Boundary value problem, partial differential equations

The problem of determining in some region $ D $ with points $ x = (x _ {1} \dots x _ {n} ) $ a solution $ u (x) $ to an equation

$$ \tag{1 } (Lu) (x) = f (x),\ \ x \in D, $$

which satisfies certain boundary conditions on the boundary $ S $ of $ D $( or on a part of it):

$$ \tag{2 } (Bu) (y) = \phi (y),\ \ y \in S. $$

As a rule, the boundary conditions relate the boundary values of the solution to its derivatives up to a certain order, i.e. $ B $ is a differential operator. However, boundary conditions of other types also occur.

Given a differential equation, the question whether a specific boundary value problem should be studied is frequently settled by the concept of what is known as well-posedness. Namely, a boundary value problem is well-posed if it is solvable, and if its solution is unique and depends continuously on the data of the problem. Different types of differential equations require different well-posed boundary value problems; and conversely, well-posed boundary value problems may sometimes serve as a basis for the classification of types of differential equations.

A boundary value problem is said to be linear if the operators $ L $ and $ B $ are linear, and homogeneous if $ f $ and $ \phi $ in (1), (2) vanish. A linear boundary value problem is said to be Noetherian if: a) the homogeneous problem has a finite number $ k $ of linearly independent solutions; b) the inhomogeneous problem is solvable if and only if $ f $ and $ \phi $ satisfy $ l $ linearly independent orthogonality conditions; and c) on the assumption that the problem is uniquely solvable, the solution depends continuously on $ f $ and $ \phi $.

If $ k = l $, the problem is called a Fredholm problem. The difference $ k - l $ defines the index of the problem.

A broad range of boundary value problems for linear second-order differential equations

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

$$ = \ f (x) $$

falls into the category of the Poincaré problem. In this type of problem the boundary conditions are prescribed on the entire boundary, which is assumed to be an $ (n - 1) $- dimensional manifold, and the boundary operator $ B $ in (2) has the form

$$ \tag{4 } (Bu) (y) = \ \sum _ {i = 1 } ^ { n } p _ {i} (y) \frac{\partial u }{\partial x _ {i} } + p (y) u = \phi (y),\ \ y \in S. $$

The Poincaré problem for bounded regions $ D $ with sufficiently-smooth boundaries have been thoroughly investigated in the case of uniformly-elliptic operators $ L $.

On the assumption that the coefficients of the operators $ L $ and $ B $ in (3), (4) are sufficiently smooth, and that the boundary of $ D $ is sufficiently smooth, the Poincaré problem is Noetherian if $ n = 2 $, $ \sum p _ {i} ^ {2} > 0 $, and is a Fredholm problem if $ n > 2 $, $ \sum p _ {i} ^ {2} > 0 $ and the vector $ \mathbf p = (p _ {1} \dots p _ {n} ) $ is not tangent to $ S $. Investigations of the Poincaré problem in the two-dimensional case make extensive use of the theory of functions of a complex variable (see Boundary value problem, complex-variable methods).

Boundary conditions for the general elliptic equation of order $ 2m $

$$ \tag{5 } Lu = \ \sum _ {| \alpha | \leq 2m } a _ \alpha (x) \frac{\partial ^ {| \alpha | } u }{\partial x ^ \alpha } = f (x),\ \ x \in D, $$

may be prescribed in terms of linear differential operators

$$ \tag{6 } B _ {j} u = \ \sum _ {| \alpha | \leq m _ {j} } b _ {j \alpha } (y) \frac{\partial ^ {| \alpha | } u }{\partial x ^ \alpha } = \phi _ {j} (y),\ \ y \in S,\ \ 1 \leq j \leq m, $$

of order $ m _ {j} < 2m $, with coefficients defined on the boundary $ S $ of $ D $. Here $ \alpha = ( \alpha _ {1} \dots \alpha _ {n} ) $, $ \alpha _ {i} $ is a non-negative integer, $ | \alpha | = \sum _ {i = 1 } ^ {n} \alpha _ {i} $ and

$$ \frac{\partial ^ {| \alpha | } }{\partial x ^ \alpha } = \ \frac{\partial ^ {| \alpha | } }{\partial x _ {1} ^ {\alpha _ {1} } \dots \partial _ {x _ {n} } ^ {\alpha _ {n} } } . $$

If the operators $ L $ and $ B _ {j} $ satisfy the so-called complementarity conditions, then bounds for the derivatives of order $ 2m $ of the function $ u $ satisfying the boundary conditions can be formulated (relative to an appropriate norm) in terms of the norm of $ f $ in (5) and suitable norms of the boundary functions $ \phi _ {j} $ in (6). Boundary value problems of this type are said to be coercive.

