ON SOLVING THE PLATEAU PROBLEM IN PARAMETRIC FORM BARUCH CAHLON

This paper presents a numerical method for finding the solution of Plateau’s problem in parametric form. Using the properties of minimal surfaces we succeded in transfering the problem of finding the minimal surface to a problem of minimizing a functional over a class of scalar functions. A numerical method of minimizing a functional using the first variation is presented and convergence is proven. A numerical example is given.

closed curve in 3-dimensional space.If the curve is planar then the problem reduces to that of finding a conformal mapping onto its interior.
To the numerical analyst the Plateau problem presents a formidable challenge.In the non-parametric case, when the surface and bounding curve admit of a single-valued projection onto an x,y plane, the problem reduces to solving the minimal surface equation 2 2 (i + z )z 2z z xy + (i + z )z 0 y xx x y x yy (.i) for the height z(x,y) of the surface above the x,y plane, and for boundary values defining the given bounding curve.Finite difference iterative schemes for(l.l)havebeen examined by Concus [2] and Greenspan [3], [4 ].
In the parametric case, where the surface is not assumed to admit a single valued planar projection, a vector function representation x(u,v) (x(u,v),y(u,v),z(u,v)) is used.Here x(u,v) is defined on a domain P in the (u,v) plane whose structure determines that of the surface.By a theorem of Weierstrass [5] the problem becomes one of finding x(u,v) such that C. x maps the boundary of onto the bounding curve(s) of the surface in a monotonic fashion.
A numerical scheme for simultaneously attaining A,B,C cannot be easily derived, for although B is in fact a boundary condition for A [I], it is not clear how one may work with C In the following we introduce a method for computing the solution of this problem.The method depends on our recognizing the solution as a function minimizing the Dirichlet integral over all functions satisfying C Similar to the method of safe descent of R. Courant [I], we define the Dirichlet integral as a functional d(g) on a class G of scalar functions g which determine the manner in which the surface is "sewed" onto its bounding curve.The percise definition of this functional is given in section 2. In section 3 the derivation of the first variation of d(g) is performed, and as an obvious consequence we see that a stationary value for d(g) defines a minimal surface.In section 4 we define a method for minimizing d(g) based on the use of the first variation; we then prove the convergence of the method and discuss its computational implementation, which is described in section 5.

DEFINITION OF d(g).
Let C be a simple closed curve in (x,y,z) space of length 2 given by C: x h() (2.1) for arc length o.We assume that h is twice continuously differentiable with

U F
A vector function of u,v on is denoted by a lower case letter such as x(u,v) while the same function referred to polar coordinates on , u r cos , v r sin is denoted by the corresponding upper case letter: ( x(u,v) r,) 0 <_ r <_ i 0 <_ 8 <_ 2 (2.4) For any sufficiently smooth functions x,y, the Dirichlet integral D of x over P is 2 + x )du dv known [i] that a vector function x X exists on for which conditions A,B,C hold.Horeover by the twice continuous differentiability of h and the extension of Kellogg's theorem to minimal surfaces [6], [7], (1,8) has a HIder continuous first derivative with respect to IX 8(1,8+5) X 8(1,8) <_ Sv (2.7) with ,v the HDlder constant and rdlder index, respectively.Specifically THEOREM i.There exists a function x(u,v) satisfying the following conditionm i.I. x is continuous on 1.2.&x 0 within t) 1.5.X@ (1,@) "s tt6"lder continuous and obeys (2.7) for some values of x maps F onto C in a monotonic one-to-one fashion; 1.7.D[x] < 1.8.For the function x(u,v) the Dirichlet integral (2.5) attains its least value among all functions satisfying 1.1-1.4,1.6,1.7.
LEM[A i. Condition 1.5 implies 1.7 for any harmonic function x Furthermore D with M a constant dependent only on v PROOF: By 1.5, X(I,) admits of a uniformly convergent Fourier series expansion a0 x,e.__.-+ cos j8 + 8. sin j@) j=l furthermore by a simple calculation 2 (Xe(l,8 +) (l,8))cos jede j 0 2 1 X@(l,@))sin j j = (x(, +) 0 whence (2.9)It is known [i] that the Dirichlet integral exists if and only if the series j(j2 + j j=l (2.10)   converges, and if so, its value is given by the series.However it is clear that if (2.9) holds then (2.10) does converge, and

