Problem In R3 Research Articles

The Darboux Problems in R3 for a Class of Degenerating Hyperbolic Equations

In this paper we investigate some boundary value problems for degenerate hyperbolic equations in R 3 which are the three-dimensional analogues of the Darboux-problems (or Cauchy-Goursat problems) in R 2. It is well known that the Darboux-problems in the plane are well posed, while the same is not true for the corresponding problems in R 3. It turns out that two of the considered homogeneous problems have an infinite number of classical solutions. This means that for the solvability of the adjoint problems, the function on the right hand side has to be orthogonal to all the infinitely many classical solutions of the homogeneous problems. We define appropriate generalized solution and special function spaces where uniqueness and existence theorems hold for the considered problems in R 3 and show that especially M. H. Protter′s problem in R 3 has solutions with strong singularities on one part of the boundary even for smooth functions on the righthand side.

On the positive solutions of the Matukuma equation

where p > 1 and u > 0 is the gravitational potential with f R3 (uP/4n(1 + Ix12» dx representing the total mass. His aim was to improve a model given earlier in 1915 by A. S. Eddington. (See [NY1,2] for a more detailed history of these two models.) Since the Matukuma equation (1.1)is rotationally invariant, the structure of positive radial solutions u(r, (1)of the corresponding initial-value problem in R3 , (1.1)

On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R2

In this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior Dirichlet problem in the plane. Specifically, the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region, and the Laplace equation in the corresponding unbounded exterior region, in addition to appropriate boundary and transmission conditions. The main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation. We provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence, uniqueness and approximation results are established. Then, we consider the case where the corresponding operator is strongly monotone and Lipschitz-continuous, and derive asymptotic error estimates for a boundary-finite element solution. We prove the unique solvability of the discrete operator equations, and based on a Strang type abstract error estimate, we show the strong convergence of the approximated solutions. Moreover, under additional regularity assumptions on the solution of the continous operator equation, the asymptotic rate of convergenceO (h) is obtained.

Hypoellipticity for a class of degenerate elliptic operators of second order

for Xi^O. Hormander's results in [2] can not be applicable to L when (xi) has a zero of infinite order. Compared with higher dimensional cases, the problem in R2 becomes much simpler. So one can expect that one investigates hypoellipticityfor more general operators in R2. In this paper we shall give sufficientconditions of hypoellipticity for operators of the form P(x, D)=D\+ a(x)Dl+P(x, D) in R2, where x = (x1} x2) (*,£)belongs to ST,hoc (resp. SJ%) if p(x, £)eC°°(T*i2n)and if for any compact subset K of Rn and for any multi-indices a and /3 (resp. for any multi-indices a and /3 there is Ca^ = CK-aip>0 (resp. Ca,p>0) such that !/> m-|a| for xGEifand ^Rn (resp. for (x, ^)^T*Rn), where ttkbR, p\%(x, &=d$Dlp(x, a Dx=-idx, =(1+|£I2)1/2and T*Rn isidentified with RnxRn. We denote by Lxm0 the set of the pseudodifferential operators whose symbols belong to Sj%loc. Let P(x, D)eL^0 be a properly supported pseudodifferential operator, and let z°=(x°,f°)eT*J2n\0 ( = Rnx(Rn\{0})). It is said that P(x, D) is microhypoeliiptic at z°if there is a conic neighbourhood cv of 2° in T*Rn\0 such that WF(u)r\(=V=WF(Pu)ncV if ue=3)f,(Rn). We also say that P(x, D) is microhypoeliipticin a conic in a conic set cv(cT*J2n\0) (resp. in Q(cRn) if P(x, D) is microhypoeliiptic at each (x, ^)ef (resp. at

Method of boundary integral equations for the second boundary-value problem in domains with a cut

The solution of the second boundary value problem in R3 with a cut reduces to an integrodifferential equation, for the solving of which one can use operator methods of the theory of extensions of symmetric operators. The bounded solvability of the corresponding operator equation problem is proved.

An optimal-order error estimate for the discontinuous Galerkin method

In this paper a new approach is developed for analyzing the discontinuous Galerkin method for hyperbolic equations. For a model problem in R 2 {R^2} , the method is shown to converge at a rate O ( h n + 1 ) O({h^{n + 1}}) when applied with nth degree polynomial approximations over a semiuniform triangulation, assuming sufficient regularity in the solution.

Central configurations of the N-body problem via equivariant Morse theory

In this paper we use the equivariant Morse theory to give an estimate of the minimal number of central configurations in the N-body problem in ℝ3. In the case of equal masses we prove that the planar central configurations are saddle points for the potential energy. From this we deduce the presence of non-planar central configurations, for every N ≧ 4.The principal difficulty in applying Morse theory is that the potential function is defined on a manifold on which the group O(3) does not act freely. To avoid this problem the equivariant cohomology functor is applied in order to obtain the Morse inequalities.

Boundary Integral Equations for Mixed Boundary Value Problems inR3

Both exterior and interior mixed Dirichlet-Neumann problems in R3 for the scalar Helmholtz equation are solved via boundary integral equations. The integral equations are equivalent to the original problem in the sense that the traces of the weak seolution satisfy the integral equations, and, conversely, the solution of the integral equations inserted into Green's formula yields the solution of the mixed boundary value problem. The calculus of pseudodifferential operators is used to prove existence and regularity of the solution of the integral equations. The regularity results — obtained via Wiener-Hopf technique — show the explicit “edge” behavior of the solution near the submanifold which separates the Dirichlet boundary from the Neumann boundary.

Bifurcation analysis for a class of problems with a free boundary

IN THIS PAPER we study problems in a bounded region in R” which is divided into two subregions, separated by a free boundary. We search for a function u, defined on the entire domain, which satisfies a different elliptic differential equation on each subdomain and an appropriate smoothness condition on the free boundary. A major application of our analysis is the construction of equilibrium solutions for the equations describing a plasma in a cavity. For certain values of physical parameters this plasma will not fill the entire cavity but will be concentrated in a small subdomain. The aim of physical experiments of this kind is to obtain very high concentrations of plasma inside a toroidal tube (the ‘Tokamak’ machine) and very high temperatures such that a nuclear fusion reaction is ignited (see Mercier [ 11). The free boundary problem has recently received much attention (Temam [2,3], Schaeffer [4], Puel [S], Kinderlehrer and Spruck [6,7]). For the problem in R2, Temam has proved the existence of a unique solution for a limited range of physical parameters. For a very special configuration in R*, Schaeffer proved the existence of multiple solutions for sets of parameters outside Temam’s range; these solutions may be constructed numerically. In this paper we shall analyse the set of equilibrium solutions using the techniques of the theory of bifurcation. The main features of our approach are: (1) solutions are constructed analytically, (2) many bifurcation points are found, which lead to the same number of equilibrium solutions, (3) the bifurcation points do not coincide with the eigenvalues of the elliptic operator on the big domain. Our analysis is, in its main lines, conceptually not very difficult. However, working out all steps in full rigour, and all details in full technical complexity, tends to obscure the main line of reasoning. This is why the presentation of the paper is organized as follows. In Section 1 we formulate the free boundary problem and give a descriptive outline of the method of solution. In Section 2, in order to convince the reader that the method works, we describe how a special class of problems can explicitly be tackled and, as an example, we analyse fully a cylindrical domain in R3. In Section 3, we finally give a rigorous treatment of the original general problem. Various lengthy computational details, of which the reading is not strictly necessary for the understanding of this paper, are presented separately in Ref. [S].