Showing papers on "Free boundary problem published in 1973"
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01 Nov 1973
TL;DR: In this paper, the first boundary value problem is formulated in the Hilbert space, and a priori estimates in the spaces Lp (?) are given for weak solutions of the problem.
Abstract: I. The First Boundary Value Problem.- 1. Notation. Auxiliary results. Formulation of the first boundary value problem.- 2. A priori estimates in the spaces Lp (?).- 3. Existence of a solution of the first boundary value problem in the spaces Lp (?).- 4. Existence of a weak solution of the first boundary value problem in Hilbert space.- 5. Solution of the first boundary value problem by the method of elliptic regularization.- 6. Uniqueness theorems for weak solutions of the first boundary value problem.- 7. A lemma on nonnegative quadratic forms.- 8. On smoothness of weak solutions of the first boundary value problem. Conditions for existence of solutions with bounded derivatives.- 9. On conditions for the existence of a solution of the first boundary value problem in the spaces of S. L. Sobolev.- II. On the Local Smoothness of Weak Solutions and Hypoellipticity of Second Order Differential Equations.- 1. The spaces Hs.- 2. Some properties of pseudodifferential operators.- 3. A necessary condition for hypoellipticity.- 4. Sufficient conditions for local smoothness of weak solutions and hypoellipticity of differential operators.- 5. A priori estimates and hypoellipticity theorems for the operators of Hormander.- 6. A priori estimates and hypoellipticity theorems for general second order differential equations.- 7. On the solution of the first boundary value problem in nonsmooth domains. The method of M. V. Keldys.- 8. On hypoellipticity of second order differential operators with analytic coefficients.- III. Additional Topics.- 1. Qualitative properties of solutions of second order equations with non- negative characteristic form.- 2. The Cauchy problem for degenerating second order hyperbolic equations.- 3. Necessary conditions for correctness of the Cauchy problem for second order equations.
654 citations
137 citations
TL;DR: In this paper, a variational free boundary problem in the theory of fluid flow through porous media is studied by a new method, which reduces the problems to variational inequalities: existence and uniqueness theorems are proved.
Abstract: Elliptic free boundary problems in the theory of fluid flow through porous media are studied by a new method, which reduces the problems to variational inequalities: existence and uniqueness theorems are proved.
111 citations
77 citations
TL;DR: In this paper, an approximate solution capability is developed to handle three dimensional anisotropic elastostatic boundary value problems, which depends crucially on the existence and explicit definition of a fundamental solution to the governing partial differential equations.
Abstract: An approximate solution capability is developed to handle three dimensional anisotropic elastostatic boundary value problems. The method depends crucially on the existence and explicit definition of a fundamental solution to the governing partial differential equations. The construction of this solution for the anisotropic elastostatic problem is presented as is the derivation of the expression for the surface tractions necessary to maintain the fundamental solution in a bounded region. After the fundamental solution and its associated surface tractions are determined, a real variable boundary integral formula is generated which can be solved numerically for the unknown surface tractions and displacements in a well-posed boundary value problem. Once all boundary quantities are known, the field solution is given by a Somigliana type integral formula. Techniques for numerically solving the integral equations are discussed.
71 citations
TL;DR: Two types of computationally efficient semi-implicit and leap-frog schemes for integrating the shallow-water equations are presented in this article, both for the pure initial value problem and for limited-area forecasts.
Abstract: Two types of computationally efficient semi-implicit and leap-frog schemes for integrating the shallow-water equations are presented. Their accuracy and stability are investigated, both for the pure initial value problem and for limited-area forecasts. The proper formulation of boundary conditions for the latter case is also discussed. Two sets of stable and sufficiently accurate boundary conditions are given. The practical usefulness of these conditions is also supported by computer experiments. DOI: 10.1111/j.2153-3490.1973.tb01601.x
70 citations
TL;DR: In this paper, a numerical method is treated for solving singular boundary value problems with solutions that can be represented as series expansions on a subinterval near the singularity, for which a difference method is used.