Let
be the collection of functions satisfying I.i, 1.2, 1.5, 1.6.Let x X be any function of By 1.6 there exists a monotonic function g(0)
4. AN ALGORITHM FOR MINIMIZING dg.) We now derive an algorithm for solving the problem min.d(g) (4.1) gG numerically.The algorithm rests upon a Raylelgh-Ritz type of approach, in which we solve (4.1)over a sequence of finite dimensional subsets of G yielding a sequence of functions converging to the solution.At some stage in the algorithm we will need to require that a "three-points condition" in which three given points of F are mapped into three given points of C is obeyed.Since can be mapped conformal- ly onto itself by a Mobius transformation in which the images of three given points can be preassigned, while a function g G can be considered as mapping F onto itself, a three points condition can always be attained through the composition of two elements of G In addition the Dirichlet integral is invariant under the Mobius transformation.In order to guarantee convergence of the algorithm we impose an additional smoothness assumption on the curve C and as a consequence, on the functions of G We also assume that a minimal surface solving (4.1) has no branch points on the boundary.
For the purposes of this section, we will assume that the function h of (2.1) has a HDlder continuous second derivative.Again using the extension of Kellogg's theorem to minimal surfaces, we see that a solution g* to (4.1) has a HIder continuous second derivative.We now redefine the collection G as the set of all monotonic twice differentiable functions g obeying (2.12) and having a HIder continuous derivative of second order.Let Let go be any function of G Then g* go vanishes for 8 0,2 and has a HDlder continuous second derivative.Hence g* go has a uniformly convergent g* go +--+ [ (aj cos j@ + b. sin j@) j=l J ( can be differentiated termwise, since the resulting series itself converges uniform- ly, obtaining g*' (8)   g(e) + [ (-jaj sin j + jbj cos jS) j-1 (4.7) If we make the (reasonable) assumption that the minimal surface defined by g* has no branch points at the boundary, then for some possitive constant g*'(8 -+ I (aj cos j8 + b. sin j@) n j=l 3 then by (4.8) and the uniform convergence of the series n (4.7), LEMMA 4. The sequences [go + Sn] [g + S']n converge uniformly to g* g*' respectively, on [0,2] for n sufficiently large, go + Sn is monotonically increasing.
Finally, using the methods of section 2, B. CAHLON, A.D. SOLOMON and L.J. NACBMAN L4MA 5. d(g 0 + S n) d(g*) as n- PROOF.This assertion is easily proved by obtaining estimates of the form (2.9) which under the heightened soothness assumption for h attain one higher power of [/j Clearly the Fourier coefficients of h depend continuously (in the L2-norm) on the argument of h while by these estimates the series (2.10) converges uniformly; hence this series, which is equal to the Dirichlet integral, depends aontinuously upon the argument of h thus proving our assertion.
Before describing our algorithm we will turn to some properties of functions of the form ^+ S n For any constants A. In the same way, multiply.ingby (+ i + sin ke) and integrating over [0,2] we obtain and the lenna is proved.Tn defined by (4.9), (4.14).Using the methods of theorem 3 we find THEOREM 4. The function 6(2n+i) is lower semi-continuous, and has partial derivatives with respect to each of its independent variables; moreover 2 2 r(l'e)t(g 0 + Tn)COS je de J 0 (4.16) 3B.
] 2 2 f r(l'e)(go + Tn)sin je de 0 (4.17) where (l,e) (g0 + Tn) An algorithm for the solution of (I) can now be defined in the following steps: I.
This value defines a monotonic function go + T*n (by (4.9)) which V. Clearly n+l < n n 1,2,...By the monotonicity of the gn there is a subsequence of the functions {Xn} which converges uniform- ly to a function .. ([i]).
PROOF.For n sufficiently large, go + Sn belongs to Cn Hence < D[] <_ n < d(g0 + Sn) and by lemma 5, our claim is proved.

