# Minimal surfaces between two points.

1. MINIMAL SURFACES AS OPTIMAL EVOLUTIONS

The minimal surfaces are characterized by zero mean curvature. These become an area of intense mathematical and scientific study over the past 15 years, specifically in the areas of molecular engineering and materials sciences due to their anticipated nanotechnology applications. The most extensive meeting ever held on the subject, in its 250-year history, was organized in 2001 at Clay Mathematics Institute. In spite of all these efforts, the old thinking about minimality was not changed.

Recently, the first author changes the traditional geometrical viewpoint (contained in [1]-[5]), looking at a minimal surface as solution in a two-time optimal control system via the multitime maximum principle (see [6]-[23]). On the other hand, our study shows, on a classical problem, the efficiency of multitime optimal control.

Let [[OMEGA].sub.0[tau]] be a two dimensional interval fixed by the diagonal opposite points 0, [tau] [member of] [R.sup.2.sub.+]. Looking for surfaces [x.sup.i](t) = [x.sup.i]([t.sup.1],[t.sup.2]), ([t.sup.1],[t.sup.2]) [member of] [[OMEGA].sub.0[tau]], i = 1,2,3, of minimum area, that evolve between two points x(0), x([tau]) and rely transversally on two curves [[GAMMA].sub.0] and [[GAMMA].sub.1], let us show that such a surface (2-sheet) is a solution of a special PDE system, via the optimal control theory (multitime maximum principle).

In [R.sup.3] we introduce the two-time controlled dynamics

(PDE) [[partial derivative][x.sup.i]/[partial derivative][t.sup.[alpha]] (t) = [u.sup.i.sub.[alpha]] (t),

t = ([t.sup.1],[t.sup.2]) [member of] [[OMEGA].sub.0[tau]], i = 1, 2, 3; [alpha] = 1, 2,[x.sup.i](0) = [x.sup.i.sub.0], [x.sup.i]([tau]) = [x.sup.i.sub.1],

where u = ([u.sub.[alpha]]) = ([u.sup.i.sub.[alpha]]) : [[OMEGA].sub.0[tau]] [right arrow] [R.sup.6] represents two open-loop [C.sup.1] control vectors, linearly independent, eventually fixed on the boundary [partial derivative][[OMEGA].sub.0[tau]]. The complete integrability conditions of the (PDE) system, restrict the set of controls to

U = {u = ([u.sub.[alpha]]) = ([u.sup.i.sub.[alpha]]) | [[partial derivative][u.sup.i.sub.1]/[partial derivative][t.sup.2]] (t) = [[partial derivative][u.sup.i.sub.2]/[partial derivative][t.sup.1]] (t)}.

A solution of the (PDE) system is a surface (2-sheet) [sigma] : [x.sup.i] = [x.sup.i]([t.sup.1],[t.sup.2]). Suppose x(0) = [x.sub.0] belongs to the image [[GAMMA].sub.1] of a curve in [R.sup.3] and [tau] = ([[tau].sup.1],[[tau].sup.2]) is the two-time when the 2-sheet x([t.sup.1],[t.sup.2]) reaches the curve [[GAMMA].sub.1] in [R.sup.3], at x([tau]) = [x.sub.1], with [[GAMMA].sub.0] and [[GAMMA].sub.1] transversal to [sigma]. Using the area, we introduce the cost functional

(J) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Of course, the maximization of J(u(*)) is equivalent to the minimization of the area, under the constraint (PDE).

Two-time optimal control problem of minimal surfaces:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To solve the previous problem, we apply the multitime maximum principle [6]-[23]. In general notations, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the control Hamiltonian is

H (x,p,u) = [p.sup.[alpha].sub.i] [X.sup.i.sub.[alpha]] (x,u)+ [p.sub.0][X.sup.0] (x,u).

Taking [p.sub.0] = 1, the adjoint dynamics says

(ADJ) [[partial derivative][p.sup.[alpha].sub.i]/[partial derivative][t.sup.[alpha]]] = - [[partial derivative]H/[partial derivative][x.sup.i]] = 0.

