Differential equation, partial, oblique derivatives

A linear boundary value problem for elliptic equations of the second order. Let $ D $ be a domain of a real Euclidean space with Cartesian coordinates $ x _ {1} \dots x _ {n} $, the boundary $ \partial D $ of which is an $ ( n - 1 ) $- dimensional Lyapunov hypersurface (cf. Lyapunov surfaces and curves). Let in $ D $ a linear differential equation of the second order be given:

$$ \tag{1 } L ( u) = \sum _ {i , j = 1 } ^ { n } a _ {ij} u _ {x _ {i} x _ {j} } + \sum _ {i = 1 } ^ { n } b _ {i} u _ {x _ {i} } + cu = F ( x) , $$

with real coefficients $ a _ {ij} $, $ b _ {i} $, $ c $, and $ F $ which satisfy a Hölder condition on $ D \cup \partial D $ and, in addition, let the equation be uniformly elliptic in $ D $. Let $ l = ( l _ {1} \dots l _ {n} ) $ be a real continuous vector defined on $ \partial D $ which does not vanish anywhere. A problem with oblique derivatives is formulated as follows: To find a solution $ u $ of equation (1) in $ D \cup \partial D $ for which the limit

$$ \lim\limits _ {\begin{array}{c} {x \rightarrow y } \\ {x \in D } \end{array} } [ l ( y) \mathop{\rm grad} _ {x} u ] = \lambda ( u) $$

exists at all points $ y \in \partial D $, and this limit should coincide with a given continuous function $ f $ on $ \partial D $:

$$ \tag{2 } \lambda ( u) = f ( y) ,\ y \in \partial D . $$

In the boundary condition (2) it may be assumed, without loss of generality, that $ l $ is a unit vector. The Neumann problem is a special case of the problem with oblique derivatives, when the left-hand side in the boundary condition (2) is identical with the derivative of the unknown solution with respect to the exterior unit normal $ \nu $:

$$ \frac{du}{d \nu } = f ( y) ,\ y \in \partial D . $$

If the conditions

$$ \tag{3 } c ( x) \leq 0 $$

and

$$ \tag{4 } \inf _ {y \in \partial D } ( \nu l ) > 0 $$

are satisfied, then the homogeneous boundary value problem

$$ \tag{5 } L ( u) = 0 ,\ \lambda ( u) = 0 $$

corresponding to the problem (1), (2) cannot have a solution other than a constant, by virtue of the Hopf and Zaremba–Giraud principles (see, for example, [1]). In particular, if strong inequality is realized in condition (3) in at least one point, the problem (1), (2) cannot have more than one solution. The problem of the existence of solutions of the problem (1), (2) is usually studied by the method of integral equations, by a priori estimates or by methods of finite-difference calculus.

If condition (4) is met, it means that the problem (1), (2) is a Fredholm problem, i.e.: a) the dimension $ \kappa _ {1} $ of the space of solutions of the homogeneous problem (5) is finite; and b) if $ \kappa _ {1} = 0 $, the problem (1), (2) is always solvable, and the solution in unique; if $ \kappa _ {1} > 0 $, there exists a space of linear functionals the vanishing of which on $ F $ and $ f $ is a necessary and a sufficient condition for solutions of the problem (1), (2) to exist; moreover, the dimension of this space is also $ \kappa _ {1} $. The problem (1), (2) can be a non-Fredholm problem only if the set $ M $ of points $ y $ for which $ ( \nu l) = 0 $ is non-empty. In particular, if $ n = 2 $, on the assumption that

$$ \sum _ {i , j = 1 } ^ { 2 } a _ {ij} u _ {x _ {i} x _ {j} } = \mathop{\rm div} \mathop{\rm grad} u ( x) $$

(which does not restrict the generality), the problem (1), (2) is reduced to an equivalent singular integral equation with Cauchy kernel, which means that the problem is Noetherian, i.e.: a) the dimension $ \kappa _ {1} $ of the space of solutions of the homogeneous problem (5) is finite; b) the dimension $ \kappa _ {2} $ of the space of linear functionals the vanishing of which on $ F $ and $ f $ is a necessary and sufficient condition for the solvability of the problem (1), (2), is also finite; and c) the index of the problem (1), (2), i.e. the difference $ \kappa _ {1} - \kappa _ {2} = \kappa $, is given by the formula

$$ \kappa = 2 ( p + 1 ) , $$

where $ 2 \pi p $ is the increment of $ \mathop{\rm arg} ( l _ {1} - il _ {2} ) $ during one traversal of the contour $ \partial D $ of $ D $ in the positive direction. In the case here considered the problem (1), (2) is a Fredholm problem only if $ p = - 1 $. The number $ p $ characterizes the rotation of the vector field $ ( l _ {1} , l _ {2} ) $. If $ ( l) $ is a uniformly elliptic system, i.e. if $ F $ and $ u $ are $ m $- component vectors, while $ a _ {ij} $, $ b _ {i} $ and $ c $ are quadratic matrices of order $ m $ and the matrices $ a _ {ij} $ satisfy the condition

$$ k _ {0} \left ( \sum _ {j = 1 } ^ { n } \alpha _ {j} ^ {2} \right ) ^ {m} \leq \ \left | \mathop{\rm det} \sum _ {i , j = 1 } ^ { n } a _ {ij} \alpha _ {i} \alpha _ {j} \right | \leq k _ {1} \left ( \sum _ {j = 1 } ^ { n } \alpha _ {j} ^ {2} \right ) ^ {m} , $$

in defining the operator $ \lambda ( u) $ in the boundary condition (2) of the problem with oblique derivatives, $ l _ {1} \dots l _ {n} $ should be understood to mean square matrices of order $ m $, while $ f $ should be understood as a vector with $ m $ components.

The problem with oblique derivatives is Noetherian for broad classes of uniformly elliptic systems and operators $ \lambda ( u) $. For instance, if $ n = 2 $ and $ a _ {ij} = 0 $, $ i \neq j $, $ a _ {ii} = E $ where $ E $ is the unit (diagonal) matrix, the problem (1), (2) is Noetherian if the condition $ \mathop{\rm det} ( l _ {1} + il _ {2} ) \neq 0 $ holds everywhere on $ \partial D $. If this condition is satisfied, the index of the problem (1), (2) is computed by the formula $ \kappa = 2 ( p + m ) $, where $ 2 \pi p $ is the increment of $ \mathop{\rm arg} \mathop{\rm det} ( l _ {1} - il _ {2} ) $ resulting from one traversal of the contour $ \partial D $ of $ D $ in the positive direction.

The problem with oblique derivatives for $ n \geq 3 $ has been intensively studied in the 1960s [1].

References

[1]	A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian)
[2]	I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian)
[3]	N.I. Muskhelishvili, "Singular integral equations" , Wolters-Noordhoff (1972) (Translated from Russian)
[4]	G. Bouligand, G. Giraud, P. Delens, "Le problème de la dérivée oblique en théorie du potentiel" , Hermann (1935)

References

[a1]	D. Gilbarg, N.S. Trudinger, "Elliptic partial differential equations of second order" , Springer (1977)