Symmetry of the blow-up set of a porous medium type equation

We study the blow-up set of a porous medium type equation with source. Under some technical conditions, we prove that if the blow-up set is a bounded smooth region, then it must be a ball with a certain radius. This problem can be reduced to a sublinear elliptic equation coupled with an overdetermined boundary condition. Roughly speaking, the overdetermined boundary condition forces the domain to be a ball. Because the nonlinear term is sublinear and then non-Lipschitz, many difficulties arise if one wants to use the moving plane method to reach the goal. In particular, the Hopf boundary lemma is not applicable to this problem. Instead, we investigate various related problems in a half space and a problem in the first quadrant of the entire space, and then use the symmetry results obtained for these problems to overcome the obstacles encountered. ©1995 John Wiley & Sons, Inc.     

Finite element approach to the Stokes problem

We approximate the Stokes problem by using a finite element method. This method utilizes the approach of Kleiser–Schumann, in which a boundary condition for the pressure is implicitly defined by a condition on the velocity. We consider a suitable splitting of the unknowns that allows one to reduce the Stokes problem to a cascade of classical Dirichlet problems and to a boundary integral equation.     

Radial symmetry and partially overdetermined problems in a convex cone

We obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function P and integral identities. In dimension 2, we prove Serrin-type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone.     

On numerical implementations of a new iterative method with boundary condition splitting for solving the nonstationary stokes problem in a strip with periodicity condition

Based on finite-difference approximations in time and a bilinear finite-element approximation in spatial variables, numerical implementations of a new iterative method with boundary condition splitting are constructed for solving the Dirichlet initial-boundary value problem for the nonstationary Stokes system. The problem is considered in a strip with a periodicity condition along it. At each iteration step of the method, the original problem splits into two much simpler boundary value problems that can be stably numerically approximated. As a result, this approach can be used to construct new effective and stable numerical methods for solving the nonstationary Stokes problem. The velocity and pressure are approximated by identical bilinear finite elements, and there is no need to satisfy the well-known difficult-to-verify Ladyzhenskaya-Brezzi-Babuska condition, as is usually required when the problem is discretized as a whole. Numerical iterative methods are constructed that are first- and second-order accurate in time and second-order accurate in space in the max norm for both velocity and pressure. The numerical methods have fairly high convergence rates corresponding to those of the original iterative method at the differential level (the error decreases approximately 7 times per iteration step). Numerical results are presented that illustrate the capabilities of the methods developed.     

A stabilized implicit fractional-step method for the time-dependent Navier–Stokes equations using equal-order pairs

A stabilized implicit fractional-step method for numerical solutions of the time-dependent Navier–Stokes equations is presented in this paper. The time advancement is decomposed into a sequence of two steps: the first step has the structure of the linear elliptic problem; the second step can be seen as the generalized Stokes problem. The two problems satisfy the full homogeneous Dirichlet boundary conditions on the velocity. On the other hand, a locally stabilized term is added in the second step of the schemes. It allows one to enhance the numerical stability and efficiency by using the equal-order pairs. Convergence analysis and error estimates for the velocity and pressure of the schemes are established via the energy method. Some numerical experiments are also used to demonstrate the efficiency of this new method.     

A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative

We develop a new method to prove symmetry results for overdetermined boundary value problems. This method is based on the use of continuous Steiner symmetrization together with derivative with respect to the domain. It allows to consider nonlinear equations in divergence form with dependence inr=|x| in the nonlinearity. By using the notion of “local symmetry” introduced by the first author, we prove that the domain is necessarily a ball. We also give an example where the solution of the overdetermined problem is not radially symmetric.     

Boundary point lemmas and overdetermined problems

Two elliptic boundary value problems are considered: a problem of mixed type in a cylindrical domain, and a Dirichlet problem in an annular domain. Under some overdetermined conditions on the boundary gradient, symmetry results for domain and solution are proved. The method of proof involves the classical boundary point lemma by Hopf, as well as a suitable adaptation of it that works well at certain corners.     

