Problem of solving a partial differential equation subject to prescribed boundary values
In
mathematics, a Dirichlet problem asks for a
function which solves a specified
partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.[1]
The Dirichlet problem can be solved for many PDEs, although originally it was posed for
Laplace's equation. In that case the problem can be stated as follows:
Given a function f that has values everywhere on the boundary of a region in , is there a unique
continuous function twice continuously differentiable in the interior and continuous on the boundary, such that is
harmonic in the interior and on the boundary?
The Dirichlet problem goes back to
George Green, who studied the problem on general domains with general boundary conditions in his Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, published in 1828. He reduced the problem into a problem of constructing what we now call
Green's functions, and argued that Green's function exists for any domain. His methods were not rigorous by today's standards, but the ideas were highly influential in the subsequent developments. The next steps in the study of the Dirichlet's problem were taken by
Karl Friedrich Gauss, William Thomson (
Lord Kelvin) and
Peter Gustav Lejeune Dirichlet, after whom the problem was named, and the solution to the problem (at least for the ball) using the
Poisson kernel was known to Dirichlet (judging by his 1850 paper submitted to the Prussian academy). Lord Kelvin and Dirichlet suggested a solution to the problem by a
variational method based on the minimization of "Dirichlet's energy". According to Hans Freudenthal (in the Dictionary of Scientific Biography, vol. 11),
Bernhard Riemann was the first mathematician who solved this variational problem based on a method which he called
Dirichlet's principle. The existence of a unique solution is very plausible by the "physical argument": any charge distribution on the boundary should, by the laws of
electrostatics, determine an
electrical potential as solution. However,
Karl Weierstrass found a flaw in Riemann's argument, and a rigorous proof of existence was found only in 1900 by
David Hilbert, using his
direct method in the calculus of variations. It turns out that the existence of a solution depends delicately on the smoothness of the boundary and the prescribed data.
General solution
For a domain having a sufficiently smooth boundary , the general solution to the Dirichlet problem is given by
where is the
Green's function for the partial differential equation, and
is the derivative of the Green's function along the inward-pointing unit normal vector . The integration is performed on the boundary, with
measure. The function is given by the unique solution to the
Fredholm integral equation of the second kind,
The Green's function to be used in the above integral is one which vanishes on the boundary:
for and . Such a Green's function is usually a sum of the free-field Green's function and a harmonic solution to the differential equation.
Existence
The Dirichlet problem for harmonic functions always has a solution, and that solution is unique, when the boundary is sufficiently smooth and is continuous. More precisely, it has a solution when
In some simple cases the Dirichlet problem can be solved explicitly. For example, the solution to the Dirichlet problem for the unit disk in R2 is given by the
Poisson integral formula.
If is a continuous function on the boundary of the open unit disk , then the solution to the Dirichlet problem is given by
The solution is continuous on the closed unit disk and harmonic on
The integrand is known as the
Poisson kernel; this solution follows from the Green's function in two dimensions:
For bounded domains, the Dirichlet problem can be solved using the
Perron method, which relies on the
maximum principle for
subharmonic functions. This approach is described in many text books.[2] It is not well-suited to describing smoothness of solutions when the boundary is smooth. Another classical
Hilbert space approach through
Sobolev spaces does yield such information.[3] The solution of the Dirichlet problem using
Sobolev spaces for planar domains can be used to prove the smooth version of the
Riemann mapping theorem.
Bell (1992) has outlined a different approach for establishing the smooth Riemann mapping theorem, based on the
reproducing kernels of Szegő and Bergman, and in turn used it to solve the Dirichlet problem. The classical methods of
potential theory allow the Dirichlet problem to be solved directly in terms of
integral operators, for which the standard theory of
compact and
Fredholm operators is applicable. The same methods work equally for the
Neumann problem.[4]
They are one of several types of classes of PDE problems defined by the information given at the boundary, including
Neumann problems and
Cauchy problems.
Example: equation of a finite string attached to one moving wall
Consider the Dirichlet problem for the
wave equation describing a string attached between walls with one end attached permanently and the other moving with the constant velocity i.e. the
d'Alembert equation on the triangular region of the
Cartesian product of the space and the time:
As one can easily check by substitution, the solution fulfilling the first condition is
Bers, Lipman; John, Fritz; Schechter, Martin (1979), Partial differential equations, with supplements by Lars Gårding and A. N. Milgram, Lectures in Applied Mathematics, vol. 3A, American Mathematical Society,
ISBN0-8218-0049-3.
Taylor, Michael E. (2011), Partial differential equations I. Basic theory, Applied Mathematical Sciences, vol. 115 (2nd ed.), Springer,
ISBN978-1-4419-7054-1.
Zimmer, Robert J. (1990), Essential results of functional analysis, Chicago Lectures in Mathematics, University of Chicago Press,
ISBN0-226-98337-4.
Folland, Gerald B. (1995), Introduction to partial differential equations (2nd ed.), Princeton University Press,
ISBN0-691-04361-2.
Chazarain, Jacques; Piriou, Alain (1982), Introduction to the Theory of Linear Partial Differential Equations, Studies in Mathematics and Its Applications, vol. 14, Elsevier,
ISBN0444864520.
Bell, Steven R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press,
ISBN0-8493-8270-X.
Warner, Frank W. (1983), Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, vol. 94, Springer,
ISBN0387908943.
Griffiths, Phillip; Harris, Joseph (1994), Principles of Algebraic Geometry, Wiley Interscience,
ISBN0471050598.
Courant, R. (1950), Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, Interscience.
Schiffer, M.; Hawley, N. S. (1962), "Connections and conformal mapping", Acta Math., 107 (3–4): 175–274,
doi:10.1007/bf02545790