Stability of anisotropic capillary surfaces between two parallel planes

We study the stability of capillary surfaces without gravity for anisotropic free surface energies. For a large class of rotationally symmetric energy functionals, it is shown that the only stable equilibria supported on parallel planes are either cylinders or a part of the Wulff shape. The first author is partially supported by Grant-in-Aid for Scientific Research (C) No. 16540195 of the Japan Society for the Promotion of Science.     

Axially symmetric volume constrained anisotropic mean curvature flow

We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume in comparison with their initial surface energy, we show that the flow exists for all time. Numerical simulations of the curvature flow are presented.     

Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape

We investigate ground state configurations of atomic systems in two dimensions interacting via short range pair potentials. As the number of particles tends to infinity, we show that low-energy configurations converge to a macroscopic cluster of finite surface area and constant density, the latter being given by the density of atoms per unit volume in the triangular lattice. In the special case of the Heitmann–Radin sticky disc potential and exact ground states, we show that the macroscopic cluster has a (unique) Wulff shape. This is done by showing that the atomistic energy of crystalline configurations, after subtracting off a bulk part and re-scaling, Gamma-converges to a macroscopic anisotropic surface energy.     

A mass transportation approach to quantitative isoperimetric inequalities

A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially Gromov’s proof of the isoperimetric inequality and the Brenier-McCann Theorem. A sharp quantitative version of the Brunn-Minkowski inequality for convex sets is proved as a corollary.     

Anisotropic capillary surfaces with wetting energy

We study the stability of capillary surfaces for anisotropic energies having boundaries supported in horizontal planes. A wetting energy term for the surface to plane interface is included. The first author is partially supported by Grant-in-Aid for Scientific Research (C) No. 16540195 of the Japan Society for the Promotion of Science.     

BETAS,a spectral code for three-dimensional magnetohydrodynamic equilibrium and nonlinear stability calculations

A fully three-dimensional spectral code has been developed and implemented. It computes three-dimensional equilibria using the variational approach, by minimizing the potential energy subject to appropriate constraints. A second minimization, with an additional constraint, is used to examine the stability of solutions. The magnetic field representation allows for non-nested flux surfaces. Thus it can be used to study solutions with islands and variation of the potential energy with respect to topology changes. A spectral representation in the toroidal and poloidal angles, coupled with a special choice of collocation points in the radial direction results in greatly enhanced resolution. A fast iterative method was developed, with the number of iterations required for convergence independent of the mesh size. Residuals converge exponentially to the round-off error, allowing the potential energy to be computed to the eight-digit accuracy required for nonlinear stability analysis. A small amount of artificial viscosity may be needed for convergence in cases where low-order resonances are present in the plasma region. Numerical results show convergence to known axially symmetric equilibrium solutions, as well as close agreement with eigenvalue calculations in helically symmetric stellarator configurations. Solutions exhibiting island formations are found to have lower potential energy than that of the nearby nested case. Internal modes are found to localize inside the resonant surface, with the eigenfunction having sharp gradients at the resonant surface.     

Inverse Anisotropic Curvature Flow from Convex Hypersurfaces

In this paper, we study the inverse anisotropic curvature flow from strictly convex hypersurfaces. We show the long-time existence and the convergence to the Wulff shape after rescaling, under certain conditions on the general speed functions.     

Constructing convex limit surfaces in material mechanics

In our previous paper [3], a method of constructing convex surfaces in 6D space of symmetric second-rank tensors by means of deforming the unit sphere S⁵ into a conoidal surface has been considered. Now we extend this method to parabolic surfaces and to the case where the shape of the surfaces depends also on the third tensor invariant. The resulting equations can be utilized for specifying the limit surfaces in the mechanics of isotropic and anisotropic solids. Some examples on approximating data on the experimental strength are presented.Institute of Polymer Mechanics, Latvian Academy of Sciences. Riga, LV-1006 Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 3, pp. 339–349, May–June, 1996.     

Instability Paths in the Kirchhoff–Plateau Problem

The Kirchhoff–Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a soap film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. Adopting a variational approach, we define an energy associated with shape deformations of the system and then derive general equilibrium and (linear) stability conditions by considering the first and second variations of the energy functional. We analyze in detail the transition to instability of flat circular configurations, which are ground states for the system in the absence of surface tension, when the latter is progressively increased. Such a theoretical study is particularly useful here, since the many different perturbations that can lead to instability make it challenging to perform an exhaustive experimental investigation. We generalize previous results, since we allow the filament to possess a curved intrinsic shape and also to display anisotropic flexural properties (as happens when the cross section of the filament is noncircular). This is accomplished by using a rod energy which is familiar from the modeling of DNA filaments. We find that the presence of intrinsic curvature is necessary to obtain a first buckling mode which is not purely tangent to the spanning surface. We also elucidate the role of twisting buckling modes, which become relevant in the presence of flexural anisotropy.     

