Boundary value problem, elliptic equations

The problem of finding a solution $ u $, regular in a domain $ D $, to an elliptic equation

$$ \tag{1 } \sum _ {i, k = 0 } ^ { n } a _ {ik} \frac{\partial ^ {2} u }{\partial x _ {i} \partial x _ {k} } + \sum _ {i = 0 } ^ { n } b _ {i} \frac{\partial u }{\partial x _ {i} } + cu = f, $$

which satisfies certain additional conditions on the boundary $ \Gamma $ of $ D $. Here $ a _ {ik} $, $ b _ {i} $, $ c $ and $ f $ are given functions on $ D $.

The classical boundary value problems are special cases of the following problem: Find a solution to equation (1), regular in a domain $ D $ and satisfying on $ \Gamma $

$$ \tag{2 } a \frac{du }{dl } + bu = g $$

where $ d/dl $ denotes differentiation in some direction, and $ a, b $ and $ g $ are given continuous functions on $ \Gamma $ with $ | a | + | b | > 0 $ everywhere on $ \Gamma $( see [1]).

Putting $ a = 0, b = 1 $, one obtains the Dirichlet problem; with $ b = 0, a = 1 $ one has a problem with oblique derivative (see Differential equation, partial, oblique derivatives), which becomes a Neumann problem if $ l $ is the direction of the conormal. If $ \Gamma = \overline \Gamma \; _ {1} \cup \overline \Gamma \; _ {2} $, where $ \Gamma _ {1} $ and $ \Gamma _ {2} $ are disjoint open subsets of $ \Gamma $, and $ \overline \Gamma \; _ {1} \cap \overline \Gamma \; _ {2} $ is either empty or an $ (n - 2) $- dimensional manifold, with $ a = 1 $, $ b = 0 $ on $ \Gamma _ {1} $, $ a = 0 $, $ b = 1 $ on $ \Gamma _ {2} $, one obtains a mixed problem.

Problem (2) has been studied for elliptic equations in two independent variables (see [2]). Fairly complete investigations have been made of the Dirichlet problem for elliptic equations in any finite number of independent variables (see [1], [3], [4]) and the problem with oblique derivative in case the direction $ l $ is not contained in a tangent plane to $ \Gamma $ at any point of $ \Gamma $. In that case the problem with oblique derivative is a Fredholm problem and the solution is smooth to the same order as the field of directions $ l $ and the function $ g $( see [1]). The case in which $ l $ lies in a tangent plane to $ \Gamma $ at certain points of $ \Gamma $ has been studied (see [3]). The local properties of solutions to the problem with oblique derivative have been investigated (see [5]). At points where the field $ l $ lies in a tangent plane to $ \Gamma $, the solution of the problem is less smooth than $ l $ and $ g $. This has been used as a basis for investigating the problem in a generalized setting (see [7], [8]).

Consider the following boundary problem for harmonic functions regular in the unit ball $ \Sigma \subset \mathbf R ^ {3} $:

$$ au _ {x} + bu _ {y} + cu _ {z} = g; $$

let $ K $ be the set of points of the unit sphere $ S $ at which the function $ \omega = ax + by + cz $ vanishes. The vector field $ P (a, b, c) $ lies in a tangent plane to $ S $ at the points of $ K $. Suppose in addition that $ K $ is the union of a finite number of disjoint curves; let $ K ^ {+} $ be the subset of $ K $ consisting of those points at which $ \mathop{\rm grad} \omega $ makes an acute angle with the projection of the field $ P $ on $ S $, and let $ K ^ {-} $ be the remaining part of $ K $. A generalized formulation of the problem is obtained when the values of $ u $ are also prescribed on $ K ^ {+} $, whereas on $ K ^ {-} $ the solution $ u $ is allowed to have integrable singularities. If $ K ^ {-} $ is empty, the solution to the generalized problem may be made arbitrarily smooth by increasing the smoothness of the additional data of the problem. Generally speaking, a solution to the mixed problem on the set $ \Gamma _ {0} = \overline \Gamma \; _ {1} \cap \overline \Gamma \; _ {2} $ has singularities (see [1]). In order to eliminate such singularities on $ \Gamma _ {0} $, one must impose additional conditions on the data (see [11]).

A large category of boundary value problems is constituted by what are known as problems with free boundaries. In these problems one must find not only a solution of equation (1), but also the domain in which it is regular. The boundary $ \Gamma $ of the domain is unknown, but two boundary conditions must be satisfied on it. An example of this type of problem is the problem of wave motions of an ideal fluid: Find a harmonic function $ u $, regular in some domain $ D $, where part of the boundary, $ \Gamma _ {1} $ say, is known and the normal derivative $ \partial u/ \partial n $ is given on $ \Gamma _ {1} $; the other part of the boundary, $ \Gamma _ {2} $, is unknown and on it one gives two boundary conditions:

$$ \frac{\partial u }{\partial n } = 0,\ \ u _ {x} ^ {2} + u _ {y} ^ {2} + u _ {z} ^ {2} = q (x, y, z), $$

where $ q > 0 $ is a given function.

For harmonic functions of two independent variables, one uses conformal mapping (see [12], [13], [14]). See also Differential equation, partial, free boundaries.

The following problem has been investigated: Find a harmonic function $ u $, regular in a domain $ D $ and satisfying the condition

$$ | \mathop{\rm grad} u | ^ {2} = q, $$

where $ q > 0 $ is a given function, on the boundary $ \Gamma $. There is a complete solution of this problem for harmonic functions of two independent variables (see [14]).

