Showing papers in "Communications in Partial Differential Equations in 1991"
TL;DR: In this article, uniform estimates and blow-up behavior for solutions of −δ(u) = v(x)eu in two dimensions are presented, with a focus on partial differential equations.
Abstract: (1991). Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)eu in two dimensions. Communications in Partial Differential Equations: Vol. 16, No. 8-9, pp. 1223-1253.
679 citations
TL;DR: In this article, the natural generalization of the natural conditions of ladyzhenskaya and uralľtseva for elliptic equations is discussed. But the natural condition of uralization is not defined.
Abstract: (1991). The natural generalizationj of the natural conditions of ladyzhenskaya and uralľtseva for elliptic equations. Communications in Partial Differential Equations: Vol. 16, No. 2-3, pp. 311-361.
644 citations
TL;DR: In this paper, weak continuity of quadratic forms on spaces of L 2 solutions of systems of partial differential equations is studied. But defect measures on the space of positions and frequencies are not defined.
Abstract: In order to study weak continuity of quadratic forms on spaces of L2 solutions of systems of partial differential equations, we define defect measures on the space of positions and frequencies.A sy...
515 citations
TL;DR: In this article, the analysis of general elliptic edge operators with constant indicide roots was studied and conditions were given to guarantee that the coefficients of this expansion are smooth. But the analysis was restricted to the case when the edge operator is semi-Fredholm.
Abstract: Examples of edge operators include Laplacians on asymptotically flat and asymptotically hyperbolic manifolds. Edge operators also arise in boundary problems around higher condimension boundaries. This paper is concerned with the analysis of general elliptic edge operators with constant indicide roots. We determine when such an operator has a distributional asymptotic expansion. Conditions are given to guarantee that the coefficients of this expansion are smooth. In Part I of this paper we only study the case when the operator is semi-Fredholm. Part II will examine edge operators with infinite dimensional kernel and cokernel, as well as develop the theory of Poisson edge operators.
503 citations
TL;DR: In this article, the existence theorem of the Vlasov-poisson system in 3D was established by a simpler method, and it was shown that the solution of the problem remains smooth for all time.
Abstract: Recently Plaffelmoser has shown that solutions of the Vlasov–Poisson system in three dimensions remain smooth for all time. This paper establish the same existence theorem by a simpler method
439 citations
TL;DR: In this paper, the authors studied the symmetry and monotonicity of solutions of fully nonlinear elliptic equations on unbounded domains and showed that they are monotonically and symmetrically symmetric.
Abstract: (1991). Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains. Communications in Partial Differential Equations: Vol. 16, No. 4-5, pp. 585-615.
191 citations
143 citations
126 citations
TL;DR: In this article, the authors considered a class of degenerate parabolic equaitons on a bounded domain with mixed boundary conditions and established the existence of a nonegative solution which is obtainable as a monotone limit of solutions of quasilinear parabolic equations.
Abstract: We consider a class of degenerate parabolic equaitons on a bounded domain with mixed boundary conditions. These problems arise, for example, in the study of flow through porous media. Under appropriate hypotheses, we establish the existence of a nonegative solution which is obtainable as a monotone limit of solutions of quasilinear parabolic equations. This construction is used establish uniqueness, cinparison, and L1 continuous dependence theorems, as well some results on blow up of solutions in finite time
125 citations
TL;DR: In this article, the uniqueness of ground state solutions of △u+f(u) = 0 in Rn, n≥3 was studied and the ground state solution was shown to be unique.
Abstract: (1991). Uniqueness of the ground state solutions of △u+f(u)=0 in Rn, n≥3. Communications in Partial Differential Equations: Vol. 16, No. 8-9, pp. 1549-1572.
124 citations
TL;DR: Theoreme d'unicite adapte au controle des solutions des problemes hyperboliques as discussed by the authors, adapting to controllable solutions of hyperbolique problems.
Abstract: (1991). Theoreme d'unicite adapte au controle des solutions des problemes hyperboliques. Communications in Partial Differential Equations: Vol. 16, No. 4-5, pp. 789-800.
TL;DR: In this article, the authors deal with the long time asymptotics of the Vlasov-mdash; Poisson-MDash; Boltzmann equation and prove existence and uniqueness for the equation.
Abstract: We deal with the long time asymptotics of the Vlasovmdash;Poissonmdash;Boltzmann equation. We prove existence and uniqueness for the equation giving the electric potential at the limit.
TL;DR: In this article, the vertial averages of the incompressible Navier-Stokes equaitons are studied from the point of view of numerical analysis:existence of solution and converagence of algorithms.
Abstract: The vertial averages of the incompressible Navier-Stokes equaitons are studied from the point of view of numerical analysis:existence of solution and converagence of algorithms. Three formulations are analysed; existence theorems are obtained when the Reynolds number is small. Convergene of a time implict algorithm is shown, while discretization in space is achived with the finite element method Resume On etudie, due vue de l'analyse numerique, le proble obtenu en Prenant la moyenne verticale des equations de Navier-stokes: on s'interesse a l'existence de solutions et a la convergence d'algorithmes. Trois formulations sont analysees; on deor$eacute;mes d'existence pour de faibles valeurs du nombrede Reynolds. On prouve la convergence d'un schema implicite en temps, tandis que la discr$eacute;tisation en espace est effectuee par elements finis
TL;DR: A free boundary problem related to singular stochastic control: the parabolic case was studied in this article. But the problem was not addressed in this paper, since it is a special case of the problem.
