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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 [[#References|[1]]]).

Putting  $  a = 0, b = 1 $, 
one obtains the [[Dirichlet problem|Dirichlet problem]]; with  $  b = 0, a = 1 $
one has a problem with oblique derivative (see [[Differential equation, partial, oblique derivatives|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|mixed problem]].

Problem (2) has been studied for elliptic equations in two independent variables (see [[#References|[2]]]). Fairly complete investigations have been made of the Dirichlet problem for elliptic equations in any finite number of independent variables (see [[#References|[1]]], [[#References|[3]]], [[#References|[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 [[#References|[1]]]). The case in which  $  l $
lies in a tangent plane to  $  \Gamma $
at certain points of  $  \Gamma $
has been studied (see [[#References|[3]]]). The local properties of solutions to the problem with oblique derivative have been investigated (see [[#References|[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 [[#References|[7]]], [[#References|[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 [[#References|[1]]]). In order to eliminate such singularities on  $  \Gamma _ {0} $, 
one must impose additional conditions on the data (see [[#References|[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 [[#References|[12]]], [[#References|[13]]], [[#References|[14]]]). See also [[Differential equation, partial, free boundaries|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 [[#References|[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 [[#References|[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 [[#References|[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 [[#References|[17]]], [[#References|[18]]]) and the generalization of Morse theory to the infinite-dimensional case (see [[#References|[19]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) {{MR|0284700}} {{ZBL|0198.14101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian) {{MR|0152665}} {{MR|0150320}} {{MR|0138774}} {{ZBL|0127.03505}} {{ZBL|0100.07603}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian) {{MR|0226183}} {{ZBL|0167.09401}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M.V. Keldysh, "On the solvability and stability of the Dirichlet problem" ''Uspekhi Mat. Nauk'' : 8 (1941) pp. 171–231 (In Russian) {{MR|}} {{ZBL|0179.43901}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> L. Hörmander, "Pseudo-differential operators and non-elliptic boundary value problems" ''Ann. of Math.'' , '''83''' : 1 (1966) pp. 129–209 {{MR|233064}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R.L. Borrelli, "The singular, second order oblique derivative problem" ''J. Math. and Mech.'' , '''16''' : 1 (1966) pp. 51–81 {{MR|0203217}} {{ZBL|0143.14603}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> Yu.V. Egorov, V.A. Kondrat'ev, "The oblique derivative problem" ''Mat. Sb.'' , '''78''' : 1 (1969) pp. 148–176 (In Russian) {{MR|0237953}} {{ZBL|0186.43202}} {{ZBL|0165.12202}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> V.G. Maz'ya, "The degenerate problem with oblique derivative" ''Mat. Sb.'' , '''87''' : 3 (1972) pp. 417–453 (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A. Yanushauskas, ''Dokl. Akad. Nauk SSSR'' , '''164''' : 4 (1965) pp. 753–755  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> 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)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> G. Giraud, ''Ann. Soc. Math. Polon.'' , '''12''' (1934) pp. 35–54  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> M.A. Lavrent'ev, "Variational methods for boundary value problems for systems of elliptic equations" , Noordhoff (1963) (Translated from Russian) {{MR|}} {{ZBL|0121.06701}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> A.I. Nekrasov, "Exact theory of waves of stationary type on the surface of a heavy fluid" , ''Collected works'' , '''1''' , Moscow (1961) (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian) {{MR|0198152}} {{ZBL|0141.08001}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> S.L. Sobolev, ''Mat. Sb.'' , '''2''' : 3 (1937) pp. 465–499  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> 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 {{MR|162050}} {{ZBL|}} </TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> J. Schauder, ''Math. Z.'' , '''33''' (1931) pp. 602–640  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> J. Leray, J. Schauder, ''Ann. Sci. Ecole Norm. Sup. Ser. 3'' , '''51''' (1934) pp. 45–78  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top"> R.S. Palais, "Morse theory on Hilbert manifolds" ''Topology'' , '''2''' : 4 (1963) pp. 299–340 {{MR|0158410}} {{ZBL|0122.10702}} </TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience (1965) (Translated from German) {{MR|0195654}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.R. Garabedian, "Partial differential equations" , Wiley (1964) {{MR|0162045}} {{ZBL|0124.30501}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Friedman, "Partial differential equations" , Holt, Rinehart &amp; Winston (1969) {{MR|0445088}} {{ZBL|0224.35002}} </TD></TR></table>