On the other hand, we have to maximize H(x,p,u) with respect to the control u, hence

[partial derivative]H/[partial derivative][u.sup.i.sub.[alpha]] = [partial derivative][X.sup.0]/[partial derivative][u.sup.i.sub.[alpha]] + [p.sup.[alpha].sub.i] = 0.

Of course, [x.sup.i.sub.[alpha]] = [u.sup.i.sub.[alpha]] and [X.sup.0] (or H) does not depend on [x.sup.i]. We eliminate [p.sup.[alpha].sub.i] using the adjoint PDE. It follows the two-time Euler-Lagrange PDEs

[partial derivative]/[partial derivative][t.sup.[alpha]] ([partial derivative][X.sup.0]/[partial derivative][x.sup.i.sub.[alpha]]) = 0.

Summarizing, we obtain

Theorem 1.1. The solution of the previous optimal control problem is a minimal surface.

Since explicitly

H(x,p,u) = [p.sup.[alpha].sub.i][u.sup.i.sub.[alpha]] - [square root of [[parallel][u.sub.1][parallel].sup.2] [[parallel][u.sub.2][parallel].sup.2] - [<[u.sup.1],[u.sub.2]>.sup.2],

the critical points u of H are solutions of the algebraic system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

Lemma 1.1. The dual vectors [p.sup.1] and [p.sup.2] determine the areal energy density [[parallel][p.sup.1][parallel].sup.2] [[parallel][p.sup.2][parallel].sup.2] - [<[p.sup.1],[p.sup.2]>.sup.2] = [([X.sup.0]).sup.3].

System (1.1) is equivalent to the system

[u.sup.i.sub.1] [p.sup.1.sub.i] = -[X.sup.0], [u.sup.i.sub.1] [p.sup.2.sub.i] = 0, [u.sup.i.sub.2] [p.sup.1.sub.i] = 0, [u.sup.i.sub.2][p.sup.2.sub.i] = -[X.sup.0]. (1.2)

In this way, [u.sub.1] is orthogonal to [p.sup.2], and [u.sub.2] is orthogonal to [p.sup.1]. Also

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

System (1.1) is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or, via relations (1.3), we obtain the unique solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consequently, we have the following

Theorem 1.2. A minimal surface is a solution of the PDEs

(PDE) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ADJ) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where B means boundary condition.

The simplest minimal surface is a planar quadrilateral fixed by the starting point [x.sup.i] (0) = [x.sup.i.sub.0] on [[GAMMA].sub.0] and the terminal point [x.sup.i] ([tau]) = [x.sup.i.sub.1] on [[GAMMA].sub.1]. In this case the vectors [u.sub.[alpha]] (*) = [u.sub.[alpha]0] = ([u.sup.i.sub.[alpha]0]) are constant in time, and consequently, the parametric representation

[x.sup.i](t) = [u.sup.i.sub.10][t.sup.1] + [u.sup.i.sub.20][t.sup.2] + [x.sup.i.sub.0], i = 1, 2, 3,

depends on six arbitrary constants. We fix these constants by the following conditions: [x.sup.i]([tau]) = [x.sup.i.sub.1] which implies det[[u.sub.1],[u.sub.2],[x.sub.0] - [x.sub.1]] = 0, and the transversality conditions

([tau]) p(0) [perpendicular to] [T.sub.x0][[GAMMA].sub.0], p([tau]) [perpendicular to] [T.sub.X1][[GAMMA].sub.1],

which show that the optimal plane is orthogonal to [[GAMMA].sub.0] and [[GAMMA].sub.1], and so the tangent lines [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are parallel.

2. MINIMAL EVOLUTION PASSING THROUGH TWO POINTS

2.1. Two-time minimal evolution avoiding an obstacle. Let us apply the multitime maximum principle to search the surface of minimum area satisfying the following conditions: it contains two diagonal points x(0), x([tau]) and it avoids an obstacle A whose boundary [partial derivative]A is a surface. For that we start with the controlled dynamics problem and its solution in Section 1. Suppose x(t) [not member of] [partial derivative]A for t [member of] [[OMEGA].sub.0[tau]]. In this hypothesis, the multitime maximum principle applies, and hence the initial dynamics (PDE) and the adjoint dynamics (ADJ) are those in Section 1.