Another type of boundary value problems are known as mixed problems (cf. Mixed problem), in which different boundary conditions are prescribed on adjacent sections of the boundary.

Characteristic for boundary value problems of differential equations that are uniformly elliptic in $ \overline{D}\; $ is that the boundary conditions are prescribed on the entire boundary. If the operator $ L $ in (3) is elliptic in the interior of the region and parabolically degenerates on a section $ S _ {0} \subseteq S $, then, depending on the type of degeneracy, $ S _ {0} $ can be eliminated from the specification of the boundary conditions. See also Boundary value problem, elliptic equations.

If equation (3) is not elliptic, part of the boundary may usually be eliminated from the specification of the boundary conditions. For example, given the simplest equation of parabolic type, the heat equation

$$ Lu = \ \frac{\partial ^ {2} u }{\partial x ^ {2} } - \frac{\partial u }{\partial t } = 0, $$

the Dirichlet problem in the domain $ D $ bounded by the straight lines $ x = \pm 1, $ $ t = \pm 1 $ amounts to the specification of boundary values $ u $ on the intervals $ \{ t = \pm 1, -1 \leq x \leq +1 \} $ and $ \{ x = \pm 1, -1 \leq t \leq 1 \} $.

Boundary value problems for equations of mixed elliptic-hyperbolic type are formulated in a special manner.

Boundary value problems occupy an important position in the theory of analytic functions. Let $ S $ be a piecewise-smooth curve in the plane, i.e. a union of a finite number of simple oriented arcs. The linear conjugation problem consists in determining a function $ \phi (z) $, analytic outside $ S $, having limit values $ \phi ^ \pm (t) $, $ t \in S $, from both sides of $ S $ and satisfying the boundary condition

$$ \phi ^ {+} (t) - G (t) \phi ^ {-} (t) = g (t),\ \ t \in S, $$

where $ G (t) $ and $ g (t) $ are given functions. The solution to this problem can be specified in explicit form with the help of representations of analytic functions by Cauchy-type integrals, subject to certain additional assumptions concerning the functions $ G $, $ g $ and the curve $ S $. See Boundary value problems of analytic function theory.

Various methods are available for the investigation of boundary value problems. The Schwarz alternating method, Poincaré's related balayage method and the Perron method are based on an application of the maximum principle. A solution using integral equations is based on various integral representations of the solutions. Among functional methods is the investigation of boundary value problems using a priori estimates. The theory of distributions is widely applied. In practical applications, various finite-difference methods are quite frequently used.

References

[1]	L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964) MR0163043 Zbl 0126.00207
[2]	A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian) MR0226183 Zbl 0167.09401
[3]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654
[4]	O.A. Ladyzhenskaya, "The boundary value problems of mathematical physics" , Springer (1985) (Translated from Russian) MR0793735 Zbl 0588.35003
[5]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) MR0284700 Zbl 0198.14101
[6]	N.I. Muskhelishvili, "Singular integral equations" , Wolters-Noordhoff (1972) (Translated from Russian) MR0355494 Zbl 0488.45002 Zbl 0174.16202 Zbl 0174.16201 Zbl 0103.07502 Zbl 0108.29203 Zbl 0051.33203 Zbl 0041.22601
[7]	A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) MR104888

Comments

Useful additional references are given below.

References

[a1]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 1–2 , Interscience (1953–1962) (Translated from German) MR1013360 MR0195654 MR1532618 MR0140802 MR0065391 MR0046886
[a2]	A. Friedman, "Partial differential equations" , Holt, Rinehart & Winston (1969) MR0445088 Zbl 0224.35002
[a3]	P.R. Garabedian, "Partial differential equations" , Wiley (1964) MR0162045 Zbl 0124.30501
[a4]	L. Hörmander, "Linear partial differential operators" , Springer (1963) pp. 29; 31 MR0161012 Zbl 0108.09301
[a5]	O.A. Ladyzhenskaya, N.N. Ural'tseva, "Linear and quasilinear elliptic equations" , Acad. Press (1968) (Translated from Russian) MR0244627 Zbl 0177.37404 Zbl 0164.13002
[a6]	J.L. Lions, E. Magenes, "Non-homogenous boundary value problems and applications" , 1–3 , Springer (1972) (Translated from French)
[a7]	M.H. Protter, H.F. Weinberger, "Maximum principles in differential equations" , Prentice-Hall (1967) MR0219861 Zbl 0549.35002
[a8]	G.E. Forsythe, W.R. Wasow, "Finite-difference methods for partial differential equations" , Wiley (1960) MR0130124 Zbl 0099.11103