# 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 ).

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 ). Fairly complete investigations have been made of the Dirichlet problem for elliptic equations in any finite number of independent variables (see , , ) 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 ). The case in which $l$ lies in a tangent plane to $\Gamma$ at certain points of $\Gamma$ has been studied (see ). The local properties of solutions to the problem with oblique derivative have been investigated (see ). 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 , ).

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 ). In order to eliminate such singularities on $\Gamma _ {0}$, one must impose additional conditions on the data (see ).

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 , , ). 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 ).

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 ).

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 ).

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 , ) and the generalization of Morse theory to the infinite-dimensional case (see ).

How to Cite This Entry:
Boundary value problem, elliptic equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary_value_problem,_elliptic_equations&oldid=46132
This article was adapted from an original article by A.I. Yanushauskas (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article