To simplify the problem, we accept as obstacle a 2-dimensional cylinder (that supports a global tangent frame {[u.sub.1],[u.sub.2]}), respectively a 2-dimensional sphere (that does not support a global tangent frame because any continuous vector field on such sphere vanishes somewhere).

2.2. Two-time evolution touching an obstacle. The points 0 [less than or equal to] [s.sub.0] [less than or equal to] [s.sub.1] [less than or equal to] [tau] determine a decomposition of the two dimensional interval [[OMEGA].sub.0[tau]] in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. To simplify the problem, suppose the sheet x(t) [not member of] [partial derivative]A for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a union of two planar quadrilaterals (one starting from x(0) and ending in x([s.sub.0]) and the other starting from x([s.sub.1]) and ending at x([tau])). If x(t) [member of] [partial derivative]A for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then we need the study in Section 3 and Section 4, which emphasizes the controls and the dual variables capable of keeping the evolution on the obstacle. Furthemore, suitable smoothness conditions on boundaries are necessary.

3. TOUCHING, APPROACHING AND LEAVING A CYLINDER

3.1. Touching a cylinder. Let us take the cylinder C : [([x.sup.1]).sup.2] + [([x.sub.2]).sup.2] [less than or equal to] [r.sup.2], x = ([x.sup.1],[x.sup.2],[x.sup.3]), as obstacle. Suppose x(t) [member of] [partial derivative]C for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this case we use the modified version of two-time maximum principle.

We introduce the set N = [R.sup.3]\C : f (x) = [r.sup.2] - ([([x.sup.1]).sup.2] + [([x.sup.2]).sup.2]) [less than or equal to] 0, and we build the functions

[c.sub.[alpha]] (x,u) = [[partial derivative]f/[partial derivative][x.sup.i]] (x) [X.sup.i.sub.[alpha]] (x,u), [alpha] = 1,2,

i.e., [c.sub.[alpha]](x,u) = -2([x.sup.1][u.sup.1.sub.[alpha]] + [x.sup.2][u.sup.2.sub.[alpha]]). Let us use the two-time maximum principle adding the constraints

[c.sub.1] (x,u) = - 2([x.sup.1][u.sup.1.sub.1] + [x.sup.2][u.sup.2.sub.1]) = 0, [c.sub.2] (x, u) = - 2([x.sup.1][u.sup.1.sub.2] + [x.sup.2][u.sup.2.sub.2]) = 0.

[partial derivative][p.sup.[alpha].sub.i]/[partial derivative][t.sup.[alpha]] (t) = [partial derivative]H/[partial derivative][x.sup.i] + [[lambda].sup.[gamma]] (t) [partial derivative][c.sub.[gamma]]/[partial derivative][x.sup.i]

are reduced to

(ADJ') [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The critical point condition with respect to the control u is

[partial derivative]H/[partial derivative]u = [[lambda].sup.[gamma]] [[partial derivative][c.sub.[gamma]]/[partial derivative]u],

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

We recall that x(t) [member of] [partial derivative]C means [([x.sup.1]).sup.2] + [([x.sup.2]).sup.2] = [r.sup.2]. Consequently,

[x.sup.i][p.sup.1.sub.i] = [[lambda].sup.1] (-2[r.sup.2]), [x.sup.i][p.sup.2.sub.i] = [[lambda].sup.2] (-2[r.sup.2]), i = 1,2.

To develop further our ideas, we accept that the cylinder C is represented by the parametrization [x.sup.1] = r cos [t.sup.1],[x.sup.2] = r sin [t.sup.1],[x.sup.3] = [t.sup.2]. We use the partial velocities (orthogonal vectors)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the area formula is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, the evolution PDEs emphasize the controls

[u.sub.1] : [u.sup.1.sub.1] = -r sin [t.sup.1],[u.sup.2.sub.1] = r cos [t.sup.1],[u.sup.3.sub.1] = 0

