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## Über dieses Buch

Vallombrosa Center was host during the week September 7-12, 1985 to about 40 mathematicians, physical scientists, and engineers, who share a common interest in free surface phenomena. This volume includes a selection of contributions by participants and also a few papers by interested scientists who were unable to attend in person. Although a proceedings volume cannot recapture entirely the stimulus of personal interaction that ultimately is the best justification for such a gathering, we do offer what we hope is a representative sampling of the contributions, indicating something of the varied and interrelated ways with which these classical but largely unsettled questions are currently being attacked. For the participants, and also for other specialists, the 23 papers that follow should help to establish and to maintain the new ideas and insights that were presented, as active working tools. Much of the material will certainly be of interest also for a broader audience, as it impinges and overlaps with varying directions of scientific development. On behalf of the organizing committee, we thank the speakers for excellent, well-prepared lectures. Additionally, the many lively informal discussions did much to contribute to the success of the conference.

## Inhaltsverzeichnis

### Optimal Crystal Shapes

Abstract
Associated with any Borel function Ф defined on the unit sphere S n in R n+1 with values in R ⋃ {∞} (and, say, bounded from below) and any n-dimensional oriented rectifiable surface S in R n+1 is the integral
$$\Phi(S) = \int_{x\,\epsilon\,S}\, \Phi(v_S\,(x))\,dH^nx;$$
here v S (·) denotes the unit normal vectorfield orienting S, and H n is Hausdorff n-dimensional surface measure. If, for example, S is composed of polygonal pieces S i with oriented unit normals v i , then $$\Phi(S) = \Sigma_{i}\,\Phi(v_i)$$ area(S i ) Perhaps the most important integrands Φ: S 2R arise as the surface free energy density functions for interfaces S between an ordered material A (hereafter called a crystal) and another phase or a crystal of another orientation. In this case v S (p) is the unit exterior normal to A at pєS and Ф(S) gives the surface free energy of S. Other interesting Ф’s need not be continuous or even bounded. See the sailboat example of [T1], in which Ф(v) is the time required to sail unit distance in direction v rotated by 90°.
Jean E. Taylor, F. J. Almgren

### Immersed Tori of Constant Mean Curvature in R 3

Abstract
In this chapter we show how to construct immersions of tori into Euclidean space R 3 which have constant mean curvature H ≠ 0. We thus exhibit an example of a “non-round” soap bubble (although it does self-intersect) providing a counterexample to a conjecture attributed to H. Hopf. We shall carefully state the theorems involved in the construction and also provide a geometric description (with suggestive sketches) of the desired surfaces. An expanded version complete with proofs appeared in a recent paper of the author [11].
Henry C. Wente

### The Construction of Families of Embedded Minimal Surfaces

Abstract
In this chapter I would like to briefly describe some new results in the classical theory of minimal surfaces. These discoveries represent joint work with William H. Meeks III. Our research made critical use of the graphics programming software developed by James T. Hoffman at the University of Massachusetts. The central theorem is the following existence result [4], [5], [6].
David A. Hoffman

### Boundary Behavior of Nonparametric Minimal Surfaces—Some Theorems and Conjectures

Abstract
Suppose D is a domain in the plane which is locally convex at every point of its boundary except possibly one, say (0,0), and φ is continuous on ∂D except possibly at (0,0), where it might have a jump discontinuity. Then for all directions from (0,0) into D, the radial limits of f exist, where f is the solution of the minimal surface equation in D or of an equation of prescribed (bounded) mean curvature in D with $$f\,\epsilon\,C^0\,(\bar D\,\backslash\{(0,0)\})$$ and $$f=\phi\,\text{on}\,\partial D\backslash\{(0,0)\})$$. Some conjectures which would generalize this result are mentioned.
Kirk E. Lancaster

### On Two Isoperimetric Problems with Free Boundary Conditions

Abstract
During the last years, free boundary problems for minimal surfaces have found much attention. Beautiful new existence results were discovered by Struwe [25], Grüter—Jost [11], and Jost [19]. The question of boundary regularity was discussed by Hildebrandt—Nitsche [14], [15], [16], [17], Grüter—Hildebrandt—Nitsche [9], Dziuk [3], [4], Küster [20], and Ye [26]. Various regularity theorems are optimal, although several questions are still open. A survey of some of these results can be found in [12].
S. Hildebrandt

### Free Boundary Problems for Surfaces of Constant Mean Curvature

Abstract
This survey describes a new existence result [21] for (disk-type) surfaces of prescribed constant mean curvature with free boundaries, and relates this result to some other well-known variational problems arising in differential geometry.
Michael Struwe

