} $$ Said that, let's call $\mathcal{IV}=\mathscr{A}([0,\pi])\times\mathscr{A}([0,\pi])$ and $\mathcal{U}$ respectively the space of Cauchy (initial value) data $\Phi=(\varphi_0,\varphi_1)$ and the space of the solution to the problem \eqref{1} and see what happens to the continuity of solution operator by adopting different topologies on these spaces.

The choice of the topology on $\mathscr{A}([0,\pi])$ and its influence on well posedness.

1. Define respectively on $\mathcal{IV}$ and $\mathcal{U}$ the norms
   $$ \|\Phi\|_\mathcal{IV}=\sum_{k=0}^M\left(\max_{[0,\pi]}\left|\frac{\Drv^k\! \varphi_0}{\Drv\! x^k}\right| + \max_{[0,\pi]}\left|\frac{\Drv^k\! \varphi_1}{\Drv\! x^k}\right|\right),\label{3}\tag{$N_A$} $$ and $$ \|u\|_\mathcal{U} =\max_{[0,\pi]\times[-\sigma, \sigma]} |u(x,y)|,\label{4}\tag{$N$} $$ where $M$ is a preassigned positive integer. Then it is easy to see, that while $$ \|(0,e^{-\sqrt{n}}\sin nx)\|_\mathcal{IV}=e^{-\sqrt{n}}\sum_{k=0}^M m^k\underset{n\to\infty}{\longrightarrow} 0 $$ we have that $$ \lim_{n\to\infty} \|u_n\|_\mathcal{U}=+\infty, $$ thus the solution operator is not continuous respect to the topology defined by the two chosen norms.
2. On the other hand, since we are working in $\mathscr{A}([0,\pi])$ again by the Cauchy-Kovalevskaya theorem we can consider consider problem \eqref{1} from the point of view of functions of a complex variable. Precisely, we can see the data $\Phi=(\varphi_0,\varphi_1)$ as traces on $[0,\pi]$ of holomorphic function in the rectangle $R_\sigma = [\sigma,\pi + \sigma] \times [-\sigma, \sigma]$ and we can define $$ \mathcal{IV}=\left\{\Phi\equiv(\varphi_0,\varphi_1) : \varphi_i\text{ is holomorphic in }R_\sigma\setminus\partial R_\sigma\text{ and continuous on }R_\sigma,\;i=0, 1\right\} $$ Now consider $u$ harmonic in $R_\sigma\setminus\partial R_\sigma$ and continuous in $R_\sigma$ and $v$ is its harmonic conjugate satisfying the condition $v(0,0)=0$ and define $f= u(x,y) + iv(x,y)$. Then if $u$ is a solution to \eqref{1} we have $$ f^\prime(x+i0) = \varphi^\prime_0(x+i0)- i \varphi^\prime_1(x+i0)\quad\forall x\in[0,\pi], $$ and this means that $f^\prime(z) = \varphi^\prime_0(z)- i \varphi^\prime_1(z)$ for all $z\in R_\sigma$ and implies $$ f(z) = \varphi_0(z) -i\int\limits_0^z\varphi_1(\zeta)\operatorname{d\!}\zeta.\label{2'}\tag{2'} $$ On the other hand we see that $u=\Re f$ is a solution to the Cauchy problem \eqref{1}: these considerations allow us to change the definition of the norm on $\mathcal{IV}$ as follows $$ \|\Phi\|_\mathcal{IV}^\prime=\max_{R_\sigma} |\varphi_0(z)|+ (\pi+\sigma)\max_{R_\sigma} |\varphi_1(z)|,\label{5}\tag{$N_B$} $$ Now we have the relationship (a priori estimate) $$ \|u\|_\mathcal{U}\le \|\Phi\|_\mathcal{IV}^\prime $$ that implies the continuous dependence on the Cauchy data for the solution of \eqref{1}.

Problem \eqref{1} is thus incorrectly posed respect to the couple of norms \eqref{3}, \eqref{4}, while is properly posed respect to the couple of norms \eqref{5}, \eqref{4}.

Notes

• When $n=1$, the Cauchy problem for the Laplace equation become well posed since it reduces to the initial value problem for a linear ordinary differential operator: this implies that all the function spaces involved are finite dimensional, thus in particular $\mathcal{IV}\equiv \Bbb R^2$ and all the norms on this space are equivalent, thus the original example cannot take place.

• The title of reference [1] (a free translation of which is "The difficult relationships between Functional analysis and Mathematical Physics") is a pun to emphasize the difficulties of applying the modern approach to functional analysis, based on topological spaces and on the geometry of such spaces, to the mathematical modeling of the physical word based on partial differential (or more generally functional) equations: apart from Hadamard well-posedness the Author describes other occurrences of these "difficulties" and for mathematical models of physical problems he not only advocates for a frame indifference of the PDE formulation, but also for a "functional topology indifference" of the same formulation.

• More specifically, references [2] and [3] deal extensively with incorrectly posed Cauchy problems respectively for the Laplace and for more general elliptic operators.

Reference

[1] Gaetano Fichera, "I difficili rapporti fra l'Analisi funzionale e la Fisica Matematica" (in Italian), Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei, Matematica e Applicazioni, Serie IX, 1 (1990), issue n.2, p. 161-170, MR1081399, Zbl 0709.45004.

[2] Vladimir Gilelevich Maz’ya, Viktor Petrovich Khavin, "On the solutions of the Cauchy problem for Laplace’s equation (uniqueness, normality, approximation)" (English), Transactions of the Moscow Mathematical Society 30(1974), 65-117 (1976), MR0385135, Zbl 0347.31008.

[3] Nikolai N. Tarkhanov,The Cauchy problem for solutions of elliptic equations (English) Mathematical Topics, 7, Berlin: Akademie-Verlag. 478 p. (1995), ISBN: 3-05-501663-7, MR1334094, Zbl 0831.35001.