Inmathematics, aDirichlet problem asks for afunction which solves a specifiedpartial 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 forLaplace's equation. In that case the problem can be stated as follows:

Given a functionf that has values everywhere on the boundary of a region in${\displaystyle {\mathbb{R}}^n}$, is there a uniquecontinuous function${\displaystyle u}$ twice continuously differentiable in the interior and continuous on the boundary, such that${\displaystyle u}$ isharmonic in the interior and${\displaystyle u=f}$ on the boundary?

This requirement is called theDirichlet boundary condition. The main issue is to prove the existence of a solution; uniqueness can be proven using themaximum principle.

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The Dirichlet problem goes back toGeorge Green, who studied the problem on general domains with general boundary conditions in hisEssay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, published in 1828.^[2] He reduced the problem into a problem of constructing what we now callGreen'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 byKarl Friedrich Gauss, William Thomson (Lord Kelvin) andPeter Gustav Lejeune Dirichlet, after whom the problem was named, and the solution to the problem (at least for the ball) using thePoisson 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 avariational method based on the minimization of "Dirichlet's energy". According toHans Freudenthal (in theDictionary of Scientific Biography, vol. 11),Bernhard Riemann was the first mathematician who solved this variational problem based on a method which he calledDirichlet'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 ofelectrostatics, determine anelectrical potential as solution. However,Karl Weierstrass found a flaw in Riemann's argument, and a rigorous proof of existence was found only in 1900 byDavid Hilbert, using hisdirect 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.

For a domain${\displaystyle D}$ having a sufficiently smooth boundary${\displaystyle \mathrm{\partial }D}$, the general solution to the Dirichlet problem is given by

${\displaystyle u(x)={\int }_{\mathrm{\partial }D}\nu (s)\frac{\mathrm{\partial }G(x,s)}{\mathrm{\partial }n}ds,}$

where${\displaystyle G(x,y)}$ is theGreen's function for the partial differential equation, and

${\displaystyle \frac{\mathrm{\partial }G(x,s)}{\mathrm{\partial }n}=\hat{n}\cdot {\mathrm{\nabla }}_{s}G(x,s)=\sum _i{n}_i\frac{\mathrm{\partial }G(x,s)}{\mathrm{\partial }{s}_i}}$

is the derivative of the Green's function along the inward-pointing unit normal vector${\displaystyle \hat{n}}$. The integration is performed on the boundary, withmeasure${\displaystyle ds}$. The function${\displaystyle \nu (s)}$ is given by the unique solution to theFredholm integral equation of the second kind,

${\displaystyle f(x)=-\frac{\nu (x)}{2}+{\int }_{\mathrm{\partial }D}\nu (s)\frac{\mathrm{\partial }G(x,s)}{\mathrm{\partial }n}ds.}$

The Green's function to be used in the above integral is one which vanishes on the boundary:

${\displaystyle G(x,s)=0}$

for${\displaystyle s\in \mathrm{\partial }D}$ and${\displaystyle x\in D}$. Such a Green's function is usually a sum of the free-field Green's function and a harmonic solution to the differential equation.

The Dirichlet problem for harmonic functions always has a solution, and that solution is unique, when the boundary is sufficiently smooth and${\displaystyle f(s)}$ is continuous. More precisely, it has a solution when

${\displaystyle \mathrm{\partial }D\in C^{1,\alpha }}$

for some${\displaystyle \alpha \in (0,1)}$, where${\displaystyle C^{1,\alpha }}$ denotes theHölder condition.

In some simple cases the Dirichlet problem can be solved explicitly. For example, the solution to the Dirichlet problem for theunit disk inR² is given by thePoisson integral formula.^[3]

If${\displaystyle f}$ is a continuous function on the boundary${\displaystyle \mathrm{\partial }D}$ of the open unit disk${\displaystyle D}$, then the solution to the Dirichlet problem is${\displaystyle u(z)}$ given by

The solution${\displaystyle u}$ is continuous on the closed unit disk${\displaystyle \overline{D}}$ and harmonic on${\displaystyle D.}$

The integrand is known as thePoisson kernel; this solution follows from the Green's function in two dimensions:

${\displaystyle G(z,x)=-\frac{1}{2\pi }\log |z-x|+\gamma (z,x),}$

where${\displaystyle \gamma (z,x)}$ isharmonic (${\displaystyle {\mathrm{\Delta }}_x\gamma (z,x)=0}$) and chosen such that${\displaystyle G(z,x)=0}$ for${\displaystyle x\in \mathrm{\partial }D}$.