The Method of Singular Equations in Boundary Value Problems in Kinetic Theory

We develop a new effective method for solving boundary value problems in kinetic theory. The method permits solving boundary value problems for mirror and diffusive boundary conditions with an arbitrary accuracy and is based on the idea of reducing the original problem to two problems of which one has a diffusion boundary condition for the reflection of molecules from the wall and the other has a mirror boundary condition. We illustrate this method with two classical problems in kinetic theory: the Kramers problem (isothermal slip) and the thermal slip problem. We use the Bhatnagar-Gross-Krook equation (with a constant collision frequency) and the Williams equation (with a collision frequency proportional to the molecular velocity).__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 437–454, June, 2005.     

ON BOUNDARY VALUE PROBLEMS FOROVERDETERMINED ELLIPTIC SYSTEMS OF TWO COMPLEX VARIABLES

In this paper,based on previous results,the Riemann-Hilbert boundary valueproblems with general forms for overdetermined elliptic equations of first order are consi-dered.The characteristic of modified function space is given.It is proved that there existsa unique solution for modified problem of the problem which we discuss.By the way,itis pointed out that there are great differences between overdetermined elliptic systems andfirst order elliptei systems in the plane.     

Two-parameter extremum problems of boundary control for stationary thermal convection equations

Two-parameter extremum problems of boundary control are formulated for the stationary thermal convection equations with Dirichlet boundary conditions for velocity and with mixed boundary conditions for temperature. The cost functional is defined as the root mean square integral deviation of the desired velocity (vorticity, or pressure) field from one given in some part of the flow region. Controls are the boundary functions involved in the Dirichlet condition for velocity on the boundary of the flow region and in the Neumann condition for temperature on part of the boundary. The uniqueness of the extremum problems is analyzed, and the stability of solutions with respect to certain perturbations in the cost functional and one of the functional parameters of the original model is estimated. Numerical results for a control problem associated with the minimization of the vorticity norm aimed at drag reduction are discussed.     

On a Class of Overdetermined Eigenvalue Problems

In this paper we present some new results of symmetry for inhomogeneous Dirichlet eigenvalue problems overdetermined by a condition involving the gradient of the first eigenfunction on the boundary. One specificity of the problem studied is the dependence of the equation and the boundary condition on the distance to the origin. The method of investigation is based on the use of continuous Steiner symmetrization together with some domain derivative tools. An application is given to the study of an overdetermined eigenvalue problem for a wedge-like membrane. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Stability estimates for the solutions of control problems for the stationary magnetohydrodynamic equations

We study control problems for the stationary magnetohydrodynamic equations. In these problems, one has to find an unknown vector function occurring in the boundary condition for the magnetic field and the solution of the boundary value problem in question by minimizing a performance functional depending on the velocity and pressure. We derive new a priori estimates for the solutions of the original boundary value problem and the extremal problem and prove theorems on the local uniqueness and stability of solutions for specific performance functionals.     

Multipole fast algorithm for the least‐squares approach of the method of fundamental solutions for three‐dimensional harmonic problems

In this article we describe an improvement in the speed of computation for the least‐squares method of fundamental solutions (MFS) by means of Greengard and Rokhlin's FMA. Iterative solution of the linear system of equations is performed for the equations given by the least‐squares formulation of the MFS. The results of applying the method to test problems from potential theory with a number of boundary points in the order of 80,000 show that the method can achieve fast solutions for the potential and its directional derivatives. The results show little loss of accuracy and a major reduction in the memory requirements compared to the direct solution method of the least squares problem with storage of the full MFS matrix. The method can be extended to the solution of overdetermined systems of equations arising from boundary integral methods with a large number of boundary integration points. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 828–845, 2003.     

Partially overdetermined elliptic boundary value problems

We consider semilinear elliptic Dirichlet problems in bounded domains, overdetermined with a Neumann condition on a proper part of the boundary. Under different kinds of assumptions, we show that these problems admit a solution only if the domain is a ball. When these assumptions are not fulfilled, we discuss possible counterexamples to symmetry. We also consider Neumann problems overdetermined with a Dirichlet condition on a proper part of the boundary, and the case of partially overdetermined problems on exterior domains.     

