Dirichlet problem

To find a harmonic function $ u $ which is regular in a domain $ D $ and which coincides with a given continuous function $ \phi $ on the boundary $ \Gamma $ of $ D $. The problem of finding the solution of a second-order elliptic equation which is regular in the domain is also known as the Dirichlet or first boundary value problem. Problems connected with this task were studied as early as 1840 by C.F. Gauss, and then by P.G.L. Dirichlet [1].

The solution $ u $ of the Dirichlet problem for a domain $ D $ with a sufficiently smooth boundary $ \Gamma $ can be represented by the integral formula

$$ \tag{1 } u( x) = \int\limits _ \Gamma \phi ( x _ {0} ) \frac{\partial G ( x, x _ {0} ) }{\partial n _ {0} } d \sigma , $$

where $ \partial G ( x , x _ {0} ) / \partial n _ {0} $ is the derivative in the direction of the interior normal at the point $ x _ {0} \in \Gamma $ of the Green function $ G ( x , x _ {0} ) $, which is characterized by the following properties:

1) $ G ( x , x _ {0} ) = s _ {n} ^ {-} 1 r ^ {2-} n + \gamma ( x, x _ {0} ) $ if $ n \geq 3 $, or

$$ G ( x , x _ {0} ) = \frac{1}{2 \pi } \mathop{\rm ln} \frac{1}{r} + \gamma ( x , x _ {0} ) \ \textrm{ if } n = 2 , $$

where $ r = | x - x _ {0} | $ is the distance between the points $ x $ and $ x _ {0} $, $ s _ {n} $ is the surface area of the unit sphere in $ \mathbf R ^ {n} $, $ \gamma ( x , x _ {0} ) $ is a harmonic function regular in $ D $ both with respect to the coordinate $ x $ and with respect to the coordinate $ x _ {0} $;

2) $ G ( x , x _ {0} ) = 0 $ for $ x _ {0} \in D $, $ x \in \Gamma $.

For the sphere, the half-space and certain other most simple domains the Green function is constructed explicitly, and formula (1) yields an effective solution of the Dirichlet problem. The formulas thus obtained for the sphere and the half-space are known as the Poisson formulas (cf. Poisson formula).

The Dirichlet problem is one of the fundamental problems in potential theory. It has served, up to this day, as a touchstone for new methods being developed which then contribute to a greater or lesser extent to the advances in the general theory of partial differential equations.

The following methods are employed in the study of the Dirichlet problem.

The variational method is based on the fact that of all functions $ u $ defined in $ D $ and assuming given values on $ \Gamma $, it is a harmonic function which minimizes the Dirichlet integral

$$ I ( u) = \int\limits _ { D } \sum _ {i = 1 } ^ { n } \left ( \frac{\partial u }{\partial x _ {i} } \right ) ^ {2} d \omega . $$

A special minimizing sequence is constructed for $ I ( u) $, after which convergence is demonstrated. Since for the sought solution $ u $ of the Dirichlet problem the existence of the integral $ I ( u) $ is required, the variational method is applicable only to functions $ \phi $ which are traces on $ \Gamma $ of functions $ F $ defined on $ \overline{D}\; $ for which $ I ( F ) $ exists and is bounded.

In the method of potentials (cf. Potentials, method of) the solution of the Dirichlet problem is sought as the double-layer potential of an unknown density defined on $ \Gamma $. With the aid of the jump formulas with respect to this density one obtains a Fredholm equation, from which the existence of the solution of the Dirichlet problem follows, taking into account that the uniqueness of this solution follows from the maximum principle. It is assumed that $ \Gamma \in H ^ {( 1 , \lambda ) } $.

In the Schwarz alternating method one considers two domains $ D _ {1} $ and $ D _ {2} $ with a non-empty intersection $ D _ {0} $ such that a manner of solving the Dirichlet problem is known for $ D _ {1} $ and $ D _ {2} $ separately. A process is then carried out to find the solution of the Dirichlet problem for $ D = D _ {1} \cup D _ {2} $. The boundaries $ \Gamma _ {1} $ and $ \Gamma _ {2} $ are assumed to be piecewise smooth, and at all intersections points of $ \Gamma _ {1} $ with $ \Gamma _ {2} $, both $ \Gamma _ {1} $ and $ \Gamma _ {2} $ are assumed to be smooth and intersecting at a non-zero angle. Sequences of harmonic functions regular in $ D _ {1} $ and $ D _ {2} $ and satisfying special boundary conditions are constructed; it is then shown that these sequences converge uniformly and that their limits coincide in $ D _ {0} $. The harmonic limit function is regular in $ D $ and is the sought solution of the Dirichlet problem. Schwarz's method may be employed for the union or the intersection of any finite number of domains.