[u.sup.2] : [u.sup.1.sub.2] = 0, [u.sup.2.sub.2] = 0,[u.sup.3.sub.2] = 1,

with <[u.sub.1],[u.sub.2]> = 0, [parallel][u.sub.1][parallel] = r, [parallel][u.sub.2][parallel] = 1. Using equalities (3.1), we obtain [u.sup.i.sub.1][p.sup.1.sub.i] = 2,[u.sup.i.sub.1][p.sup.2.sub.i] = 0,[u.sup.i.sub.2][p.sup.1.sub.i] = 0,[u.sup.i.sub.2][p.sup.2.sub.i] = 2, which confirm that [u.sub.1] is orthogonal to [p.sup.2] and [u.sub.2] is orthogonal to [p.sup.1]. The relations [p.sup.1.sub.i] = [[lambda].sup.1] (-2[x.sup.i]) + 2[u.sup.i.sub.1],i = 1,2,[p.sup.1.sub.3] = 0 produce

[p.sup.1.sub.1] = -2[[lambda].sup.1]r cos [t.sup.1] - 2r sin [t.sup.1], [p.sup.1.sub.2] = -2[[lambda].sup.1] r sin [t.sup.1] + 2r cos [t.sup.1], [p.sup.1.sub.3] = 0.

Similarly, the relations [p.sup.2.sub.i] = [[lambda].sup.2](-2[x.sup.i]) + 2[u.sup.i.sub.2],i = 1, 2,[p.sup.2.sub.3] = 2 give

[p.sup.2.sub.1] = -2[[lambda].sup.2]r cos [t.sup.1], [p.sup.2.sub.2] = -2[[lambda].sup.2]r sin [t.sup.1],[p.sup.2.sub.3] = 2.

[partial derivative][[lambda].sup.1]/[partial derivative][t.sup.1] + [partial derivative][[lambda].sup.2]/[partial derivative][t.sup.2] = -1.

We use the particular solution [[lambda].sup.1] = [k.sup.1] - [t.sup.1],[[lambda].sup.2] = [k.sup.2] of this PDE.

The evolution (PDE) on the interval [[omega].sub.0] [less than or equal to] [t.sup.1] [less than or equal to] [[omega].sub.1] shows that the surface of evolution is a cylindric quadrilateral fixed by the initial point [x.sup.i] ([[omega].sub.0],[t.sup.2]) = [x.sup.i.sub.0] generator and with the terminal point [x.sup.i] ([[omega].sub.1],[t.sup.2]) = [x.sup.i.sub.1] generator.

3.2. Approaching and leaving the cylinder. Now we must put together the previous results. Suppose x(t) [member of] N = [R.sup.3]\C for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[FIGURE 1 OMITTED]

For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the 2-sheet of evolution x(*) consists in two pieces. To simplify, we accept it as a union of two planar sheets. Suppose the first planar sheet touches the cylinder at the point x([s.sub.0]). In this case, we can take

[p.sup.1.sub.1] = - 2rcos [[phi].sub.0],[p.sup.1.sub.2] = 2rsin [[phi].sub.0],[p.sup.1.sub.3] = 0,

for the angle [[phi].sub.0] as shown in Fig. 1, and

[p.sup.2.sub.1] = 0,[p.sup.2.sub.2] = 0,[p.sup.2.sub.3] = 2.

By the jump conditions, the vectors [p.sup.1] (*), [p.sup.2](*) are continuous when the evolution 2-sheet x(*) hits the boundary [partial derivative]C at the two-time [s.sub.0]. In other words, we must have the identities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e., [k.sup.1] = [t.sup.1.sub.0], [t.sup.1.sub.0] + [[phi].sub.0] = [pi]/2, [k.sup.2] = 0. The last two equalities show that the optimal (particularly, planar) 2-sheet is tangent to the cylinder along the generator [x.sup.1] = [x.sup.1]([s.sub.0]), [x.sup.2] = [x.sup.2]([s.sub.0]),[x.sup.3] [member of] R.

