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Grzegorz ukaszewicz Witold Sadowski 《Mathematical Methods in the Applied Sciences》2000,23(12):1023-1035
We consider a boundary value problem describing the stationary flow of a non‐Newtonian fluid through the frozen ground, with a free interface between the liquid and the solid phases. We prove the existence of at least one weak solution of the problem. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
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Yong Liu Kelei Wang Juncheng Wei 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2018,35(4):993-1017
From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov–Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that the theorem of Valdinoci et al. [41], [42] is optimal. 相似文献
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M. Struwe 《Inventiones Mathematicae》1984,75(3):547-560
ForC
4-embedded manifoldsS 3 which are differmorphic to the standard sphere in 3 the existence of non-constant minimal surfaces bounded byS and intersectingS orthogonally along their boundaries is deduced.This research was supported by the Sonderforschungsbereich 72 of the Deutsche-Forschungs-Gemeinschaft 相似文献
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Let p
*
=n/(n−2) and n≥3. In this paper, we first classify all non-constant solutions of
We then establish a sup + inf and a Moser-Trudinger type inequalities for the equation −Δu=u
+
p*
. Our results illustrate that this equation is much closer to the Liouville problem −Δu=e
u
in dimension two than the usual critical exponent equation, namely is.
Received: 11 March 2002; in final form: 8 July 2002 /
Published online: 16 May 2003 相似文献
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In the paper, the solvability of the free boundary problem of magnetohydrodynamics for a viscous incompressible fluid in a
simply connected domain is proved. The solution is obtained in the Sobolev–Slobodetskii spaces W22 + l,1 + l/2,1/2 < l < 1 W_2^{2 + l,1 + l/2},1/2 < l < 1 . Bibliography: 15 titles. 相似文献
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We formulate the filtration problem with free boundary as a problem with discontinuous nonlinearity for a degenerate elliptic
or parabolic system. We prove that a solution of the Dirichlet problem exists in both cases. We study some qualitative properties
of these solutions, e.g., the existence of “dead cores”.
Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk. Translated from Ukrainskii Matematicheskii
Zhurnal, Vol. 48, No. 9, pp. 1155–1165, September, 1996. 相似文献
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Theoretical and Mathematical Physics - We study the free boundary problem with no initial conditions for a third-order relaxation transfer equation. First, we reduce the problem to a second-order... 相似文献
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《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1999,328(4):357-362
We consider a class of solutions to the 2D Navier-Stokes equations in a strip such that the longitudinal component of velocity is a linear function of the longitudinal coordinate while the transversal component and the pressure do not depend on this coordinate. One of the strip boundaries is free and the other boundary can be a solid wall or free too. We formulate conditions for a global solvability in time of corresponding initial boundary value problems, describe their asymptotic properties, give examples of exact solutions, study blowing up solutions in the case when the both strip boundaries are free. 相似文献
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This paper deals with a certain condenser capacity in an anisotropic environment. More precisely, we are going to investigate a free boundary problem for a class of anisotropic equations on a ring domain N≥2. Our aim is to show that if the problem admits a solution in a suitable weak sense, then the underlying domain Ω is a Wulff‐shaped ring. The proof makes use of a maximum principle for an appropriate P‐function, in the sense of L. E. Payne, a Rellich type identity and some geometric arguments involving the anisotropic mean curvature of the free boundary. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
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Let u be the Newtonian potential of a real analytic distribution in an open set Ω. In this paper we assume u is analytically
continuable from the complement of Ω into some neighborhood of a point x0 ∈ ∂Ω, and we study conditions under which the analytic continuation implies that ∂Ω is a real analytic hypersurface in some
neighborhood of x0. 相似文献
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In this paper, we study the uniqueness problem of a two-phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data. We show that in general the comparison principle between the sub- and super-solutions does not hold, and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter-examples in various cases in any dimension. In one-dimension, a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed. In fact, the critical case signifies the change from uniqueness to non-uniqueness of a solution of the free boundary problem. Non-uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes, such as melting of ice or solidification of water, even with the same prescribed boundary data. However, we prove that a uniqueness theorem is true for the initial-boundary value problem of an ε-evolutionary problem which is the smoothed two-phase parabolic free boundary problem. 相似文献
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We study the equation with a discontinuous nonlinearity: ?Δu = λH(u ? 1) (H is Heaviside's unit function) in a square subject to various boundary conditions. We expect to find a curve dividing the harmonic (Δu = 0) region from the superharmonic (Δu = ?λ) region, defined by the equation u(x, y) = 1. This curve is called the free boundary since its location is determined by the solution to the problem. We use the implicit function theorem to study the effect of perturbation of the boundary conditions on known families of solutions. This justifies rigorously a formal scheme derived previously by Fleishman and Mahar. Our method also discovers bifurcations from previously known solution families. 相似文献
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Antoine Henrot 《Arkiv f?r Matematik》1994,32(1):79-98
We begin by giving some results of continuity with respect to the domain for the Dirichlet problem (without any assumption
of regularity on the domains). Then, following an idea of A. Beurling, a technique of subsolutions and supersolutions for
the so-called quadrature surface free boundary problem is presented. This technique would apply to many free boundary problems
inR
N,N≥2, which have overdetermined Cauchy data on the free boundary. Some applications to concrete examples are also given.
This work was done while the author was at the University of Nancy I (France) supported by URA 750 CNRS and project Numath
INRIA Lorraine. 相似文献
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Joachim Escher Gieri Simonett 《NoDEA : Nonlinear Differential Equations and Applications》1995,2(4):463-510
This paper is concerned with the motion of an incompressible fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed, impermeable layer and above by a free surface moving under the influence of gravity. The laminar flow is governed by Darcy's law.We prove existence of a unique maximal classical solution, using methods from the theory of maximal regularity, analytic semigroups, and Fourier multipliers. Moreover, we describe a state space which can be considered as domain of parabolicity for the problem under consideration.Supported by Schweizerischer Nationalfonds 相似文献
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In this paper, we consider a free boundary problem with volume constraint. We show that positive minimizer is locally Lipschitz
and the free boundary is analytic away from a singular set with Hausdorff dimension at most n − 8. 相似文献
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