A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary

In this paper, we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact Riemannian 3‐manifold M with boundary ?M. These minimal surfaces are either disjoint from ?M or meet ?M orthogonally. The main feature of our result is that there is no assumptions on the curvature of M or convexity of ?M. We prove the boundary regularity of the minimal surfaces at their free boundaries. Furthermore, we define a topological invariant, the filling genus, for compact 3‐manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold M. Our proof employs a variant of the min‐max construction used by Colding and De Lellis on closed embedded minimal surfaces, which were first developed by Almgren and Pitts.© 2014 Wiley Periodicals, Inc.     

Ruin theory for classical risk process that is perturbed by diffusion with risky investments

In this paper, we study the ruin theory for classical risk process that is perturbed by diffusion with risky investments. We obtain the upper bound for the minimal ruin probability. We also investigate the relationships between the adjustment coefficient and the diffusion volatility parameter, the risk‐free rate and the correlation coefficient by numerical calculation. We give the relationships between ruin and investment. Copyright © 2008 John Wiley & Sons, Ltd.     

Well-posedness of the free boundary problem in compressible elastodynamics

We study the free boundary problem for the flow of a compressible isentropic inviscid elastic fluid. At the free boundary moving with the velocity of the fluid particles the columns of the deformation gradient are tangent to the boundary and the pressure vanishes outside the flow domain. We prove the local-in-time existence of a unique smooth solution of the free boundary problem provided that among three columns of the deformation gradient there are two which are non-collinear vectors at each point of the initial free boundary. If this non-collinearity condition fails, the local-in-time existence is proved under the classical Rayleigh–Taylor sign condition satisfied at the first moment. By constructing an Hadamard-type ill-posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non-collinearity condition and the Rayleigh–Taylor sign condition leads to Rayleigh–Taylor instability.     

Gaussian Curvature Estimates for the Convex Level Sets of Solutions for Some Nonlinear Elliptic Partial Differential Equations

We give lower bound estimates for the Gaussian curvature of convex level sets of minimal surfaces and the solutions to semilinear elliptic equations in terms of the norm of boundary gradient and the Gaussian curvature of the boundary.     

Pseudo‐reflection phenomena for singularities in thin elastic shells

We consider problems of statics of thin elastic shells with hyperbolic middle surface subjected to boundary conditions ensuring the geometric rigidity of the surface. The asymptotic behaviour of the solutions when the relative thickness tends to zero is then given by the membrane approximation. It is a hyperbolic problem propagating singularities along the characteristics. We address here the reflection phenomena when the propagated singularities arrive to a boundary. As the boundary conditions are not the classical ones for a hyperbolic system, there are various cases of reflection. Roughly speaking, singularities provoked elsewhere are not reflected at all at a free boundary, whereas at a fixed (or clamped) boundary the reflected singularity is less singular than the incident one. Reflection of singularities provoked along a non‐characteristic curve C are also considered. Copyright © 2003 John Wiley & Sons, Ltd.     

Hamiltonian long‐wave expansions for free surfaces and interfaces

The theory of internal waves between two bodies of immiscible fluid is important both for its interest to ocean engineering and as a source of numerous interesting mathematical model equations that exhibit nonlinearity and dispersion. In this paper we derive a Hamiltonian formulation of the problem of a dynamic free interface (with rigid lid upper boundary conditions), and of a free surface and a free interface, this latter situation occurring more commonly in experiment and in nature. From the formulation, we develop a Hamiltonian perturbation theory for the long‐wave limits, and we carry out a systematic analysis of the principal long‐wave scaling regimes. This analysis provides a uniform treatment of the classical works of Peters and Stoker (28), Benjamin (3, 4), Ono (26), and many others. Our considerations include the Boussinesq and Korteweg–de Vries (KdV) regimes over finite‐depth fluids, the Benjamin‐Ono regimes in the situation in which one fluid layer is infinitely deep, and the intermediate long‐wave regimes. In addition, we describe a novel class of scaling regimes of the problem, in which the amplitude of the interface disturbance is of the same order as the mean fluid depth, and the characteristic small parameter corresponds to the slope of the interface. Our principal results are that we highlight the discrepancies between the case of rigid lid and of free surface upper boundary conditions, which in some circumstances can be significant. Motivated by the recent results of Choi and Camassa (6, 7), we also derive novel systems of nonlinear dispersive long‐wave equations in the large‐amplitude, small‐slope regime. Our formulation of the dynamical free‐surface, free‐interface problem is shown to be very effective for perturbation calculations; in addition, it holds promise as a basis for numerical simulations. © 2005 Wiley Periodicals, Inc.     