Let us analyse what happen with the evolution 2-sheet as it leaves the boundary [partial derivative]C at the point x([s.sub.1]). Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The jump theory gives

[p.sup.[alpha]] ([t.sup.1+.sub.1],[t.sup.2]) = [p.sup.[alpha]] ([t.sup.1-.sub.1],[t.sup.2]) - [[lambda].sup.[alpha]]([t.sup.1.sub.1],[t.sup.2]) [nabla]f(x([t.sup.1.sub.1],[t.sup.2]))

for f(x) = [r.sup.2] - [([x.sup.1]).sup.2] - [([x.sup.2]).sup.2]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this way, [p.sup.1.sub.1] ([t.sup.1+.sub.1],[t.sup.2]) = -2r sin [t.sup.1.sub.1],[p.sup.1.sub.2]([t.sup.1+.sub.1],[t.sup.2]) = 2r cos [t.sup.1.sub.1] and so the planar 2-sheet of evolution is tangent to the boundary [partial derivative]C along the generator by the point x([s.sub.1]). If we apply the usual two-time maximum principle after x(*) leaves the cylinder C, we find (see Fig. 1)

[p.sup.1.sub.1] = const. = -2r cos [[phi].sub.1],[p.sup.1.sub.2] = const. = -2r sin [[phi].sub.1]; [p.sup.2.sub.1] = 0,[p.sup.2.sub.2] = 0.

Therefore -cos [[phi].sub.1] = - sin [t.sup.1.sub.1], -sin [[phi].sub.1] = -cos [t.sup.1.sub.1] and so [[phi].sub.1] + [t.sup.1.sub.1] = [pi].

Open Problem. What happen when the surface in the exterior of the cylinder is a non-planar minimal sheet?

4. TOUCHING, APPROACHING AND LEAVING A SPHERE

4.1. Touching a sphere. Let us take as obstacle the sphere B : f(x) = [r.sup.2] - [[delta].sub.ij][x.sup.i][x.sup.j] [greater than or equal to] 0, x = ([x.sup.1],[x.sup.2],[x.sup.3]), i,j = 1,2,3. Suppose x(t) [member of] [partial derivative]B for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this case we use the modified version of two-time maximum principle.

We introduce the set N = [R.sup.3]\B : f (x) = [r.sup.2] - [[delta].sub.ij][x.sup.i][x.sup.j] [less than or equal to] 0 and we build the functions [c.sub.[alpha]] (x,u) = [partial derivative]f/[partial derivative][x.sup.i] (x) [X.sup.i.sub.[alpha]] (x,v), [alpha] = 1,2, i.e., [c.sub.[alpha]] (x,v) = -2[[delta].sub.ij][x.sup.i][u.sup.j.sub.[alpha]]. Let us use the two-time maximum principle adding the constraints

[c.sub.1] (x,u) = -2[[delta].sub.ij][x.sup.i][u.sup.j.sub.1] = 0,[c.sub.2](x,u) = - 2[[delta].sub.ij][x.sup.i][u.sup.j.sub.2] = 0.

Then the condition

[partial derivative][p.sup.[alpha].sub.i]/[partial derivative][t.sup.[alpha]] (t) = [partial derivative]H/[partial derivative][x.sup.i] + [[lambda].sup.[gamma]] [[partial derivative][c.sub.[gamma]]/[partial derivative][x.sup.i]]

is reduced to

(ADJ") [partial derivative][p.sup.[alpha].sub.i]/[partial derivative][t.sup.[alpha]] (t) = [[lambda].sup.[gamma]] (t) (-2[u.sup.i.sub.[gamma]]).

The condition of critical point for the control u becomes

[partial derivative]H/[partial derivative]u = [[lambda].sup.[gamma]] [partial derivative][c.sub.[gamma]]/[partial derivative]u,

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

We recall that x(t) [member of] [partial derivative]B means [[delta].sub.ij][x.sup.i][x.sup.j] = [r.sup.2]. Consequently,

[x.sup.i][p.sup.1.sub.i] = [[lambda].sup.1] (-2[r.sup.2]), [x.sup.i][p.sup.2.sub.i] = [[lambda].sup.2](-2[r.sup.2]).

To develop further our ideas, we accept that the sphere B is represented by the parametrization

[x.sup.1] = r cos [t.sup.1] cos [t.sup.2],[x.sup.2] = r sin [t.sup.1] cos [t.sup.2],[x.sup.3] = r sin [t.sup.2].

