Coercive boundary value problem

A boundary value problem satisfying a coerciveness inequality. Coercive boundary value problems for elliptic equations are sometimes called elliptic boundary value problems [4].

Let $ P ( \xi _{1} \dots \xi _{n} ) $ be a homogeneous polynomial of degree $ 2m $ so that

$$ \tag{1} P (D) u \ = \ f,\ \ D \ = \ \left ( \frac \partial {\partial x _ 1} \dots \frac \partial {\partial x _ n} \right ) , $$

is an elliptic equation of order $ 2m $. Consider the boundary value problem for equation (1) in the half-plane $ \mathbf R _ n^{+} $: $ \{ x _{n} > 0 \} $, with boundary conditions

$$ \tag{2} q _{j} (D) u \ = \ \phi _{j} ,\ \ j = 1 \dots m,\ \ \textrm{ for } \ x _{n} = 0, $$

where the $ q _{j} ( \xi _{1} \dots \xi _{n} ) $, $ j = 1 \dots m $, are homogeneous polynomials of degrees $ \mu _{j} $.

The problem (1), (2) is coercive in $ \mathbf R _ n^{+} $ if the orders of all the operators $ q _{j} $ with respect to $ \partial / \partial x _{n} $ are less than $ 2m $ and if the problem has no bounded solutions of the type

$$ \tag{3} u (x) \ = \ w (x _{n} ) \mathop{\rm exp}\nolimits \ i (x _{1} \xi _{1} + \dots + x _{ {n - 1}} \xi _{ {n - 1}} ), $$

$$ \xi^ \prime \ = \ ( \xi _{1} \dots \xi _{ {n - 1}} ) \ \neq \ 0. $$

As a polynomial in $ \tau $, $ P ( \xi^ \prime ,\ \tau ) $ has exactly $ m $ roots $ \tau _{1} \dots \tau _{m} $ in the upper half-plane $ \mathop{\rm Im}\nolimits \ \tau > 0 $. If these roots are distinct, the condition that problem (1), (2) has no bounded solutions of type (3) is equivalent to

$$ \tag{4} \left [ \prod _ {l < j} ( \tau _{j} - \tau _{l} ) \right ]^{-1} \mathop{\rm det}\nolimits (q _{j} ( \xi^ \prime ,\ \tau _{i} )) \ \neq \ 0,\ \ \xi^ \prime \neq 0. $$

This condition is sometimes called the complementarity condition.

The boundary value problem

$$ \tag{5} P (x,\ D) u \ = \ f,\ \ x \in G,\ \ q _{i} (x,\ D) u \ = \ \phi _{j} ,\ \ x \in \partial G, $$

$$ j \ = \ 1 \dots m, $$

for a linear elliptic equation of order $ 2m $ in a domain $ G $ is coercive at a point $ x _{0} $ on the boundary $ \Gamma $ of $ G $ if the problem

$$ P _{2m} (x _{0} ,\ D) u \ = \ f,\ \ q _ j^{0} (x _{0} ,\ D) u \ = \ \phi _{j} ,\ \ j = 1 \dots m, $$

where $ P _{2m} $ and $ q _ j^{(0)} $ are the homogeneous components of the highest degree of the corresponding polynomials, is coercive.

A good tool for the investigation of coercive boundary value problems is the method of boundary rectification. This method involves a homeomorphic mapping, defined in a neighbourhood $ U (X) $ of a point $ X $ on the boundary $ \Gamma $ of $ G $, with the property that the intersection $ U (X) \cap \Gamma $ is mapped onto some domain of the plane $ y _{n} = 0 $, while $ U (X) \cap G $ is mapped onto the half-plane $ \mathbf R _ n^{+} $ with points $ (y _{1} \dots y _{n} ) $. As a result, a coercive boundary value problem for an arbitrary domain $ G $ with sufficiently smooth boundary $ \Gamma $ is reduced to a problem for $ \mathbf R _ n^{+} $.