ON EXISTENCE AND UNIQUENESS IN A FREE BOUNDARY PROBLEM FROM COMBUSTION

ABSTRACT

We study a free boundary problem for the heat equation describing the propagation of laminar flames under certain geometric assumptions on the initial data. The problem arises as the limit of a singular perturbation problem, and generally no uniqueness of limit solutions can be expected. However, if the initial data is starshaped, we show that the limit solution is unique and coincides with the minimal classical supersolution. Under certain convexity assumption on the data, we prove first that the limit solution is a classical solution of the free boundary problem for a short time interval, and then that the solution, in fact, stays classical as long as it does not vanish identically.     

Local Uniformization and Free Boundary Regularity of Minimal Singular Surfaces

We continue the study of minimal singular surfaces obtained by a minimization of a weighted energy functional in the spirit of J. Douglas’s approach to the Plateau problem. Modeling soap films spanning wire frames, a singular surface is the union of three disk-type surfaces meeting along a curve which we call the free boundary. We obtain an a priori regularity result concerning the real analyticity of the free boundary curve. Using the free boundary regularity of the harmonic map, we construct local harmonic isothermal coordinates for the minimal singular surface in a neighborhood of a point on the free boundary. Applications of the local uniformization are discussed in relation to H. Lewy’s real analytic extension of minimal surfaces.     

Projector preconditioning for partially bound‐constrained quadratic optimization

Preconditioning by a conjugate projector is combined with the recently proposed modified proportioning with reduced gradient projection (MPRGP) algorithm for the solution of bound‐constrained quadratic programming problems. If applied to the partially bound‐constrained problems, such as those arising from the application of FETI‐based domain decomposition methods to the discretized elliptic boundary variational inequalities, the resulting algorithm is shown to have better bound on the rate of convergence than the original MPRGP algorithm. The performance of the algorithm is illustrated on the solution of a model boundary variational inequality. Copyright © 2007 John Wiley & Sons, Ltd.     

Existence and geometric structure of metrics on surfaces which extremize eigenvalues

This is an exposition of the research area around our lecture at the 60^th anniversary conference of IMPA which was held in October of 2012. It is a survey of results which have been obtained over many years concerning sharp upper bounds on the first eigenvalue of a surface, either with or without boundary, in terms or area or boundary length and the surface topology. It is mostly expository, but contains a new coarse upper bound for non-orientable surfaces with boundary. It also contains a classical reformulation of recent results in [10].     

Minimal surfaces,corners, and wires

Weierstrass representations are given for minimal surfaces that have free boundaries on two planes that meet at an arbitrary dihedral angle. The contact angles of a surface on the planes may be different. These surfaces illustrate the behavior of soapfilms in convex and nonconvex comers. They can also be used to show how a boundary wire can penetrate a soapfilm with a free end, as in the overhand knot surface. They should also cast light on the behavior of capillary surfaces.     

A Bernoulli problem with non-constant gradient boundary constraint

In this article, we present a result about the existence and convexity of solutions to a free boundary problem of Bernoulli type, with non-constant gradient boundary constraint depending on the outer unit normal. In particular, we prove that, in the convex case, the existence of a subsolution guarantees the existence of a classical solution, which is proved to be convex.     

