# 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 [1]).

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

#### References

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