### On the Existence of Embedded Minimal Surfaces of Higher Genus with Free Boundaries in Riemannian Manifolds

Abstract
In this chapter we consider the following configuration: a Riemannian manifold X of bounded geometry, some closed Jordan curves Γ j , and a supporting surface ∂K, disjoint from the Γ j . We further assume that the Γ j are contained in a suitable barrier ∂A of nonnegative mean curvature (cf. §2 for details).
Jürgen Jost

### Free Boundaries in Geometric Measure Theory and Applications

Abstract
The most famous problem in the theory of minimal surfaces is the so called Plateau problem, where one is looking for a minimal surface spanning a given boundary. This is a problem with a fixed boundary and was essentially solved around 1930 by Douglas and Radó.
Michael Grüter

### A Mathematical Description of Equilibrium Surfaces

Abstract
The central point in many problems of mathematical physics is answering questions about the boundary of a region, using as little information as possible about the region itself.
Mario Miranda

### Interfaces of Prescribed Mean Curvature

Abstract
Several questions of mathematical and physical interest lead to the consideration of an “energy functional” of the following type:
$$F[V] = \text{(weighted area of}\, S) + \int_{v}\, H dv,$$
(*)
where S is the surface bounding the region V of n-space and H is a given summable function. In the following, we shall be concerned with a problem of this type, representing in a sense a simplified physical situation, and investigate some basic properties of its solutions. The results we obtain may serve both as an illustration of the use of certain variational techniques and as an instance of results that could be obtained, under appropriate conditions, in more general cases.
I. Tamanini

### On the Uniqueness of Capillary Surfaces

Abstract
Let Ω ⊂ ℝn. Consider the equation of prescribed mean curvature
$$\text{div}\,Tu = H\, \text{in}\,\Omega$$
(1)
where
$$Tu = \frac{Du}{\sqrt{1 + |Du|^2}}$$
(2)
and Du is the gradient of u.
Luen-fai Tam

### The Behavior of a Capillary Surface for Small Bond Number

Abstract
The boundary value problem
$$\begin{array}{lc}\text{div}(Tu) = \kappa u & \text{in} \Omega \\ Tu \cdot v = \cos\gamma & \text{on} \Sigma = \partial \Omega\end{array}$$
(1)
determines the height u(x) of a capillary surface. Here κ is a positive constant, Ω is a bounded domain in R n , v is the exterior normal on Σ, and Tu is the vector operator
$$Tu = \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}}.$$
David Siegel

### Convexity Properties of Solutions to Elliptic P.D.E.’S

Abstract
How do the data of an elliptic boundary value problem (domain, boundary values, elliptic operator) affect the shape of the solution v? Estimates for v, Dv, or even D 2 v may be necessary to prove existence and regularity theorems, and they often also characterize fundamental geometric behavior of v. In this note we shall study some particular estimates involving v, Dv, D 2 v: ones that are related to convexity properties of v. The results do not usually lead to existence theorems (with some exceptions, e.g. [3]), but are surprising and have independent beauty.
Nicholas J. Korevaar

### Boundary Behavior of Capillary Surfaces Via the Maximum Principle

Abstract
Various authors have studied the boundary behavior of capillary surfaces using deep techniques. This talk is concerned with obtaining the same results via the maximum principle.
Gary M. Lieberman

### Convex Functions Methods in the Dirichlet Problem for Euler—Lagrange Equations

Abstract
In this paper we investigate a priori estimates for solutions of the second order elliptic E—L equations,* whose gradients satisfy some prescribed limitations. Such problems arise from the relativity theory and continuous mechanics and can be described in terms of variational problems for the n-dimensional multiple integrals
$$\int_{B}\, F(x, u, Du)\, dx$$
(1)
whose integrands F(x, u, p) are defined only for vectors p belonging to prescribed domain G in R n . If G coincides with the whole space R n , then we do not have any prescribed limitations for the gradient of desired solutions for the E—L equation corresponding to the functional (1). This most simple case was investigated in our paper [1].
Ilya J. Bakelman

### Stability of a Drop Trapped Between Two Parallel Planes: Preliminary Report

Abstract
This paper deals with the physical problem of a drop of liquid trapped between two homogeneous parallel planes in the absence of gravity. Explicit stability results are derived in the case of the contact angles between the liquid and the planes being equal to π/2, and a sufficient condition for stability for more general contact angles is derived. The present paper is an abbreviated version of a paper which will appear [7], The abbreviation consisted of omitting most proofs. I have been informed by Professor Stephan Hildebrandt that his student, Maria Athanassenas, has independently derived the stability results for contact angles equal to π/2, apparently using different methods.
Thomas I. Vogel