In the case of problem (5), condition (4) means that the characteristic direction of the system of differential operators

$$ q _{j} (x,\ D) u ,\ \ j = 1 \dots m, $$

defined in a neighbourhood of the points of $ \Gamma $, is not tangent to $ \Gamma $ at any point of $ \Gamma $. Under this condition, a solution $ u $ of problem (5) satisfies the coerciveness estimate

$$ \| u \| _{ {2m,\ G}} \ \leq \ C \ \left \{ \ \| f \| _{ {0,\ G}} + \sum _ {j = 1} ^ m \ \| \phi _{j} \| _{ {2m - \mu _{j} ,\ \Gamma}} + \| u \| _{ {0,\ G}} \right \} , $$

where the norm in the Sobolev space $ W _ 2^{2m} (G) $ is estimated in terms of the norm of $ u $ in $ L _{2} (G) $, the norm of $ f $ in $ L _{2} (G) $ and the norm of $ \phi _{j} $ in $ W _{2} ^ {2m - \mu _ j} ( \Gamma ) $, $ j = 1 \dots m $, $ \mu _{j} $ being the order of the operator $ q _{j} $.

The concept of a coercive boundary value problem may also be generalized to elliptic systems. The methods of investigation and the results valid for coercive boundary value problems for a single elliptic equation may be generalized to coercive boundary value problems for elliptic systems [3]. The investigation of coercive boundary value problems for elliptic equations is often reduced, e.g. by the parametrix method, to the investigation of a system of singular integral equations. This is a Noetherian system of integral equations (see [2], [6]), and condition (4) — or, in the case of a system, its analogue — guarantees normality of the system. A coercive boundary value problem is always Noetherian, i.e. the inhomogeneous problem is solvable subject to the observance of a finite number of orthogonality conditions imposed on the right-hand sides of the (system of) equations and on the functions that appear in the boundary conditions, while the homogeneous problem has finitely many linearly independent solutions.

One example of a problem which is coercive for any elliptic equation is the Dirichlet problem. In this case

$$ q _{j} (x,\ D) \ = \ \frac{\partial ^ {j - 1}}{\partial n ^ {j - 1}} ,\ \ j = 1 \dots m, $$

where $ \partial / \partial n $ denotes differentiation in the direction of the conormal for the elliptic operator in question. However, it is not true that the Dirichlet problem is a coercive boundary value problem for every elliptic system of equations. An example of systems for which the Dirichlet problem is not coercive is the system of two equations known in complex notation as the Bitsadze equation.

References

[1]	A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian)
[2]	Ya.B. Lopatinskii, "On a method of reducing problems for a system of differential equations of elliptic type to regular integral equations" Amer. Math. Soc. Transl. (2) , 89 (1970) pp. 149–183 Ukrain. Mat. Zh. , 5 : 2 (1953) pp. 123–151
[3]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[4]	L. Hörmander, "Linear partial differential operators" , Springer (1963)
[5]	L. Hörmander, Matematika , 4 : 4 (1960) pp. 37–73
[6]	Z.Ya. Shapiro, "The first boundary value problem for an elliptic system of differential equations" Mat. Sb. , 28 (1951) pp. 55–78 (In Russian)

Comments

The complementarity condition is also called the complementing, complementary or admissibility condition. A Noetherian system of integral equations is, in Western literature, now more often known as a system associated with a Fredholm operator, or a Fredholm system.

Other basic references are [a1], [a2]. Additional important references are [a3], [a4].

References

[a1]	S. Agmon, A. Douglis, L. Nirenberg, "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions" Comm. Pure Appl. Math. , 12 (1959) pp. 623–727
[a2]	S. Agmon, A. Douglis, L. Nirenberg, "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II" Comm. Pure Appl. Math. , 17 (1964) pp. 35–92
[a3]	J.L. Lions, E. Magenes, "Non-homogenous boundary value problems and applications" , 1–2 , Springer (1972) (Translated from French)
[a4]	A. Friedman, "Partial differential equations" , Holt, Rinehart & Winston (1969)