Abstract: A numerical method is treated for solving singular boundary value problems with solutions that can be represented as series expansions on a subinterval near the singularity. A regular boundary value problem is derived on the remaining interval, for which a difference method is used. Convergence theorems are given for general schemes and for schemes of positive type for second order equations.
51 citations
TL;DR: In this article, a pseudodifferential operator PELm(C2) with a principal symbol p E C ~ (T * ( ~ ) ~ O ), positively homogeneous of degree m, such that C~ # 0 everywhere on the set of zeros of p.
Abstract: In this paper we shall prove results, extending slightly those announced in [16]. The background is some work of HSrmander [9] and Egorov and Kondratev [5], which we shall first describe briefly. We shall always use the same notations for function spaces as HSrmander [7]. Let ~ be a paracompact C ~ manifold without boundary, T*(~) the cotangent space, T*(~)~,0 the space of non zero cotangent vecors and Lm(~) the space of pseudodifferential operators of type 1,0, introduced by HSrmander [8, 10]. In [9] HSrmander studied a pseudodifferential operator PELm(C2) with a principal symbol p E C ~ ( T * ( ~ ) ~ O ) , positively homogeneous of degree m, such that C~ # 0 everywhere on the set of zeros of p. Here C~,EC~(T * ( ~ ) ~ 0 ) is defined by
51 citations
TL;DR: In this article, a direct method is developed for obtaining the discrete solution of the polar coordinate form of Poisson's equation defined on a disk subject to both Dirichlet and Neumann boundary conditions.
Abstract: A direct method is developed for obtaining the discrete solution of the polar coordinate form of Poisson’s equation defined on a disk. The problem is solved subject to both Dirichlet and Neumann boundary conditions. For the Dirichlet boundary condition, the solution is obtained as the superposition of two solutions defined on an annulus. The direct method may be used to obtain one and a short additional calculation provides the other. For the Neumann boundary condition a solution may not exist; however, a method is given for obtaining a solution in the sense of least squares.
48 citations
TL;DR: In this paper, the axisymmetric MHD-equilibria obtained by superposing the magnetic field of the plasma currents and the field of current-carrying conductors situated outside the plasma were numerically calculated by several different iteration methods for solving the non-linear free boundary problem.
Abstract: Solutions are given for axisymmetric MHD-equilibria obtained by superposing the magnetic field of the plasma currents and the field of current-carrying conductors situated outside the plasma. These equilibria were numerically calculated by several different iteration methods for solving the non-linear free boundary problem. They can describe experiments in which the discharge time is long compared with the diffusion time of the magnetic field through the external conducting walls. Equilibria were obtained both for approximately circular plasma cross-sections and strongly elongated (elliptical) ones. Such calculations are of interest for constructing axisymmetric tokamak divertors, as well as for producing elongated crosssections, which have been postulated to be stable for higher values of plasma pressure. The results presented here are restricted to the case of zero plasma pressure and a particular current distribution in order to reduce the number of free parameters and allow more detailed discussion of the dependence of the equilibrium configurations on the arrangement of the external conductors and the currents.
47 citations
TL;DR: In this article, it was shown that the Dirichlet problem for the equation &U + p(u) = f has a variational solution for each f E V * in the form a(v, v) + snp(v) v dx (v E C, Q)) is coercive.
TL;DR: Upper and lower bounds with respect to different norms are given for the error and the gradient of the error introduced by approximating omega by a polygonal domain omega sub h with side-lengths at most h.
Abstract: : Consider Dirichlet's problem with vanishing boundary values for Poisson's equation in a smooth domain omega in the plane. In the paper upper and lower bounds with respect to different norms are given for the error and the gradient of the error introduced by approximating omega by a polygonal domain omega sub h with side-lengths at most h. (Author)
TL;DR: A boundary layer whose thickness is of the order of the mean free path squared divided by the radius of the curvature of the boundary is proposed to exist at the bottom of the kinetic boundary layer as mentioned in this paper.
