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| In [[mathematics]], a '''Dirichlet problem''' is the problem of finding a [[function (mathematics)|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.
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| 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:
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| :Given a function ''f'' that has values everywhere on the boundary of a region in '''R'''<sup>''n''</sup>, is there a unique [[continuous function]] ''u'' twice continuously differentiable in the interior and continuous on the boundary, such that ''u'' is [[harmonic function|harmonic]] in the interior and ''u'' = ''f'' on the boundary?
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| This requirement is called the [[Dirichlet boundary condition]]. The main issue is to prove the existence of a solution; uniqueness can be proved using the [[maximum principle]].
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| ==History==
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| The '''Dirichlet problem''' is named after [[Peter Gustav Lejeune Dirichlet]], who proposed a solution by a variational method which became known as [[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.
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| However, [[Karl Weierstrass]] found a flaw in Dirichlet's argument, and a rigorous proof of existence was found only in 1900 by [[David Hilbert]]. It turns out that the existence of a solution depends delicately on the smoothness of the boundary and the prescribed data.
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| == General solution ==
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| For a domain <math>D</math> having a sufficiently smooth boundary <math>\partial D</math>, the general solution to the Dirichlet problem is given by
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| :<math>u(x)=\int_{\partial D} \nu(s) \frac{\partial G(x,s)}{\partial n} ds</math>
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| where <math>G(x,y)</math> is the [[Green's function]] for the partial differential equation, and
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| :<math>\frac{\partial G(x,s)}{\partial n} = \widehat{n} \cdot \nabla_s G (x,s) = \sum_i n_i \frac{\partial G(x,s)}{\partial s_i}</math>
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| is the derivative of the Green's function along the inward-pointing unit normal vector <math>\widehat{n}</math>. The integration is performed on the boundary, with [[Measure (mathematics)|measure]] <math>ds</math>. The function <math>\nu(s)</math> is given by the unique solution to the [[Fredholm integral equation]] of the second kind,
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| :<math>f(x) = -\frac{\nu(x)}{2} + \int_{\partial D} \nu(s) \frac{\partial G(x,s)}{\partial n} ds.</math>
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| The Green's function to be used in the above integral is one which vanishes on the boundary:
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| :<math>G(x,s)=0</math>
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| for <math>s\in \partial D</math> and <math>x\in D</math>. Such a Green's function is usually a sum of the free-field Green's function and a harmonic solution to the differential equation. | |
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| ===Existence===
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| The Dirichlet problem for harmonic functions always has a solution, and that solution is unique, when the boundary is sufficiently smooth and <math>f(s)</math> is continuous. More precisely, it has a solution when
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| :<math>\partial D \in C^{1,\alpha}</math>
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| for some <math>\alpha\in(0,1)</math>, where <math>C^{1,\alpha}</math> denotes the [[Hölder condition]]. | |
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| == Example: the unit disk in two dimensions ==
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| In some simple cases the Dirichlet problem can be solved explicitly. For example, the solution to the Dirichlet problem for the unit disk in '''R'''<sup>2</sup> is given by the [[Poisson integral formula]].
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| If <math>f</math> is a continuous function on the boundary <math>\partial D</math> of the open unit disk <math>D</math>, then the solution to the Dirichlet problem is <math>u(z)</math> given by
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| :<math>u(z) = \begin{cases} \frac{1}{2\pi}\int_0^{2\pi} f(e^{i\psi})
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| \frac {1-\vert z \vert ^2}{\vert 1-ze^{-i\psi}\vert ^2} d \psi & \mbox{if }z \in D \\
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| f(z) & \mbox{if }z \in \partial D. \end{cases}</math>
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| The solution <math>u</math> is continuous on the closed unit disk <math>\bar{D}</math> and harmonic on <math>D.</math>
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| The integrand is known as the [[Poisson kernel]]; this solution follows from the Green's function in two dimensions:
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| :<math>G(z,x) = -\frac{1}{2\pi} \log \vert z-x\vert + \gamma(z,x)</math>
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| where <math>\gamma(z,x)</math> is harmonic
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| :<math>\Delta_x \gamma(z,x)=0</math>
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| and chosen such that <math>G(z,x)=0</math> for <math>x\in \partial D</math>.