The balayage method in the form in which it was originally introduced by H. Poincaré in 1890 is applicable to domains which can be exhausted by a denumerable set of spheres. The initial step in this method is the construction of the Newton potential which assumes the specified value $ \phi $ at the boundary $ \Gamma $, after which the problem is reduced to replacing this potential by a potential of masses located on $ \Gamma $ without changing the values of $ \phi $ on $ \Gamma $, i.e. to balayage of masses. Such a balayage process for the sphere $ D $ is readily realized in an explicit manner using Poisson's formula. The denumerable number of balayages from spheres whose union exhausts the domain $ D $ of general form yields a certain potential of masses located on the boundary $ \Gamma $, and hence also the solution of the Dirichlet problem.

The Perron method (or the method of upper and lower functions), which is applicable to domains $ D $ of a fairly general kind, resembles the balayage method. It involves the construction of sequences of upper (superharmonic) and lower (subharmonic) functions, the common limit of which is the sought solution of the Dirichlet problem. In order for this solution to assume a specified value at a point $ Q \in \Gamma $, it is necessary and sufficient that a local barrier $ \omega _ {Q} $ exists. The function $ \omega _ {Q} $ is continuous and superharmonic in the intersection $ \overline{D}\; \cap \Sigma $( where $ \Sigma $ is a sphere with its centre at the point $ Q $); $ \omega _ {Q} > 0 $ everywhere in $ \overline{D}\; \cap \Sigma $ except at the point $ Q $, where it vanishes.

The points of $ \Gamma $ for which a local barrier exists are known as regular points. If $ \Gamma $ consists exclusively of regular points, the solution of the Dirichlet problem obtained is continuous in $ D $ and assumes the specified values on $ \Gamma $. However, irregular points may also exist on $ \Gamma $. For instance, isolated points on $ \Gamma $ will be irregular in $ \mathbf R ^ {2} $, while the tip of a sufficiently thin cusp, entering inside $ D $, will be irregular in $ \mathbf R ^ {3} $. A consequence of the presence of irregular points is that the Dirichlet problem is not solvable for all continuous functions $ \phi $ on $ \Gamma $ or else that the solution is unstable with respect to changes in the boundary data [6].

A generalized solution method of the Dirichlet problem, introduced in 1924 by N. Wiener, satisfies the following conditions: a) it is applicable to all domains; and b) it yields the classical solution of the Dirichlet problem if such a solution exists. Let a domain $ D $ be the limit of a monotone increasing sequence of regular domains $ \{ D _ {n} \} $ such that $ D _ {n} \subset D _ {n + 1 } \subset D $ and such that any compact set $ K \subset D $ is contained in $ D _ {n} $ if $ n > n ( K) $. The generalized solution of the Dirichlet problem $ u $ is obtained as the limit of the sequence $ \{ u _ {n} \} $ of solutions of the Dirichlet problem for the domains $ D _ {n} $ and a boundary function $ \phi $ continuously extended inside $ D $. The solution $ u $ is independent of the choice of the exhausting sequence $ \{ D _ {n} \} $ and of the manner of continuous extension of $ \phi $ inside $ D $.

A generalized solution of the Dirichlet problem may also be based on Perron's method. Let $ {\overline{H}\; } _ \phi $ be the lower envelope of the family of all superharmonic functions $ v $ that satisfy the condition

$$ \lim\limits \inf v ( x) \geq \phi ( x _ {0} ) ,\ \ x \in D , x \rightarrow x _ {0} ; \ {\underline{H} } {} _ \phi = {\overline{H}\; } _ {- \phi } , $$

on $ \Gamma $. The inequality $ \underline{H} {} _ \phi \leq \overline{H}\; _ \phi $ applies to all domains $ D $ and functions $ \phi $. If $ \underline{H} {} _ \phi = \overline{H}\; _ \phi = u $, the function $ u $ is harmonic. It is known as the generalized solution of the Dirichlet problem, while the boundary function $ \phi $ is called resolutive. Any continuous function $ \phi $ is resolutive, and the behaviour of the generalized solution $ u $ at a point $ x _ {0} \in \Gamma $ will depend on whether $ x _ {0} $ is regular or irregular.

The Wiener-generalized solution of the Dirichlet problem satisfies an integral representation (de la Vallée-Poussin formula):

$$ \tag{2 } u ( x) = \int\limits _ \Gamma \phi ( x _ {0} ) d \omega ( x , D \mid x _ {0} ) , $$

which is generalization of formula (1). Here, $ d \omega ( x , E ) $ is the harmonic measure of a set $ E \subset \Gamma $ at a point $ x $[5].

It follows that it is possible to consider the generalized Dirichlet problem for arbitrary boundary functions $ \phi $, and the boundary condition must be satisfied in some weaker form only. For instance, if $ D $ is a domain in $ \mathbf R ^ {2} $ with sufficiently smooth boundary $ \Gamma $ and if the boundary function $ \phi $ has only discontinuity points of the first kind, the boundary condition need be satisfied at the continuity points of $ \phi $ only; the solution must be bounded at the discontinuity points in order to ensure uniqueness of the solution. N.N. Luzin generalized the Dirichlet problem for an arbitrary measurable almost-everywhere finite boundary function $ \phi $ on $ \Gamma $. A possible boundary condition would be for the boundary values of the solution along the normal to $ \Gamma $ to exist and to coincide with $ \phi $ almost-everywhere on $ \Gamma $.

