View source for Differential equation, partial, oblique derivatives

A linear boundary value problem for elliptic equations of the second order. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d0320001.png" /> be a domain of a real Euclidean space with Cartesian coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d0320002.png" />, the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d0320003.png" /> of which is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d0320004.png" />-dimensional Lyapunov hypersurface (cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]). Let in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d0320005.png" /> a linear differential equation of the second order be given:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d0320006.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

with real coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d0320007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d0320008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d0320009.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200010.png" /> which satisfy a Hölder condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200011.png" /> and, in addition, let the equation be uniformly elliptic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200012.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200013.png" /> be a real continuous vector defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200014.png" /> which does not vanish anywhere. A problem with oblique derivatives is formulated as follows: To find a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200015.png" /> of equation (1) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200016.png" /> for which the limit

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200017.png" /></td> </tr></table>

exists at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200018.png" />, and this limit should coincide with a given continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200019.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200020.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

In the boundary condition (2) it may be assumed, without loss of generality, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200022.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200023.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200024.png" /></td> </tr></table>

If the conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

are satisfied, then the homogeneous boundary value problem

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

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, [[#References|[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|finite-difference calculus]].

If condition (4) is met, it means that the problem (1), (2) is a Fredholm problem, i.e.: a) the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200028.png" /> of the space of solutions of the homogeneous problem (5) is finite; and b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200029.png" />, the problem (1), (2) is always solvable, and the solution in unique; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200030.png" />, there exists a space of linear functionals the vanishing of which on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200032.png" /> is a necessary and a sufficient condition for solutions of the problem (1), (2) to exist; moreover, the dimension of this space is also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200033.png" />. The problem (1), (2) can be a non-Fredholm problem only if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200034.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200035.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200036.png" /> is non-empty. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200037.png" />, on the assumption that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200038.png" /></td> </tr></table>

(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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200039.png" /> of the space of solutions of the homogeneous problem (5) is finite; b) the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200040.png" /> of the space of linear functionals the vanishing of which on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200042.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200043.png" />, is given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200044.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200045.png" /> is the increment of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200046.png" /> during one traversal of the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200047.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200048.png" /> in the positive direction. In the case here considered the problem (1), (2) is a Fredholm problem only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200049.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200050.png" /> characterizes the rotation of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200051.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200052.png" /> is a uniformly elliptic system, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200054.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200055.png" />-component vectors, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200058.png" /> are quadratic matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200059.png" /> and the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200060.png" /> satisfy the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200061.png" /></td> </tr></table>

in defining the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200062.png" /> in the boundary condition (2) of the problem with oblique derivatives, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200063.png" /> should be understood to mean square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200064.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200065.png" /> should be understood as a vector with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200066.png" /> components.

The problem with oblique derivatives is Noetherian for broad classes of uniformly elliptic systems and operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200067.png" />. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200071.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200072.png" /> is the unit (diagonal) matrix, the problem (1), (2) is Noetherian if the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200073.png" /> holds everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200074.png" />. If this condition is satisfied, the index of the problem (1), (2) is computed by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200075.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200076.png" /> is the increment of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200077.png" /> resulting from one traversal of the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200078.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200079.png" /> in the positive direction.

The problem with oblique derivatives for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032000/d03200080.png" /> has been intensively studied in the 1960s [[#References|[1]]].

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



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Gilbarg,   N.S. Trudinger,   "Elliptic partial differential equations of second order" , Springer  (1977)</TD></TR></table>