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| ==Methods of solution== | |
| For bounded domains, the Dirichlet problem can be solved using the [[Perron method]], which relies on the [[maximum principle]] for [[subharmonic function]]s. This approach is described in many text books.<ref> See for example:
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| *{{harvnb|John|1982}}
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| *{{harvnb|Bers|John|Schechter|1979}}
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| *{{harvnb|Greene|Krantz|2006}}
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| </ref> It is not well-suited to describing smoothness of solutions when the boundary is smooth. Another classical [[Hilbert space]] approach through [[Sobolev space]]s does yield such information.<ref> See for example:
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| *{{harvnb|Bers|John|Schechter|1979}}
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| *{{harvnb|Chazarain|Piriou|1982}}
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| *{{harvnb|Taylor|2011}}
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| </ref> 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]]. {{harvtxt|Bell|1992}} has outlined a different approach for establishing the smooth Riemann mapping theorem, based on the [[reproducing kernel]]s 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 operator]]s, for which the standard theory of [[compact operator|compact]] and [[Fredholm operator]]s is applicable. The same methods work equally for the [[Neumann problem]].
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| <ref>See:
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| *{{harvnb|Folland|1995}}
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| *{{harvnb|Bers|John|Schechter|1979}}</ref>
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| ==Generalizations== | |
| Dirichlet problems are typical of [[elliptic partial differential equation]]s, and [[potential theory]], and the [[Laplace equation]] in particular. Other examples include the [[biharmonic equation]] and related equations in [[elasticity theory]].
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| They are one of several types of classes of PDE problems defined by the information given at the boundary, including [[Neumann problem]]s and [[Cauchy problem]]s.
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| ==Example - equation of a finite string attached to one moving wall==
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| Let us consider the Dirichlet problem for the [[wave equation]] which describes a string attached between walls with one end
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| attached permanently and with the other moving with the constant velocity i.e. the [[d'Alembert equation|d’Alembert equation]]
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| on the triangular region of the [[Cartesian product]] of the space and the time:
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| :: <math>\frac{\partial{}^2}{\partial t^2}u(x,t) - \frac{\partial{}^2}{\partial x^2} u(x,t) = 0 </math>
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| :: <math>u(0,t)= 0</math>
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| :: <math>u(\lambda t, t)=0 </math>
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| As one can easily check by substitution that the solution fulfilling the first condition is
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| :: <math>u(x,t)= f(t-x) - f(x+t)</math>
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| Additionally we want
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| :: <math>f(t-\lambda t) - f(\lambda t+t)=0</math>
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| Substituting
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| :: <math>\tau=(\lambda +1) t</math>
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| we get the condition of [[self-similarity]]
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| <math>f(\gamma \tau) = f(\tau)</math>
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| where
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| <math>\gamma= \frac{1-\lambda}{\lambda +1} </math>
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| It is fulfilled for example by the [[composite function]]
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| <math>\sin[\log(e^{2 \pi} x)]= \sin[\log(x)]</math>
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| with
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| <math>\lambda=e^{2\pi}=1^{-i}</math>
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| thus in general
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| <math>f(\tau) = g[\log(\gamma \tau)]</math>
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| where <math>g</math> is a [[periodic function]] with a period <math>\log(\gamma)</math>
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| <math>g[\tau+\log(\gamma)]= g(\tau)</math>
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| and we get the general solution
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| :: <math>u(x,t)=g[\log(t-x)] - g[\log(x+t)]</math>.
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| * {{springer|author=A. Yanushauskas|id=d/d032910|title=Dirichlet problem}}
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| * S. G. Krantz, ''The Dirichlet Problem.'' §7.3.3 in ''Handbook of Complex Variables''. Boston, MA: Birkhäuser, p. 93, 1999. ISBN 0-8176-4011-8.
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| * S. Axler, P. Gorkin, K. Voss, ''[http://www.ams.org/mcom/2004-73-246/S0025-5718-03-01574-6/home.html The Dirichlet problem on quadratic surfaces]'' Mathematics of Computation '''73''' (2004), 637-651.