Abstract: (1991). A free boundary problem related to singular stochastic control: the parabolic case. Communications in Partial Differential Equations: Vol. 16, No. 2-3, pp. 373-424.
TL;DR: In this paper, the existence and weak stability of a Vlasov-Poisson syste with two typs of particles is studied, in which the electrons are supposed to be at thermal equilibrium.
Abstract: We are studying the existence and weak stability of a Vlasov–Poisson syste with two typs of particles , in which the electrons are supposed to be at thermal equilibrium. This modifies the source term in the Poisson equaitonm\, and estimates in the Marcinkiewicz space M3 for the potential are used to get the strong compactness of approximations using a new regularized kernal which preservs an approriate energy inequality.
TL;DR: Harnack's Inequality for Degenerate Parabolic Equations as discussed by the authors is a seminal work in Partial Differential Equations (PDE) and is based on the Harnack inequality.
Abstract: (1991). Harnack's Inequality for Degenerate Parabolic Equations. Communications in Partial Differential Equations: Vol. 16, No. 4-5, pp. 745-770.
TL;DR: In this paper, lower bounds of the life-span of classical solutions to the Cauchy problems for fully nonlinear wave equaitons of the form kappav;u=F(u,Du,DxDu) for the space dimension n 3.
Abstract: In an unified and simple way we get lower bounds of the life-span of classical solutions to the Cauchy problems for fully nonlinear wave equaitons of the form kappav;u=F(u,Du,DxDu) for the space dimension n 3
TL;DR: In this paper, it was shown that the partial differential operator in R3 fails to be analytic hypoelliptic, i.e., if m {3,4,5,...} is a constant.
Abstract: If m {3,4,5,...} then the partial differential operator in R3 fails to be analytic hypoelliptic. This results from the existence of parameters C such that the ordinary differential equation has a nontrivial solution which remains bounded as
TL;DR: In this article, the existence theorem of global solutions for the Cauchy of hyperbolic systems of conservaiton laws with a symmetry is established, and admission criteria for solutions to such systems are discussed.
Abstract: Hyperbolic systems of conservaiton laws with a symmetry are studied. Some peculiar phenomena for such systems are shown. Admissibility criteria for solutions to such systems are discussed. propagation and cancellation of initial oscillations for the systems are classified. As a byproduct of this study, an existence theorem of global solutions for the Cauchy of the systems is established.
TL;DR: In this article, a positive solution for the neumann problem with critical non linearity on boundary is proposed. But it is not a solution to the non-linearity on the boundary.
Abstract: (1991). Positive solution for neumann problem with critical non linearity on boundary. Communications in Partial Differential Equations: Vol. 16, No. 11, pp. 1733-1760.
TL;DR: In this paper, the inverse of the formal derivative of the coefficients-to-solutions map is used to partially invert the latter, which is suggested by seismic exploration of the earth.
Abstract: Waves travelling through a medium carry information about the medium to distant locations. This fact is fundamental to many methods in science and engineering for exploring structural properties and material parameters of media. In mathematical formulations of such situations one considers the map which sends the coefficients of a hyperbolic differential equation to the boundary values of its solutions. The inversion of this non-linear map is desired. In this paper the author uses the inverse of the formal derivative of the coefficients-to-solutions map to partially invert the latter. The problem he studies is suggested by seismic exploration of the earth.
TL;DR: On the boundary of a smooth bounded convex domain in Cn (more generally, a domain that admits a defining funtion that is plurisubharmonic on the boundary), the canonicl (narm-minimizing) solution operator of the -equation, the inverse of □b (the complex Gereen Operator) and the Szego projection are continuous is Sobolev norms as mentioned in this paper.
Abstract: On the boundary of a smooth bounded convex domain in Cn (more generally, a domain that admits a defining funtion that is plurisubharmonic on the boundary), the canonicl (narm-minimizing) solution operator of the -equation, the inverse of □b (the complex Gereen Operator) and the Szego projection are continuous is Sobolev norms.
TL;DR: In this article, the authors considered the initial boundary value problem for the nonlinear Schr-dinger equations in an exterior domain and established the existence theorem of smooth solutions by using a-priori decay estimates of solutions which were obtained by the pseudoconformal indentity.
Abstract: We consider the initial–boundary value problem for the nonlinear Schr‐dinger equations in an exterior domain. Global existence theorem of smooth solutions is established by using a–priori decay estimates of solutions which are obtained by the pseudoconformal indentity
TL;DR: In this paper, the smooth regularity of solutions of double obstacle problems involving degenerate elliptic equations was investigated, and the authors showed that the smoothness of the solution of the double obstacle problem depends on the complexity of the problem.
Abstract: (1991). Smooth regularity of solutions of double obstacle problems involving degenerate elliptic equations. Communications in Partial Differential Equations: Vol. 16, No. 4-5, pp. 821-843.
TL;DR: In this article, the authors propose a global solution to the klein-gordon equation for the two-dimensional klein gord equation, which they call the global solution of the KG equation.
Abstract: (1991). Global solution to the two–dimensional klein–gordon equation. Communications in Partial Differential Equations: Vol. 16, No. 6-7, pp. 941-995.