Stefan-like problems with curvature

We study a two-phase free boundary problem in which the speed of the free boundary depends also on its curvature. It is assumed that the free boundary is Lipschitz and it is proved that the solution as well as the free boundary are classical.     

论修正的Castigliano定理

本文给出了修正的Castigliano定理,与经典的Castigliano定理相比较,修正的Castigliano定理应用方便而且广泛。对经典Castigliano定理的修正是在两个方面,第一个修正是以外载荷与影响函数乘积的表达式代替经典Castigliano定理中余能密度对集中力偏导数的表达式,这一修正为我们在复杂载荷作用下的计算带来极大的简便;第二个修正是在经典Castigliano定理中引入了非齐次边界位移与影响函数乘积的表达式,这一修正为求解复杂边界条件的问题提供了理论基础。我们还说明了如何应用修正的Castigliano定理求解表面结构力学问题的方法。最后,作为修正的Castigliano定理的应用算例,我们求解了两邻边固定另两邻边自由的矩形板的挠曲面方程。     

Stabilität Mehrfach Zusammenhängender Minimalflächen

Multiply connected minimal surfaces of genus 0 with only simple interior branch points, for which the corresponding boundary value problem $$\Delta h - K|x_z |^2 h = 0; h_{|\partial \Omega } = 0$$ (K is the Gauss curvature and x_z is the complex gradient of the surface x) is uniquely solvable and which have the property, that the condition K|x_z|²≠0 holds in the branch points, are always isolated and stable solutions of the Plateau problem, corresponding to their boundary curves. To achieve these results one has to consider the conformal type as a variable. We give a method to perform the variation of the conformal type for holomorphic functions. Using the Weierstrass representation we thus obtain a differentiable structure on the set of multiply connected minimal surfaces. We find interesting connections between the classical Riemann-Hilbert problem and Fredholm properties of a projection operator on this manifold.     

Harmonic map heat flow with free boundary

In this paper, we study the harmonic map heat flow with free boundary from a Riemannian surface with smooth boundary into a compact Riemannian manifold. As a consequence, we get at least one disk-type minimal surface in a compact Riemannian manifold without minimal 2-sphere.     

Convergence of coercive approximations for a model of gradient type in poroplasticity

We study the existence theory to the quasi‐static initial‐boundary‐value problem of poroplasticity. In this article the classical quasi‐static Biot model is considered for soil consolidation coupled with a nonlinear system of differential equations. This work, for the poroplasticity model of monotone‐gradient type, presents a convergence result of the coercive approximation to the solution of the original noncoercive problem. Copyright © 2008 John Wiley & Sons, Ltd.     

Gaussian curvature estimates for the convex level sets of p‐harmonic functions

We give a positive lower bound for the Gaussian curvature of the convex level sets of p‐harmonic functions with the Gaussian curvature of the boundary and the norm of the gradient on the boundary. Combining the deformation process, this estimate gives a new approach to studying the convexity of the level sets of the p‐harmonic function. © 2010 Wiley Periodicals, Inc.     

A finite volume scheme for a model coupling free surface and pressurised flows in pipes

A model is derived for the coupling of transient free surface and pressurized flows. The resulting system of equations is written under a conservative form with discontinuous gradient of pressure. We treat the transition point between the two types of flows as a free boundary associated to a discontinuity of the gradient of pressure. The numerical simulation is performed by making use of a Roe-like finite volume scheme that we adapted to such discontinuities in the flux. The validation is performed by comparison with experimental results.     

Non-parametric radially symmetric mean curvature flow with a free boundary

We study the mean curvature flow of radially symmetric graphs with prescribed contact angle on a fixed, smooth hypersurface in Euclidean space. In this paper we treat two distinct problems. The first problem has a free Neumann boundary only, while the second has two disjoint boundaries, a free Neumann boundary and a fixed Dirichlet height. We separate the two problems and prove that under certain initial conditions we have either long time existence followed by convergence to a minimal surface, or finite maximal time of existence at the end of which the graphs develop a curvature singularity. We also give a rate of convergence for the singularity.