Abstract: A boundary layer whose thickness is of the order of the mean free path squared divided by the radius of the curvature of the boundary is proposed to exist at the bottom of the kinetic boundary layer (or Knudsen layer) when the boundary is convex in shape.
TL;DR: In this article, general boundary value problems for second-order elliptic differential equations are considered on manifolds with edges, where in the neighborhood of an edge point the manifold is diffeomorphic to the interior of a convex dihedral angle.
Abstract: In this paper general boundary value problems for second-order elliptic differential equations are considered on manifolds with edges. It is assumed that in the neighborhood of an edge point the manifold is diffeomorphic to the interior of a convex dihedral angle. Effective conditions for normal solvability of these boundary value problems are obtained and the parametrix is constructed. The methods make use of the theory of analytic functions of several variables and automorphic functions. Bibliography: 17 items.
TL;DR: In this article, the boundary conditions for the finite difference approximation of the 1-D linearized shallow water wave equation are tested. And two types of boundary treatments are found to avoid or suppress the unfavorable oscillation due to the computational modes.
TL;DR: In this article, a free boundary problem for a parabolic equation in one space variable was studied, which arises from the problem of selecting an optimal stopping strategy for the diffusion process connected with the equation.
Abstract: This paper deals with a free boundary problem for a parabolic equation in one space variable which arises from the problem of selecting an optimal stopping strategy for the diffusion process connected with the equation I is shown that a solution of the free boundary problem yields the solution of a minimum problem concerning supersolutions of the parabolic equation as well as the solution of the optimal stopping problem Theorems regarding the exis- tence, uniqueness, regularity, and approach to the steady state of solutions of the free boundary problem are established
TL;DR: In this article, it was shown that the function of least area among all functions defined in a convex domain, vanishing on its boundary, and constrained to lie above a concave analytic obstacle leaves the obstacle along an analytic curve.
Abstract: We announce that the function of least area among all functions defined in a convex domain, vanishing on its boundary, and constrained to lie above a concave analytic obstacle leaves the obstacle along an analytic curve. We announce a result about the curve of separation determined by the solution to a variational inequality. A strictly convex domain Q with smooth boundary <3Q is given in the z = xx + ix2 plane together with a smooth function \\j/(z) which assumes a positive maximum in Q and is negative on 3Q. Let K denote the closed convex set of Lipschitz functions v satisfying v ^ i// in Q and v = 0 on <9Q. Let us denote by u the function of K which minimizes area among all functions of K\\ that is (1) ueK: (\\ _L \\u | 2 \\ l /2 n (I + \\ux\\ ) (v — u)x dx ^ 0, v e K. The existence of such w, actually satisfying ueH(Q) n C(Q), 1 ^ q < oo, 0 < A < 1, was shown in the work of H. Lewy and G. Stampacchia [7] and also in M. Giaquinta and L. Pepe [1]. For u there is a set of coincidence / consisting of the points zeQ where u(z) = il/(z). Let us call (2) r(u) = r = {(xi,X2,x3):x3 = u(z) = \\\\j{z\\zedl} the \"curve\" of separation. Up to this time it has only been known that when i// is smooth and strictly concave, T is a Jordan curve [2], On the other hand, the corresponding problem for the ueK minimizing the Dirichlet integral has been thoroughly studied by H. Lewy and G. Stampacchia [6]. We wish to announce here the THEOREM. Let ij/ be analytic and strictly concave. Let u be the solution of(\\). Then F(u) is an analytic Jordan curve (as a function of its arc length parameter). The demonstration relies on the resolution of a system of differential equations and the utilization of the system to extend analytically a conAMS 1970 subject classifications. Primary 35J20; Secondary 53A10.
TL;DR: In this paper, the independence of the thermodynamic limit on the boundary conditions is considered in the framework of functional integration, and a functional measure is constructed and the Feynman-Kac-like formula for the statistical operator written down.