We use the partial velocities (orthogonal vectors)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the area formula is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, the evolution PDEs emphasize the controls

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with <[u.sub.1],[u.sub.2]> = 0, [parallel][u.sub.1][parallel] = r cos [t.sup.2], [parallel][u.sub.2][parallel] = r. Using equalities (4.1), we obtain [u.sup.i.sub.1][p.sup.1.sub.i] = -[X.sup.0],[u.sup.i.sub.2][p.sup.2.sub.i] = 0,[u.sup.i.sub.2][p.sup.1.sub.i] = 0,[u.sup.i.sub.2][p.sup.2.sub.i] = -[X.sup.0], which confirm that [u.sub.1] is orthogonal to [p.sup.2] and [u.sub.2] is orthogonal to [p.sup.1]. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or explicitly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[partial derivative][[lambda].sup.1]/[partial derivative][t.sup.1] + [partial derivative][[lambda].sup.2]/[partial derivative][t.sup.2] = cos [t.sup.2].

We use the particular solution [[lambda].sup.1] = 0, [[lambda].sup.2] = sin [t.sup.2] + [k.sup.2] of this PDE.

4.2. Approaching and leaving the sphere. We must now put together the previous results. So suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the 2-sheet of evolution x(*) consists in two pieces. To simplify, we accept a union of two planar sheets. Suppose the first planar sheet touches the sphere at the point x([s.sub.0]). In this case, we can take

[p.sup.1.sub.1] = - r cos [[phi].sub.0], [p.sup.1.sub.2] = r sin [[phi].sub.0],[p.sup.1.sub.3] = 0,

for the angle [[phi].sub.0] as shown in Fig. 1, and

[p.sup.2.sub.1] = 0, [p.sup.2.sub.2] = 0, [p.sup.2.sub.3] = -r.

By the jump conditions, the vectors [p.sup.1](*), [p.sup.2](*) are continuous when the evolution 2-sheet x(*) hits the boundary [partial derivative]B at the two-time [s.sub.0]. In other words, we must have the identities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e., [t.sup.1.sub.0] + [[phi].sub.0] = [pi]/2, [k.sup.2] = 0, [t.sup.2.sub.0] = 0. The last two equalities show that the optimal (particularly, planar) 2-sheet is tangent to the sphere at the point ([x.sup.1] ([s.sub.0]), [x.sup.2] ([s.sub.0]), [x.sup.3]([s.sub.0])).

Let us analyse what happen with the evolution 2-sheet as it leaves the boundary [partial derivative]B at the point x([s.sub.1]). We then have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The jump theory gives

[p.sup.[alpha]] ([t.sup.1+.sub.1],[t.sup.2+.sub.1]) = [p.sup.[alpha]] ([t.sup.1-.sub.1],[t.sup.2-.sub.1]) - [[lambda].sup.[alpha]] ([t.sup.1.sub.1],[t.sup.2.sub.1]) [nabla]f (x([t.sup.1.sub.1],[t.sup.2.sub.1]))

for f (x) = [r.sup.2] - [[delta].sub.ij] [x.sup.i][x.sup.j]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this way, [p.sup.1.sub.1] ([t.sup.1+.sub.1],[t.sup.2+.sub.2]) = -rsin [t.sup.1.sub.1], [p.sup.1.sub.2]([t.sup.1+.sub.1],[t.sup.2+.sub.2]) = r cos [t.sup.1.sub.1], and so the planar 2-sheet of evolution is tangent to the boundary [partial derivative]B at the point x([s.sub.1]). If we apply the usual two-time maximum principle after x(*) leaves the sphere C, we find (see Fig. 1)

[p.sup.1.sub.1] = const. = -r cos [[phi].sub.1],[p.sup.1.sub.2] = const. = -r sin [[phi].sub.1]; [p.sup.2.sub.1] = 0, [p.sup.2.sub.2] = 0.

Therefore - cos [[phi].sub.1] = -sin [t.sup.1.sub.1], - sin [[phi].sub.1] = -cos [t.sup.1.sub.1] and so [[phi].sub.1] + [[theta].sup.1.sub.1] = [pi].

Open Problem. What happen when the surface in the exterior of the sphere is a non-planar minimal sheet?

