Editing Partial differential equation (section)

== Introduction ==
A function {{math|''u''(''x'', ''y'', ''z'')}} of three variables is "[[Harmonic function|harmonic]]" or "a solution of the [[Laplace's equation|Laplace equation]]" if it satisfies the condition
<math display="block">\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}=0.</math>
Such functions were widely studied in the 19th century due to their relevance for [[classical mechanics]], for example the equilibrium temperature distribution of a homogeneous solid is a harmonic function. If explicitly given a function, it is usually a matter of straightforward computation to check whether or not it is harmonic. For instance
<math display="block">u(x,y,z) = \frac{1}{\sqrt{x^2 - 2x + y^2 + z^2 + 1}}</math> and <math display="block">u(x,y,z) = 2x^2 - y^2 - z^2</math>
are both harmonic while
<math display="block">u(x,y,z)=\sin(xy)+z</math>
is not. It may be surprising that the two examples of harmonic functions are of such strikingly different form. This is a reflection of the fact that they are ''not'', in any immediate way, special cases of a "general solution formula" of the Laplace equation. This is in striking contrast to the case of [[ordinary differential equation]]s (ODEs) [[Linear differential equation#Homogeneous equation with constant coefficients|roughly similar]] to the Laplace equation, with the aim of many introductory textbooks being to find algorithms leading to general solution formulas. For the Laplace equation, as for a large number of partial differential equations, such solution formulas fail to exist.

The nature of this failure can be seen more concretely in the case of the following PDE: for a function {{math|''v''(''x'', ''y'')}} of two variables, consider the equation
<math display="block">\frac{\partial^2v}{\partial x\partial y}=0.</math>
It can be directly checked that any function {{mvar|v}} of the form {{math|1=''v''(''x'', ''y'') = ''f''(''x'') + ''g''(''y'')}}, for any single-variable functions {{mvar|f}} and {{mvar|g}} whatsoever, will satisfy this condition. This is far beyond the choices available in ODE solution formulas, which typically allow the free choice of some numbers. In the study of PDEs, one generally has the free choice of functions.

The nature of this choice varies from PDE to PDE. To understand it for any given equation, ''existence and uniqueness theorems'' are usually important organizational principles. In many introductory textbooks, the role of [[Picard–Lindelöf theorem|existence and uniqueness theorems for ODE]] can be somewhat opaque; the existence half is usually unnecessary, since one can directly check any proposed solution formula, while the uniqueness half is often only present in the background in order to ensure that a proposed solution formula is as general as possible. By contrast, for PDE, existence and uniqueness theorems are often the only means by which one can navigate through the plethora of different solutions at hand. For this reason, they are also fundamental when carrying out a purely numerical simulation, as one must have an understanding of what data is to be prescribed by the user and what is to be left to the computer to calculate.

To discuss such existence and uniqueness theorems, it is necessary to be precise about the [[Domain of a function|domain]] of the "unknown function". Otherwise, speaking only in terms such as "a function of two variables", it is impossible to meaningfully formulate the results. That is, the domain of the unknown function must be regarded as part of the structure of the PDE itself.

The following provides two classic examples of such existence and uniqueness theorems. Even though the two PDE in question are so similar, there is a striking difference in behavior: for the first PDE, one has the free prescription of a single function, while for the second PDE, one has the free prescription of two functions.
* Let {{mvar|B}} denote the unit-radius disk around the origin in the plane. For any continuous function {{mvar|U}} on the unit circle, there is exactly one function {{mvar|u}} on {{mvar|B}} such that <math display="block">\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0</math> and whose restriction to the unit circle is given by {{mvar|U}}.
* For any functions {{mvar|f}} and {{mvar|g}} on the real line {{math|'''R'''}}, there is exactly one function {{mvar|u}} on {{math|'''R''' × (−1, 1)}} such that <math display="block">\frac{\partial^2u}{\partial x^2} - \frac{\partial^2u}{\partial y^2} = 0</math> and with {{math|1=''u''(''x'', 0) = ''f''(''x'')}} and {{math|1={{sfrac|∂''u''|∂''y''}}(''x'', 0) = ''g''(''x'')}} for all values of {{mvar|x}}.
Even more phenomena are possible. For instance, the [[Bernstein's problem|following PDE]], arising naturally in the field of [[differential geometry]], illustrates an example where there is a simple and completely explicit solution formula, but with the free choice of only three numbers and not even one function.
* If {{mvar|u}} is a function on {{math|'''R'''<sup>2</sup>}} with <math display="block">\frac{\partial}{\partial x} \frac{\frac{\partial u}{\partial x}}{\sqrt{1 + \left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial y}\right)^2}} +
\frac{\partial}{\partial y} \frac{\frac{\partial u}{\partial y}}{\sqrt{1 + \left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial y}\right)^2}}=0,</math> then there are numbers {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} with {{math|1=''u''(''x'', ''y'') = ''ax'' + ''by'' + ''c''}}.
In contrast to the earlier examples, this PDE is '''nonlinear''', owing to the square roots and the squares. A '''linear''' PDE is one such that, if it is homogeneous, the sum of any two solutions is also a solution, and any constant multiple of any solution is also a solution.