NONPARAMETRIC MINIMAL SURFACES IN R WHOSE BOUNDARIES HAVE A JUMP DISCONTINUITY

Let be a domain in R 2 which is locally convex at each point of its boundary except possibly one, say (0,0), be continuous on /{(0,0)} with a jump discontinuity at (0,0) and f be the unique variational solution of the minimal surface equation with boundary values . Then the radial limits of f at (0,0) from all directions in exist. If the radial limits all lie between the lower and upper limits of at (0,0), then the radial limits of f are weakly monotonic; if not, they are weakly increasing and then decreasing (or the reverse). Additionally, their behavior near the extreme directions is examined and a conjecture of the author's is proven.


I. INTRODUCTION.
How does the generalized solution of the Dirichlet problem for the minimal surface equation with boundary values # behave when has a jump discontinuity (say at the origin) Under certain mild conditions on the domain R2 we shall show that the radial limits at (0,0) of the solution, denoted Rf (8), exist for all 8 (,8), where {(r,8) l< 8 < 8, 0< r < r(8)}.Further, on at most three intervals (i.e.[e,'], [8 L, OR], [8",8]) Rf (8) is constant and elsewhere it is strictly monotonic.
If Rf(8) lies between the lower and upper limits of # at (0,0), then Rf is weakly monotonic on [,8].If not, then Rf is not monotonic on [,8] but it is weakly monotonic on [e, e+n and on [8-,8].Under some smoothness and nontangency assumptions, we shall show that e" =e or e" =e+ and 8" 8 or 8" 8-n   We shall also show that O R 8 L + when O L and O R occur.Thus there is at most one interval on which Rf(8) is constant.
2. PRELIMINARIES.By we will mean a bounded open subset of 2 with the following properties: (a) is connected and simply connected.(b) 28 is Lipschitz and N (0,0) e .
Let S be the closure of SO, r be the closure of 0' F+ be the closure of FO{(x,y) y> 0}, and Fbe the closure of F a{(x,y) y< 0}.
Throughout this paper, we will make the following ASSUMPTION.f C().
REMARK.If 8 a 7, it follows from Lemma or from standard barrier arguments that * holds for all C (8).
In the case to be eliminated, Rf is weakly monotonic (say increasing) on [,], strictly increasing on [a',0 L], constant on [0L,eR], and strictly increasing on [e R,8"], for some " 0 L < 0 L+ =< O R < " <--.We may rotate the x-y plane so that O R 0 and (by a conformal map of B into B fixing (-I,0) and (I,0)) we may assume that u(0) 0. As in [5], there exist neighborhoods U and U" of 0 in E and a c-l-diffeomorphism F: U" U with DF(0) e'id for some 0 # e such that where 0 # A a + ib and n > m > are integers.Suppose we set s + it F(w) and x x F - for m C U. Let y be the image of the real axis under F. Then y is tangent to the real axis at the origin and, since x(w) 0 for w real, x(m) 0 for m y.If m re then x(r, 6) rmsin(m6) and the only curves on which x vanishes are 6 k/m for all integers k.Thus y must be the real axis in U. Since y(w) 0 for w real, y() 0 for m real.This means that b 0 and g(m) alm(mn) + o(Imln).If o is a curve in U from (r,6) (e,0) to (r,6) (,) ( small) such that (x(o), g(o)) is star-shaped with respect to the origin, then the sign pattern of x(o) is +,-and 9() is +,-,+.Thus m must 2 be 2, n must be 3, z(s,o) s and so Rf(8) z(F(u (O))) cannot be monotonic on (a,B).Q.E.D.
In [I], the case C() and a > is considered and the conjecture that O R 8 L w is mentioned.The following theorem proves that this is always true.
THEOREM 2. In case (ii) of Theorem I, O R O L .
A question of interest is to determine the asymptotic behavior of Rf(0) for 0 > O R near O R A discussion of the asymptotic behavior of Rf(0) for 0 < 0 L near 0 L is similar.We may assume that Rf is increasing on [0R, B].
As in the proof of Theorem I, let us assume that O R 0 and u(0) 0; then Next, if 0 O R < 0 < 8", then Rf(0) is equal to that value of z > o for which yX(z) tan(0).For this value of z z + o([z I) (2 tan(0)/3a2) and so asymptotically as 0/0+, Rf(0) (2/3a) 2 0 2 We wish to examine the behavior of Rf(0) near O and O 8.
Let us say that a "fan" exists at 0 0 when Rf( 8) is constant on a nontrivial interval containing 00.Since 8 e < 2, we get COROLLARY.Suppose that the hypotheses of Theorem 3 are satisfied for F + and F Then no more than one "fan" can occur. 4. EXAMPLES.
This last part shows that for any angle (0,), we can set -/2 and find an example in which " + , Rf(O) is (weakly) increasing on [,8], and F intersects the positive z-axis in an angle of REMARK.In [2], the behavior of a (nonparametric) solution of an equation of prescribed mean curvature with prescribed boundary values in a domain with a reentrant corner is examined.The results of [2] can be extended to the case in which has a jump discontinuity.In fact, by combining the work in [2] with the techniques used above, Theorems I, 2, and 3 and the Corollary can be proven in this new situation.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.

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) C(')/{N}) be the variational solution of the Dirichlet problem (for f the minimal surface equation) in " with boundary data .Next let 0 < e < /4

First
Round of Reviews March 1, 2009