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[4] Ariana Pitea: Null Lagrangian forms on 2nd order jet bundles, J. Adv. Math. Stud., 3(2010), No. 1, 73-82.

[5] W. Schief: On a maximum principle for minimal surfaces and their integrable discrete counterparts, J. Geom. Phys.,5 6(2006), No. 9, 1484-1485.

[6] C. Udriste: Multi-time maximum principle, Short Communication, Int. Congress of Mathematicians, Madrid, August 22-30, ICM Abstracts, p. 47, 2006.

[7] C. Udriste and I. Tevy: Multi-Time Euler-Lagrange-Hamilton Theory,WSEAS Trans. Math., 6(2007), No. 6, 701-709.

[8] C. Udriste: Controllability and observability of multitime linear PDE systems,Proc. 6-th Congress of Romanian Mathematicians, Bucharest, June 28-July 4, 2007, vol. 1, 313-319.

[9] C. Udriste and I. Tevy: Multi-time Euler-Lagrange dynamics, Proc. 7-th WSEAS Int. Conf. Syst. Theory Sci. Comp. (ISTASC'07), Vouliagmeni Beach, Athens, Aug. 24-26 (2007), 66-71.

[10] C. Udriste: Multi-time stochastic control theory, Proc. 6-th WSEAS Int. Conf. Circuits, Systems, Electronics, Control & Signal Processing (CSECS'07), Cairo, December 29-31, 2007, 171-176.

[11] C. Udriste: Multitime controllability, observability and bang-bang principle, J. Optim. Theory Appl., 139(2008), No. 1, 141-157.

[12] C. Udriste and L. Matei: Lagrange-Hamilton theories, Monographs and Textbooks 8, Geometry Balkan Press, Bucharest, 2008 (in Romanian).

[13] C. Udriste: Finsler optimal control and Geometric Dynamics, Proc. American Conference Appl. Math., Cambridge, Massachusetts, 2008, 33-38.

[14] C. Udriste: Lagrangians constructed from Hamiltonian systems, Proc. 9-th WSEAS Int. Conf. Math. Comp. Business and Economics (MCBE-08), Bucharest, June 24-26, 2008, 30-33.

[15] C. Udriste, O. Dogaru and I. Tevy: Null Lagrangians forms and Euler-Lagrange PDEs,J.Adv. Math. Stud., 1(2008), No. 1-2, 143-156.

[16] C. Udriste: Simplified multitime maximum principle,Balkan J. Geom. Appl., 14(2009), No. 1, 102-119.

[17] C. Udriste: Nonholonomic approach of multitime maximum principle, Balkan J. Geom. Appl., 14(2009), No. 2, 111-126.

[18] C. Udriste and I. Tevy: Multitime linear-quadratic regulator problem based on curvilinear integral, Balkan J. Geom. Appl., 14(2009), No. 2, 127-137.

[19] C. Udriste and I. Tevy: Multitime dynamic programming for curvilinear integral actions, J. Optim. Theory Appl., 146(2010), No. 1, 189-207.

[20] Monica Pirvan and C. Udriste: Optimal control of electromagnetic energy, Balkan J. Geom. Appl., 15(2010), No. 1, pp. 131-141.

[21] C. Udriste: Equivalence of multitime optimal control problems, Balkan J. Geom. Appl., 15(2010), No. 1, pp. 155-162.

[22] C. Udriste, O. Dogaru, I. Tevy and D. Billa: Elementary work, Newton law and Euler-Lagrange equations, Balkan J. Geom. Appl., 15(2010), No. 2, 92-99.

[23] C. Udriste, V. Arsinte and C. Cipu: Von Neumann analysis of linearized discrete Tzitzeica PDE, Balkan J. Geom. Appl., 15(2010), No. 2, 100-112.

CONSTANTIN UDRISTE, IONEL TEVY AND VASILE ARSINTE

University "Politehnica" of Bucharest

Faculty of Applied Sciences

Splaiul Independentei, No. 313, 060042 Bucharest, Romania

University "Politehnica" of Bucharest

Faculty of Applied Sciences

Splaiul Independenttei, No. 313, 060042 Bucharest, Romania

Callatis High School ofMangalia

Negru Voda, No. 11, Mangalia, Romania