Given an equation $ Lu = f $, where $ L $ is an operator of order $ 2m $, uniformly elliptic in the closure $ \overline{D}\; $ of a domain $ D $, consider the problem of determining a solution $ u $, regular in $ D $ and satisfying on the boundary $ \Gamma $ of $ D $ the conditions

$$ \tag{3 } B _ {j} u = \Phi _ {j} ,\ \ j = 1 \dots m, $$

where $ B _ {j} (x, D), j = 1 \dots m $, are differential operators satisfying the following complementarity condition.

Let $ L ^ \prime (x, \partial / \partial x _ {1} \dots \partial / \partial x _ {n + 1 } ) $ be the principal part of $ L $, let $ B _ {j} ^ { \prime } $ be the principal part of $ B _ {j} $, $ n $ the normal to $ \Gamma $ at a point $ x $ and $ \lambda \neq 0 $ an arbitrary vector parallel to $ \Gamma $. Let $ \tau _ {k} ^ {+} ( \lambda ) $ denote the roots of $ L ^ \prime (x, \lambda + \tau n) $ with positive imaginary parts. The polynomials $ B _ {j} ^ { \prime } (x, \lambda + \tau n) $, $ j = 1 \dots m $, as functions of $ \tau $, must be linearly independent modulo the polynomial $ \prod _ {k = 1 } ^ {m} ( \tau - \tau _ {k} ^ {+} ( \lambda )) $. In this case, too, the problem is normally solvable. Violation of the complementarity condition may entail an essential change in the nature of the problem (see [17]).

Problem (2) is a special case of problem (3). For problem (2) with $ a \equiv 1 $, the complementarity condition is equivalent to the condition that there be no point on the boundary of the domain at which the direction $ l $ lies in a tangent plane to the boundary.

Another particular case of problem (3) is the boundary value problem

$$ \frac{\partial ^ {j} u }{\partial n ^ {j} } = \Phi _ {j} ,\ \ j = 0 \dots m - 1, $$

which is an analogue, to some extent, of the Dirichlet problem for higher-order elliptic equations.

The boundary value problem has been studied for the poly-harmonic equation $ \Delta ^ {k} u = 0 $ when the boundary of the domain consists of manifolds of different dimensions (see [15]).

In investigations of boundary value problems for non-linear equations (e.g. the Dirichlet and Neumann problems), much importance attaches to a priori estimates, various fixed-point principles (see [17], [18]) and the generalization of Morse theory to the infinite-dimensional case (see [19]).

References

[1]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) MR0284700 Zbl 0198.14101
[2]	I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian) MR0152665 MR0150320 MR0138774 Zbl 0127.03505 Zbl 0100.07603
[3]	A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian) MR0226183 Zbl 0167.09401
[4]	M.V. Keldysh, "On the solvability and stability of the Dirichlet problem" Uspekhi Mat. Nauk : 8 (1941) pp. 171–231 (In Russian) Zbl 0179.43901
[5]	L. Hörmander, "Pseudo-differential operators and non-elliptic boundary value problems" Ann. of Math. , 83 : 1 (1966) pp. 129–209 MR233064
[6]	R.L. Borrelli, "The singular, second order oblique derivative problem" J. Math. and Mech. , 16 : 1 (1966) pp. 51–81 MR0203217 Zbl 0143.14603
[7]	Yu.V. Egorov, V.A. Kondrat'ev, "The oblique derivative problem" Mat. Sb. , 78 : 1 (1969) pp. 148–176 (In Russian) MR0237953 Zbl 0186.43202 Zbl 0165.12202
[8]	V.G. Maz'ya, "The degenerate problem with oblique derivative" Mat. Sb. , 87 : 3 (1972) pp. 417–453 (In Russian)
[9]	A. Yanushauskas, Dokl. Akad. Nauk SSSR , 164 : 4 (1965) pp. 753–755
[10]	M.I. Vishik, G.I. Eskin, "Sobolev–Slobodinsky spaces of variable order with weighted norm, and their applications to mixed boundary value problems" Sibirsk. Mat. Zh. , 9 : 5 (1968) pp. 973–997 (In Russian)
[11]	G. Giraud, Ann. Soc. Math. Polon. , 12 (1934) pp. 35–54
[12]	M.A. Lavrent'ev, "Variational methods for boundary value problems for systems of elliptic equations" , Noordhoff (1963) (Translated from Russian) Zbl 0121.06701
[13]	A.I. Nekrasov, "Exact theory of waves of stationary type on the surface of a heavy fluid" , Collected works , 1 , Moscow (1961) (In Russian)
[14]	F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian) MR0198152 Zbl 0141.08001
[15]	S.L. Sobolev, Mat. Sb. , 2 : 3 (1937) pp. 465–499
[16]	S. Agmon, A. Douglis, L. Nirenberg, "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, II." Comm. Pure Appl. Math. , 17 (1964) pp. 35–92 MR162050
[17]	J. Schauder, Math. Z. , 33 (1931) pp. 602–640
[18]	J. Leray, J. Schauder, Ann. Sci. Ecole Norm. Sup. Ser. 3 , 51 (1934) pp. 45–78
[19]	R.S. Palais, "Morse theory on Hilbert manifolds" Topology , 2 : 4 (1963) pp. 299–340 MR0158410 Zbl 0122.10702

References

[a1]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654
[a2]	P.R. Garabedian, "Partial differential equations" , Wiley (1964) MR0162045 Zbl 0124.30501
[a3]	A. Friedman, "Partial differential equations" , Holt, Rinehart & Winston (1969) MR0445088 Zbl 0224.35002