The Dirichlet problem for the general second-order elliptic equation

$$ \tag{3 } \sum _ {i , j = 0 } ^ { n } a _ {ij} ( X) \frac{\partial ^ {2} u }{\partial x _ {i} \partial x _ {j} } + \sum _ {i = 0 } ^ { n } b _ {i} ( X) \frac{\partial u }{\partial x _ {i} } + c ( X) u = \ f ( X) $$

is a Fredholm problem. The solution sought must be regular in the domain and must assume given values on the boundary. The above methods for studying the Dirichlet problem for harmonic functions have also been generalized to equation (3).

For uniformly elliptic systems the Dirichlet problem may not merely prove to be a non-Fredholm problem, but may have infinitely many linearly independent solutions [8].

The Dirichlet problem is also studied for certain non-elliptic equations or degenerate equations. In such cases the Dirichlet problem sometimes turns out to be ill-posed.

References

[1]	P.G.L. Dirichlet, Abh. Königlich. Preuss. Akad. Wiss. (1850) pp. 99–116
[2]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) MR0284700 Zbl 0198.14101
[3]	R. Courant, "The Dirichlet priciple, conformal mapping and minimal surfaces" , Interscience (1950) (With appendix by M. Schiffer: Some recent developments in the theory of conformal mapping)
[4]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654
[5]	M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) MR0106366 Zbl 0084.30903
[6]	M.V. Keldysh, "On the solvability and stability of Dirichlet's problem" Uspekhi Mat. Nauk , 8 (1941) pp. 171–231 (In Russian)
[7]	M.V. Keldysh, "On certain classes of degeneration of equations of elliptic type on the boundary of a domain" Dokl. Akad. Nauk. SSSR , 77 (1951) pp. 181–183
[8]	A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian) MR0226183 Zbl 0167.09401

Comments

The trace on a set $ \Gamma $ of a function $ F $ defined on a domain $ D $, $ \Gamma \subset D $, is the function $ \phi $ on $ \Gamma $ that is the restriction to $ \Gamma $ of $ F $, i.e. $ \phi = F \mid _ \Gamma $.

The notation $ \Gamma \in H ^ {( 1 , \lambda ) } $, used in the article above, is not standard. It appears in [2] and means: "the boundary G belongs to the family of sets that can locally be represented by functions having Hölder continuous first derivatives" (cf. Hölder condition).

An important tool for solving the Dirichlet problem for domains in $ \mathbf R ^ {2} $ is conformal mapping, since in $ \mathbf R ^ {2} $ the theory of harmonic functions is closely related with the theory of analytic functions of a complex variable.

Sometimes solutions can be found in the form of Fourier series.

A weak solution of the Dirichlet problem

$$ - \Delta u = f $$

with homogeneous boundary data can be defined as an element $ u \in H _ {0} ^ {1} ( D) $ such that

$$ \int\limits _ { D } \nabla u \cdot \nabla v d x = \ \int\limits _ { D } f v d x $$

for any $ v \in H _ {0} ^ {1} ( D) $, $ f $ being given in $ H ^ {-} 1 ( D) $( all symbols are standard). The definition can be easily extended to more general cases. In particular, the Laplace operator can be replaced by any elliptic operator $ L $ in divergence form. If $ L $ is uniformly elliptic (i.e. satisfies a coerciveness inequality) a fundamental result for establishing existence and uniqueness of a weak solution is the Lax–Milgram theorem.

References

[a1]	O.D. Kellogg, "Foundations of potential theory" , Springer (1929) (Re-issue: Springer, 1967) MR0222317 MR1522134 Zbl 0152.31301 Zbl 0053.07301
[a2]	P.R. Garabedian, "Partial differential equations" , Wiley (1964) MR0162045 Zbl 0124.30501
[a3]	F. Riesz, B. Szökevalfi-Nagy, "Leçons d'analyse fonctionelle" , Gauthier-Villars (1968)
[a4]	A. Friedman, "Partial differential equations" , Holt, Rinehart & Winston (1969) MR0445088 Zbl 0224.35002
[a5]	O.A. Ladyzhenskaya, N.N. Ural'tseva, "Equations aux dérivées partielles de type elliptique" , Dunod (1969) (Translated from Russian) Zbl 0164.13001
[a6]	K. Yosida, "Functional analysis" , Springer (1980) MR0617913 Zbl 0435.46002
[a7]	S. Lang, "Complex analysis" , Springer (1985) MR0788885 Zbl 0562.30001
[a8]	D. Gilbar, "Elliptic partial differential equations of second order" , Springer (1983)
[a9]	L.L. Helms, "Introduction to potential theory" , Wiley (Interscience) (1969)
[a10]	C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972) MR0419799 Zbl 0248.31011