Singularities on Free Surfaces of Fluid Flows

Isolated singularities on free surfaces of two-dimensional and axially symmetric three-dimensional steady potential flows with gravity are considered. A systematic study is presented, where known solutions are recovered and new ones found. In two dimensions, the singularities found include those described by the Stokes solution with a 120° angle, Craya's flow with a cusp on the free surface, Gurevich's flow with a free surface meeting a rigid plane at 120° angle, and Dagan and Tulin's flow with a horizontal free surface meeting a rigid wall at an angle less than 120°. In three dimensions, the singularities found include those in Garabedian's axially symmetric flow about a conical surface with an approximately 130° angle, flows with axially symmetric cusps, and flows with a horizontal free surface and conical stream surfaces. The Stokes, Gurevich, and Garabedian flows are exact solutions. These are used to generate local solutions, including perturbations of the Stokes solution by Grant and Longuet-Higgins and Fox, perturbations of Gurevich's flow by Vanden-Broeck and Tuck, asymmetric perturbations of Stokes flow and nonaxisymmetric perturbations of Garabedian's flow. A generalization of the Stokes solution to three fluids meeting at a point is also found.     

Overdetermined anisotropic elliptic problems

A symmetry result is established for solutions to overdetermined anisotropic elliptic problems in variational form, which extends Serrin’s theorem dealing with the isotropic radial case. The involved anisotropy arises from replacing the Euclidean norm of the gradient with an arbitrary norm in the associated variational integrals. The resulting symmetry of the solutions is that of the so-called Wulff shape.     

Study of Three-Dimensional Boundary Value Problems of Fluid Filtration in an Anisotropic Porous Medium

Three-dimensional boundary value problems (the first and second boundary value problems and the conjugation problem) of stationary filtration of fluids in anisotropic (orthotropic) and inhomogeneous porous media are posed and studied. A medium is characterized by a symmetric permeability tensor whose components generally depend on the coordinates of points of the space. A nonsingular affine transformation of coordinates is used and the problems are stated in canonical form, which dramatically simplifies their study. In the case of orthotropic and piecewise orthotropic homogeneous medium, the solution of the problem with canonical boundaries (plane and ellipsoid surfaces) can be obtained in finite form. In the general case, where the orthotropic medium is inhomogeneous and the boundary surfaces are arbitrary and smooth, the problem can be reduced to singular and hypersingular integral equations. The problems are topical, for example, in the practice of fluid (water, oil) recovery from natural anisotropic and inhomogeneous soil strata.     

On instability of equilibrium figures of rotating viscous incompressible self-gravitating liquid not subjected to capillary forces

We prove that axially symmetric equilibrium figures of uniformly rotating viscous incompressible liquid are unstable, when the second variation of the energy functional can take negative values. We assume that there is no surface tension. __________ Translated from Problemy Matematicheskogo Analiza, No. 33, 2006, pp. 91–101.     

An exterior boundary value problem in Minkowski space

In three‐dimensional Lorentz–Minkowski space ??³, we consider a spacelike plane Π and a round disc Ω over Π. In this article we seek the shapes of unbounded surfaces whose boundary is ? Ω and its mean curvature is a linear function of the distance to Π. These surfaces, called stationary surfaces, are solutions of a variational problem and governed by the Young–Laplace equation. In this sense, they generalize the surfaces with constant mean curvature in ??³. We shall describe all axially symmetric unbounded stationary surfaces with special attention in the case that the surface is asymptotic to Π at the infinity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symmetry and monotonicity of positive solutions of elliptic equations with mixed boundary conditions in a super-spherical cone

In this paper, we study symmetry and monotonicity properties for positive solutions of semilinear elliptic equations with mixed boundary conditions. We establish a version of the maximum principle for mixed boundary conditions in a narrow domain, and prove the positive solution in super-spherical cone is axially symmetric and monotone in some direction by the methods of moving planes.     

A Contact Problem for a Half-Space and a Rigid Base with an Axially Symmetric Recess

An axially symmetric problem of frictionless contact interaction of an elastic half-space and a rigid base that has a small surface recess is considered. A corresponding problem of elasticity theory is formulated. The problem is solved by the method of double integral equations. The contact pressure, the surface shape of the half-space after compression, and the size of the gap are found in closed form.     

Associated families of pluriharmonic maps and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are Kähler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic Kähler symmetric spaces.     

Associated families of pluriharmonic maps¶and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are K?hler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic K?hler symmetric spaces. Received: 8 July 1997     

Unstable Willmore surfaces of revolution subject to natural boundary conditions

In the class of surfaces with fixed boundary, critical points of the Willmore functional are naturally found to be those solutions of the Euler-Lagrange equation where the mean curvature on the boundary vanishes. We consider the case of symmetric surfaces of revolution in the setting where there are two families of stable solutions given by the catenoids. In this paper we demonstrate the existence of a third family of solutions which are unstable critical points of the Willmore functional, and which spatially lie between the upper and lower families of catenoids. Our method does not require any kind of smallness assumption, and allows us to derive some additional interesting qualitative properties of the solutions.     

Surface Tension and Wulff Shape for a Lattice Model without Spin Flip Symmetry

We propose a new definition of surface tension and check it in a spin model of the Pirogov-Sinai class without symmetry. We study the model at low temperatures on the phase transitions line and prove: (i) existence of the surface tension in the thermodynamic limit, for any orientation of the surface and in all dimensions $ d \geq 2 $; (ii) the Wulff shape constructed with such a surface tension coincides with the equilibrium shape of the cluster which appears when fixing the total spin magnetization (Wulff problem). Communicated by Vincent Rivasseau submitted 24/01/03, accepted: 12/04/03