For bounded domains, the Dirichlet problem can be solved using thePerron method, which relies on themaximum principle forsubharmonic functions. This approach is described in many text books.^[4] It is not well-suited to describing smoothness of solutions when the boundary is smooth. Another classicalHilbert space approach throughSobolev spaces does yield such information.^[5] The solution of the Dirichlet problem usingSobolev spaces for planar domains can be used to prove the smooth version of theRiemann mapping theorem.Bell (1992) has outlined a different approach for establishing the smooth Riemann mapping theorem, based on thereproducing kernels of Szegő and Bergman, and in turn used it to solve the Dirichlet problem. The classical methods ofpotential theory allow the Dirichlet problem to be solved directly in terms ofintegral operators, for which the standard theory ofcompact andFredholm operators is applicable. The same methods work equally for theNeumann problem.^[6]

Dirichlet problems are typical ofelliptic partial differential equations, andpotential theory, and theLaplace equation in particular. Other examples include thebiharmonic equation and related equations inelasticity theory.

They are one of several types of classes of PDE problems defined by the information given at the boundary, includingNeumann problems andCauchy problems.

Consider the Dirichlet problem for thewave equation describing a string attached between walls with one end attached permanently and the other moving with the constant velocity i.e. thed'Alembert equation on the triangular region of theCartesian product of the space and the time:

${\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}u(x,t)-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}u(x,t)=0,}$

${\displaystyle u(0,t)=0,}$

${\displaystyle u(\lambda t,t)=0.}$

As one can easily check by substitution, the solution fulfilling the first condition is

${\displaystyle u(x,t)=f(t-x)-f(x+t).}$

Additionally we want

${\displaystyle f(t-\lambda t)-f(\lambda t+t)=0.}$

Substituting

${\displaystyle \tau =(\lambda +1)t,}$

we get the condition ofself-similarity

${\displaystyle f(\gamma \tau )=f(\tau ),}$

where

${\displaystyle \gamma =\frac{1-\lambda }{\lambda +1}.}$

It is fulfilled, for example, by thecomposite function

${\displaystyle \sin [\log (e^{2\pi }x)]=\sin [\log (x)]}$

with

${\displaystyle \lambda =e^{2\pi }=1^{-i},}$

thus in general

${\displaystyle f(\tau )=g[\log (\gamma \tau )],}$

where${\displaystyle g}$ is aperiodic function with a period${\displaystyle \log (\gamma )}$:

${\displaystyle g[\tau +\log (\gamma )]=g(\tau ),}$

and we get the general solution

${\displaystyle u(x,t)=g[\log (t-x)]-g[\log (x+t)].}$

• Lebesgue spine

• A. Yanushauskas (2001) [1994],"Dirichlet problem",Encyclopedia of Mathematics,EMS Press
• S. G. Krantz,The Dirichlet Problem. §7.3.3 inHandbook of Complex Variables. Boston, MA: Birkhäuser, p. 93, 1999.ISBN 0-8176-4011-8.
• S. Axler,P. Gorkin, K. Voss,The Dirichlet problem on quadratic surfaces, Mathematics of Computation73 (2004), 637–651.
• Gilbarg, David;Trudinger, Neil S. (2001),Elliptic partial differential equations of second order (2nd ed.), Berlin, New York:Springer-Verlag,ISBN 978-3-540-41160-4.
• Gérard, Patrick;Leichtnam, Éric: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71 (1993), no. 2, 559–607.
• John, Fritz (1982),Partial differential equations, Applied Mathematical Sciences, vol. 1 (4th ed.), Springer-Verlag,ISBN 0-387-90609-6.
• 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,ISBN 0-8218-0049-3.
• Agmon, Shmuel (2010),Lectures on Elliptic Boundary Value Problems, American Mathematical Society,ISBN 978-0-8218-4910-1
• Stein, Elias M. (1970),Singular Integrals and Differentiability Properties of Functions, Princeton University Press.
• Greene, Robert E.; Krantz, Steven G. (2006),Function theory of one complex variable,Graduate Studies in Mathematics, vol. 40 (3rd ed.), American Mathematical Society,ISBN 0-8218-3962-4.
• Taylor, Michael E. (2011),Partial differential equations I. Basic theory, Applied Mathematical Sciences, vol. 115 (2nd ed.), Springer,ISBN 978-1-4419-7054-1.
• Zimmer, Robert J. (1990),Essential results of functional analysis, Chicago Lectures in Mathematics, University of Chicago Press,ISBN 0-226-98337-4.
• Folland, Gerald B. (1995),Introduction to partial differential equations (2nd ed.), Princeton University Press,ISBN 0-691-04361-2.
• Chazarain, Jacques; Piriou, Alain (1982),Introduction to the Theory of Linear Partial Differential Equations, Studies in Mathematics and Its Applications, vol. 14, Elsevier,ISBN 0444864520.
• Bell, Steven R. (1992),The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press,ISBN 0-8493-8270-X.
• Warner, Frank W. (1983),Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, vol. 94, Springer,ISBN 0387908943.
• Griffiths, Phillip; Harris, Joseph (1994),Principles of Algebraic Geometry, Wiley Interscience,ISBN 0471050598.
• 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

This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(June 2021) (Learn how and when to remove this message)

• "Dirichlet problem",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Weisstein, Eric W."Dirichlet Problem".MathWorld.

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