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| *{{Citation | last1=Gilbarg | first1=David | last2=Trudinger | first2=Neil S. | author2-link=Neil Trudinger | title=Elliptic partial differential equations of second order | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-3-540-41160-4 | year=2001}}
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| *Gérard, Patrick; [[Eric Leichtnam|Leichtnam, Éric]]: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71 (1993), no. 2, 559-607.
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| *{{citation|last=John|first= Fritz|title=Partial differential equations|edition=4th|series= Applied Mathematical Sciences|volume= 1|publisher= Springer-Verlag|year= 1982|id= ISBN 0-387-90609-6}}
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| *{{citation|last=Bers|first=Lipman|last2=John|first2=Fritz|last3= Schechter|first3= Martin|title=Partial differential equations, with supplements by Lars Gȧrding and A. N. Milgram|series= Lectures in Applied Mathematics|volume= 3A|publisher= American Mathematical Society|year=1979|id=ISBN 0-8218-0049-3}}
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| *{{citation|title=Lectures on Elliptic Boundary Value Problems|first=Shmuel|last= Agmon|authorlink=Shmuel Agmon|year=2010|publisher=American Mathematical Society|id=ISBN 0-8218-4910-7}}
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| * {{citation|first=Elias M.|last= Stein|authorlink=Elias Stein|year=1970|title=Singular Integrals and Differentiability Properties of Functions|publisher=Princeton University Press}}
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| *{{citation|last=Greene|first= Robert E.|last2= Krantz|first2= Steven G.|title= Function theory of one complex variable|edition=3rd|series= Graduate Studies in Mathematics|volume= 40|publisher= American Mathematical Society|year= 2006|id= ISBN 0-8218-3962-4}}
| |
| *{{citation| last=Taylor|first= Michael E.|authorlink=Michael E. Taylor|title= Partial differential equations I. Basic theory|edition=2nd |series= Applied Mathematical Sciences|volume= 115|publisher=Springer|year=2011|id= ISBN 978-1-4419-70}}
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| *{{citation|last=Zimmer|first= Robert J.|title= Essential results of functional analysis|series= Chicago Lectures in Mathematics|publisher= University of Chicago Press|year= 1990|id= ISBN 0-226-98337-4}}
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| *{{citation|last=Folland|first= Gerald B.|title= Introduction to partial differential equations|edition=2nd|publisher=Princeton University Press|year=1995|id= ISBN 0-691-04361-2}}
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| *{{citation|title=Introduction to the Theory of Linear Partial Differential Equations|volume=14|series= Studies in Mathematics and Its Applications|first=Jacques|last= Chazarain|first2= Alain|last2= Piriou|publisher=Elsevier|year= 1982|id=ISBN 0444864520}}
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| *{{citation|last=Bell|first=Steven R.|title= The Cauchy transform, potential theory, and conformal mapping|series= Studies in Advanced Mathematics|publisher= CRC Press|year= 1992|id=ISBN 0-8493-8270-X}}
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| *{{citation|title=Foundations of Differentiable Manifolds and Lie Groups|series=Graduate Texts in Mathematics|volume= 94|year=1983|
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| first=Frank W.|last= Warner|id=ISBN 0387908943|publisher=Springer}}
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| *{{citation|title=Principles of Algebraic Geometry|first=Phillip |last=Griffiths|first2= Joseph|last2= Harris|publisher= Wiley Interscience| year=1994|id=ISBN 0471050598}}
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| *{{citation|last=Courant|first= R.|title=Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces|
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| publisher=Interscience|year= 1950}}
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| *{{citation|last=Schiffer|first= M.|last2=Hawley|first2= N. S.|title=Connections and conformal mapping|journal=
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| Acta Math.|volume= 107|year= 1962|pages= 175–274}}
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| == External links ==
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| * {{springer|title=Dirichlet problem|id=p/d032910}}
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| * {{MathWorld | urlname=DirichletProblem | title=Dirichlet Problem}}
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| * [http://math.fullerton.edu/mathews/c2003/DirichletProblemMod.html Dirichlet Problem Module by John H. Mathews]
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| [[Category:Potential theory]]
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| [[Category:Partial differential equations]]
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| [[Category:Fourier analysis]]
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| [[Category:Mathematical problems]]
| |
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