NUMERICAL EXAMPLE
As an example of an application of the results in the previous sections we consider the followng.
Let C be a simple closed curve in (x,y,z) space of length 2 given by C X e) 0 < e < 2, where h e (I-j, -] e ([7 "' e [-j , -j 1 (0, O, - (TI(e) T2(e) T3(e)) ( Our problem is to find a minimal surface spanned by a curve C Let k A I , B I B k be given.The function g(O) will be A 0 k g(0) 0 + -+ Y. (Aj cos j0 + B. sin j0) j=l 3 The monotonicity of g (1emma 6) demands Let the function (e)= (g(e)) and (e) (Hi(e) H2(e) H3(e)) Now we solve the Laplace equation AX 0 (5.4)   on the domain D with the boundary condition X(l,e) (e) (see Eq. (2.11)).
Define the mesh points in the r e plane by the points of intersection of the circles r ih (i O, 1,2, i0, i 0 + i N) and the straight lines e jSe (j 0,1 M) X 2 X 3 for 0 < i < i 0 1 Let (ih,j6e) (X ,j, i,j' i,j The value of X,j i <_ j <_ M and 1,2,3 are obtained from Poisson's integral i f (i-(ih)2)H() d (5.5) X,j i + i2h 2 2(ih)cos( j6e) We compute the integral of Eq. (5.5) by the compound Simpson's rule.
To obtain the value of Xf for N > i > i 0 i < j < M 1,2,3 we use the following.
Consider Laplace's equation in polar coordinates 82X 1 82X Then Laplace's equation at the point (i,j) may then be approximated by Xi+i,j 2X' + X" I (Xi+l,.-.Xi-l,j) xi,j+I 0 (5.8) If these equations are written out in detail for i i 0 i 0 + I,...,N and j 1,2,...,M and by using the relation X,j Xi,j+M (5.9)   then it will be found that their matrix form is ) Hj H(J68) (5.15) The matrix A is given by i To solve Eq. ( 5.10) we use the same method as in [i0].
We factorized the supermatrices in the form A =LU (5.17) This method has been described by Wilson [Ii] though not actually for elliptic difference equations but for equation of a similar form.
To solve Eq. ( 5.10) we first solve the following equation UN-j-j + VN-jXN-j+I YN-j N i >_ j >_ i (5.27) Thus we obtain the values of X.
for all I < i < N i <_ j <_ M We do these by approximating it by a generalization of Simpson rule [12].
In the third step we compute the value N+I,j [8---)N+I,j (see Eq. (3.8)). (5.29) In our example we first compute the value of the integral D and of E We do this by choosing of {Aj } {Bj } (j I, k) in a random way and such that {Aj} {Bj} satisfies Eqs. (4.12), (4.13).For IEI not sufficiently big we stop the random process and then we use gradient method [9].
To use the gradient method we calculate the value of the gadient by approxi- mating the integrals 0 B 0 ----2 / ____OX (l,B)T(g(B))sin j dB (5.31) j =I 0 As before we approximate the integrals (5.30), (5.31) by the compound Simpson's rule.
We halted our process when the values of IEI were smaller than In the following table we see the numerical results for In Table I we present a selected result that was obtained by random choices of Aj Bj In Table II we see selected results that were obtained by using gradient method.The intial value of {Aj} {Bj} for the gradient method are the best results obtained by random selection.In Figure I, we show the =Linimal surface drawn from the values of X,j and using the closed curve C given in (5.1).
Calculations by Random Choice for the Coefficients of (Aj,Bj)

3
'-e]-f r(l'e)(g*(e))(e)de 0 LEMMA2.D[] is uniformly bounded for all s as 0 PROOF.As e 0 the function e converges uniformly on F according to(3.4).Using the Frenet formulas, we see Ye(l,e) t(g, + ) + g (o) n(o do But then if g*' (e) is H61der continuous with index v then YO is rdlder continuous with r61der index v and r61der constant independent of e which implies, by lemma i, D[e] < M(3.5)   with M a constant independent of e By the uniform convergence of e to Y in F and hence on D and the lower semi-continuity of the Dirichlet integral (we see by(3.4), (3.5), that the limit of the left hand side exists for -0 and (3.1) is proved.

LEMLA 3 .
If a stationary value for d(g) is attained for some g* G then g* defines a minimal surface.PROOF.If D[X,] 0 for all () then by the fundamental theorem of the calculus of variations (1,e)-(g(e)) 0 r (3.7) which by (23.3) implies Xr(l,8)X8(1,8) 0 H'Older constant and 61der exponent, respectively.Clearly now the series in the relation a 0

2 '
......I + cos k0 > 0 Multiplying(4.11)  by this function and integrating over [0,2] we obtain Mobius transformation of R onto itself, derive a monotonic * satisfying the three-points condition; each D and I are M x M matrices and B. CABLON, A.D. SOLOMON and L.J.