Abstract: The problem of the independence of the thermodynamic limit on the boundary conditions is considered in the framework of functional integration. For every domain and every boundary condition in a sufficiently large class a functional measure is constructed and the Feynman-Kac-like formula for the statistical operator written down. Making use of some volume-independent estimates for the Green function of the heat equation, the thermodynamic limit along convex domains for general boundary conditions is proved to exist and to be equal to that for Dirichlet conditions.
Alza1
TL;DR: In this article, an analytic procedure for the calculation of nonequilibrium boundary layer flows over surfaces of arbitrary catalycities is described, and an existing equilibrium boundary layer integral matrix code is extended to include nonequ equilibrium chemistry while retaining all of the general boundary condition features built into the original code.
Abstract: The development of an analytic procedure for the calculation of nonequilibrium boundary layer flows over surfaces of arbitrary catalycities is described. An existing equilibrium boundary layer integral matrix code was extended to include nonequilibrium chemistry while retaining all of the general boundary condition features built into the original code. For particular application to the pitch-plane of shuttle type vehicles, an approximate procedure was developed to estimate the nonequilibrium and nonisentropic state at the edge of the boundary layer.
TL;DR: In this paper, the problem of completeness of a system of elementary solutions in the space of biharmonic functions with finite energy is investigated and the necessary and sufficient conditions are formulated for the boundary values which ensure that the solution belongs to the energy space.
TL;DR: In this paper, the authors present uniqueness theorems for positive solutions of the semilinear second order elliptic boundary value problem (1) below with a nonnegative nonlinearity F(x, zu) which is monotonic and sublinear in w.r.t.
TL;DR: In this paper, the authors presented a method of analysis in the classical style for the response of a thin plate to a single frequency, forced harmonic vibration, where the plate may be quadrilateral or triangular and it may have any boundary conditions.
TL;DR: The Schwarz alternating procedure as discussed by the authors provides an explicit formula for the solution to Laplace's equation on a disk using this integral, and it can be used to compute the capacity of a lens to six significant digits.
Abstract: The Schwarz alternating procedure provides a method for the numerical solution of certain boundary value problems. The region must consist of the overlapping union of two or more simple regions, and solutions to the boundary value problem must be particularly easy to compute for the simple regions. For example, Poisson’s integral provides an explicit formula for the solution to Laplace’s equation on a disk. Using this integral, Schwarz’s method permits a solution of Laplace’s equation on the union of two disks as a convergent alternating sequence of solutions on two disjoint disks. This paper discusses the numerical implementation for this model problem, paying particular attention to the treatment of singularities in the Poisson kernel and at the corners of the region. After demonstrating the speed and accuracy of the method for this model problem, the techniques are used to compute the capacity of a lens to six significant digits. This is a classical problem of long-standing interest, for which an exact...
TL;DR: In this paper, two versions of the Wazewski retract method are proven for generalized differential equations, and these theorems are then applied to study some two-point boundary value problems.
Abstract: Two versions of the Wazewski retract method are proven for generalized differential equations. These theorems are then applied to study some two-point boundary value problems for second order generalized differential equations of the type $x'' \in G( {t,x,x'} )$, where $G( {t,x,x'} )$ is an upper semicontinuous, compact, convex set-valued mapping.
01 Jan 1973
TL;DR: In this paper, convergence and general properties of mixed finite element models of a general class of boundary-value problems of the type Au + ku + f = 0, u belongs to R, and B(u - g) = 0 on boundary (R sup 1), B*(Tu - s) = 1 on boundary(R sup 2) are considered.
Abstract: : Convergence and general properties of mixed finite element models of a general class of boundary-value problems of the type Au + ku + f = 0, u belongs to R, and B(u - g) = 0 on boundary (R sup 1), B*(Tu - s) = 0 on boundary (R sup 2) are considered here where u = u(x) is a function defined on a bounded region R of (E sup n), boundary R is the smooth boundary of R, x is a point in R, A is a linear factorable operator, k is a positive constant, and B and B* are operators describing mixed boundary conditions on boundary R. (Author)