View source for Boundary value problem, elliptic equations

The problem of finding a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173501.png" />, regular in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173502.png" />, to an elliptic equation

<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/b/b017/b017350/b0173503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

which satisfies certain additional conditions on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173504.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173505.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173507.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173509.png" /> are given functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735010.png" />.

The classical boundary value problems are special cases of the following problem: Find a solution to equation (1), regular in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735011.png" /> and satisfying on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735012.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/b/b017/b017350/b01735013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735014.png" /> denotes differentiation in some direction, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735016.png" /> are given continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735017.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735018.png" /> everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735019.png" /> (see [[#References|[1]]]).

Putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735020.png" />, one obtains the [[Dirichlet problem|Dirichlet problem]]; with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735021.png" /> one has a problem with oblique derivative (see [[Differential equation, partial, oblique derivatives|Differential equation, partial, oblique derivatives]]), which becomes a [[Neumann problem(2)|Neumann problem]] if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735022.png" /> is the direction of the conormal. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735025.png" /> are disjoint open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735026.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735027.png" /> is either empty or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735028.png" />-dimensional manifold, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735030.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735033.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735034.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735035.png" /> is not contained in a tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735036.png" /> at any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735037.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735038.png" /> and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735039.png" /> (see [[#References|[1]]]). The case in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735040.png" /> lies in a tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735041.png" /> at certain points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735042.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735043.png" /> lies in a tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735044.png" />, the solution of the problem is less smooth than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735046.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735047.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/b/b017/b017350/b01735048.png" /></td> </tr></table>

let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735049.png" /> be the set of points of the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735050.png" /> at which the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735051.png" /> vanishes. The vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735052.png" /> lies in a tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735053.png" /> at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735054.png" />. Suppose in addition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735055.png" /> is the union of a finite number of disjoint curves; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735056.png" /> be the subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735057.png" /> consisting of those points at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735058.png" /> makes an acute angle with the projection of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735059.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735060.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735061.png" /> be the remaining part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735062.png" />. A generalized formulation of the problem is obtained when the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735063.png" /> are also prescribed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735064.png" />, whereas on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735065.png" /> the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735066.png" /> is allowed to have integrable singularities. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735067.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735068.png" /> has singularities (see [[#References|[1]]]). In order to eliminate such singularities on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735069.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735070.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735071.png" />, regular in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735072.png" />, where part of the boundary, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735073.png" /> say, is known and the normal derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735074.png" /> is given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735075.png" />; the other part of the boundary, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735076.png" />, is unknown and on it one gives two boundary 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/b/b017/b017350/b01735077.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735078.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735079.png" />, regular in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735080.png" /> and satisfying 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/b/b017/b017350/b01735081.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735082.png" /> is a given function, on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735083.png" />. There is a complete solution of this problem for harmonic functions of two independent variables (see [[#References|[14]]]).

Given an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735084.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735085.png" /> is an operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735086.png" />, uniformly elliptic in the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735087.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735088.png" />, consider the problem of determining a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735089.png" />, regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735090.png" /> and satisfying on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735091.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735092.png" /> 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/b/b017/b017350/b01735093.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735094.png" />, are differential operators satisfying the following complementarity condition.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735095.png" /> be the principal part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735096.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735097.png" /> be the principal part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735099.png" /> the normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350100.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350102.png" /> an arbitrary vector parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350103.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350104.png" /> denote the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350105.png" /> with positive imaginary parts. The polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350107.png" />, as functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350108.png" />, must be linearly independent modulo the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350109.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350110.png" />, the complementarity condition is equivalent to the condition that there be no point on the boundary of the domain at which the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350111.png" /> lies in a tangent plane to the boundary.

Another particular case of problem (3) is the 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/b/b017/b017350/b017350112.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350113.png" /> 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)</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">  A.V. Bitsadze,   "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)</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)</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</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</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)</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)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A. Yanushauskas,   ''Dokl. Akad. Nauk SSSR'' , '''164''' :  4  (1965)  pp. 753–755</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)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  G. Giraud,   ''Ann. Soc. Math. Polon.'' , '''12'''  (1934)  pp. 35–54</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)</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)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  F.D. Gakhov,   "Boundary value problems" , Pergamon  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  S.L. Sobolev,   ''Mat. Sb.'' , '''2''' :  3  (1937)  pp. 465–499</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</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top">  J. Schauder,   ''Math. Z.'' , '''33'''  (1931)  pp. 602–640</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</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</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)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Garabedian,   "Partial differential equations" , Wiley  (1964)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations" , Holt, Rinehart &amp; Winston  (1969)</TD></TR></table>