A nonlinear parabolic free boundary problem

Summary When two immiscible fluids in a porous medium are in contact with one another, an interface is formed and the movement of the fluids results in a free boundary problem for determining the location of the interface along with the pressure distribution throughout the medium. The pressure satisfies a nonlinear parabolic partial differential equation on each side of the interface while the pressure and the volumetric velocity are continuous across the interface. The movement of the interface is related to the pressure through Darcy’s law. Two kinds of boundary conditions are considered. In Part I the pressure is prescribed on the known boundary. A weak formulation of the classical problem is obtained and the existence of a weak solution is demonstrated as a limit of a sequence of classical solutions to certain parabolic boundary value problems. In Part II the same analysis is carried out when the flux is specified on the known boundary, employing special techniques to obtain the uniform parabolicity of the sequence of approximating problems. Entrata in Redazione il 29 novembre 1975. This research was supported in part by the National Science Foundation, the Senior Fellowship Program of the North Atlantic Treaty Organization, the Italian Consiglio Nazionale delle Ricerche, and the Texas Tech. University.     

Radial symmetry and uniqueness for an overdetermined problem

Consider a function u, harmonic in a ring‐shaped domain and taking two constant (distinct) values on the two connected components of the boundary. If we know in advance that one of the components is a sphere, and that u satisfies some overdetermined condition on the other one, can we conclude that u is radial? This paper answers this question for certain overdetermined conditions on the gradient of u, generalizing some previous results. Conditions depending on the principal curvatures of the boundary are also investigated. Existence and uniqueness of a radial solution to the overdetermined problem are discussed. Some extensions to ellipsoidal domains, as well as to quasilinear elliptic equations, are carried out. Copyright © 2001 John Wiley & Sons, Ltd.     

Uniqueness of solutions to inverse Sturm–Liouville problems with potential using three spectra

The uniqueness of solutions to two inverse Sturm–Liouville problems using three spectra is proven, based on the uniqueness of the solution-pair to an overdetermined Goursat–Cauchy boundary value problem. We discuss the uniqueness of the potential for a Dirichlet boundary condition at an arbitrary interior node, and for a Robin boundary condition at an arbitrary interior node, whereas at the exterior nodes we have Dirichlet boundary conditions in both situations. Here we are particularly concerned with potential functions that are L²(0,a).     

Analytic investigation of features of stresses in plane nonstationary contact problems with moving boundaries

We propose a technique for the analytic investigation of features of contact stresses in the vicinity of the nonstationary moving boundary of a contact region in plane nonstationary contact problems with moving boundaries, which is based on the reduction of a boundary two-dimensional singular integral equation resolving the problem to a system of two one-dimensional singular equations. As tools of research, a method for the reduction of singular integral equations to an equivalent Riemann type problem for piecewise analytic functions and a technique of fractional integro-differentiation are used. It is shown that, on the moving boundary of the contact region, a power singularity, the order of which depends on the velocity of the boundary, takes place.     

The Least Square Method for Systems of Linear Ordinary Differential Equations

Computational Mathematics and Mathematical Physics - Some techniques for applying the least square method to solve boundary value problems for overdetermined systems of linear ordinary differential...     

Singular integrals with generalized Cauchy kernels for the velocity field in boundary value problems of filtration in an anisotropic inhomogeneous porous layer

We study two-dimensional stationary and nonstationary boundary value problems of fluid filtration in an anisotropic inhomogeneous porous layer whose conductivity is modeled by a not necessarily symmetric tensor. For the velocity field, we introduce generalized singular Cauchy and Cauchy type integrals whose kernels are expressed via the leading solutions of the main equations and have a hydrodynamic interpretation. We obtain the limit values of a Cauchy type generalized integral (Sokhotskii-Plemelj generalized formulas). This permits one to develop a method for solving boundary value problems for the filtration velocity field. The idea of the method and its efficiency are illustrated for the boundary value problem of filtration in adjacent layers of distinct conductivities and the problem of the evolution of liquid interface.