### The Limit of Stability of Axisymmetric Rotating Drops

Abstract
We consider an incompressible liquid drop Ω held together by surface tension in the absence of gravity; the drop is assumed to be small enough to make self-gravitation effects negligible. It is further assumed that the drop is driven to rotate at an imposed angular velocity. Finally, consideration is restricted to drop shapes Ω which are simply-connected and axisymmetric with respect to the axis of rotation.
Frederic Brulois

### Numerical Methods for Propagating Fronts

Abstract
In many physical problems, a key aspect is the motion of a propagating front separating two components. As fundamental as this may be, the development of a numerical algorithm to track the moving front accurately is difficult. In this report, we describe some previous theoretical and numerical work. We begin with two examples to motivate the problem, followed by some analytical results. These theoretical results are then used as a foundation for two different types of numerical schemes. Finally, we describe the application of one of these schemes to our work in combustion.
James A. Sethian

### A Dynamic Free Surface Deformation Driven by Anisotropic Interfacial Forces

Abstract
The dynamic deformation of a viscous droplet under the influence of anisotropic surface forces is examined. Asymptotic calculations are compared with the case of isotropic surface tension with possible relevance to the cleavage stage in the division of biological cells.
Daniel Zinemanas, Avinoam Nir

### Stationary Flows in Viscous Fluid Bodies

Abstract
Consider a drop of a viscous, incompressible fluid under the influence of some exterior force density f. A stationary flow inside the fluid body can be described by the Navier-Stokes system
$$\begin{array}{rclr}-v\Delta\upsilon + Dp + \upsilon \cdot D\upsilon = f\quad\quad\quad\\ \text{in}\,\Omega, \\\text{div}\,\upsilon = 0 \\\end{array}$$
(1)
together with the boundary conditions
$$\upsilon \cdot n = 0, t_{k} \cdot T \cdot n = 0\,\,\,\,\,\,\text{on}\, \Sigma, k = 1,\, 2,$$
(2)
$$n \cdot T \cdot n = p_{0} \quad\text{on}\,\Sigma$$
(3)
As usual, $$v=(v^1,v^2,v^3)=v(x), x=(x^1,x^2,x^3)$$ denotes the velocity, p = p(x) the pressure, and v > 0 is the kinematical viscosity. The unknown domain occupied by the fluid is denoted by Ω, its boundary by Σ; n is the outer normal to Σ, and t 1, t 2 span the tangent plane.
Josef Bemelmans

### Large Time Behavior for the Solution of the Non-Steady Dam Problem

Abstract
In this paper we consider two water reservoirs which are separated by a dam D consisting of an isotropic, homogeneous, porous material. The levels of the reservoirs may be different and time dependent and they are supposed to tend to fixed levels if t tends to infinity. We start with nonstationary initial conditions and we are interested in the asymptotic behavior of the pressure distribution u(t, z) of the water in the dam if t tends to infinity.
Dietmar Kröner

### New Results Concerning the Singular Solutions of the Capillarity Equation

Abstract
In this work we study the global existence and uniqueness of a singular solution of the capillarity equation in ℝ N :
$$\text{div}(D\upsilon/ \sqrt{1 + |D\upsilon|^2}) = \kappa\upsilon$$
(1)
with a κ < 0. Our results concern symmetric solutions with an isolated singularity at the origin. Then equation (1) takes the equivalent form in ]0, +∞[:
$$r^{n-1}u{^\prime}/\sqrt{1 + u{^\prime}^2})^\prime(r) = -(N - 1)r^{n-1}u(r),$$
(2)
where
$$u(r) = \sqrt{-\kappa(N-1)^{-1}}\,\upsilon(\sqrt{-(N-1)\kappa^{-1}r}).$$
Marie-Françoise Bidaut-Veron

### Continuous and Discontinuous Disappearance of Capillary Surfaces

Abstract
We consider the problem of finding a capillary surface u(x) in a cylinder Z with section Ω, in the absence of gravity. The surface is to meet Z in (constant) contact angle γ and project simply onto Ω. It is known that to each Ω there exists a critical angle γ0 ∈ [0, π/2] such that a surface exists if γ0 < γ ≤ π/2, while no surface exists if 0 ≤ γ < γ0 We show here that if 0 < γ0 < π/2 and if Ω is smooth, then there is no surface at γ = γ0, while if Ω has corners a surface can in some cases be found. In the former case, the surface disappears “continuously” and always becomes unbounded in a sub-domain Ω* of positive measure, as γ ↘ γ0. In the latter case the surface can be bounded, and even analytic with the exception of the comer points. Some applications of the result are given and the exceptional case γ0 = 0 is discussed.
Paul Concus, Robert Finn
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