共查询到20条相似文献,搜索用时 15 毫秒
1.
In this paper, we mainly introduce a general method to study the existence and uniqueness of solution of free boundary problems with partially degenerate diffusion. 相似文献
2.
In this paper, we prove a uniqueness theorem for a free boundary problem which is given in the form of a variational inequality. This free boundary problem arises as the limit of an equation that serves as a basic model in population biology. Apart from the interest in the problem itself, the techniques used in this paper, which are based on the regularity theory of variational inequalities and of harmonic functions, are of independent interest, and may have other applications.
3.
Zi-Cai Li Qing Fang Hung-Tsai Huang Yimin Wei 《Journal of Computational and Applied Mathematics》2009,233(2):293-307
In this paper, we consider the problem of solution uniqueness for the second order elliptic boundary value problem, by looking at its finite element or finite difference approximations. We derive several equivalent conditions, which are simpler and easier than the boundedness of the entries of the inverse matrix given in Yamamoto et al., [T. Yamamoto, S. Oishi, Q. Fang, Discretization principles for linear two-point boundary value problems, II, Numer. Funct. Anal. Optim. 29 (2008) 213–224]. The numerical experiments are provided to support the analysis made. Strictly speaking, the uniqueness of solution is equivalent to the existence of nonzero eigenvalues in the corresponding eigenvalue problem, and this condition should be checked by solving the corresponding eigenvalue problems. An application of the equivalent conditions is that we may discover the uniqueness simultaneously, while seeking the approximate solutions of elliptic boundary equations. 相似文献
4.
In this paper, we prove a bifurcation phenomenon in a two-phase, singularly perturbed, free boundary problem of phase transition. We show that the uniqueness of the solution for the two-phase problem breaks down as the boundary data decreases through a threshold value. For boundary values below the threshold, there are at least three solutions, namely, the harmonic solution which is treated as a trivial solution in the absence of a free boundary, a nontrivial minimizer of the functional under consideration, and a third solution of the mountain-pass type. We classify these solutions according to the stability through evolution. The evolution with initial data near a stable solution, such as the trivial harmonic solution or a minimizer of the functional, converges to the stable solution. On the other hand, the evolution deviates away from a non-minimal solution of the free boundary problem. 相似文献
5.
A. L. Karakhanyan C. E. Kenig H. Shahgholian 《Calculus of Variations and Partial Differential Equations》2007,28(1):15-31
We show that the free boundary ∂{u > 0}, arising from the minimizer(s) u, of the functional
approaches the (smooth) fixed boundary ∂Ω tangentially, at points where the Dirichlet data vanishes along with its gradient.
相似文献
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7.
F. Bozorgnia 《Applied Numerical Mathematics》2011,61(1):92-107
In this paper different numerical methods for a two-phase free boundary problem are discussed. In the first method a novel iterative scheme for the two-phase membrane is considered. We study the regularization method and give an a posteriori error estimate which is needed for the implementation of the regularization method. Moreover, an efficient algorithm based on the finite element method is presented. It is shown that the sequence constructed by the algorithm is monotone and converges to the solution of the given free boundary problem. These methods can be applied for the one-phase obstacle problem as well. 相似文献
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9.
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
10.
In this paper we complete the study of the regularity of the free boundary in two-phase problems for linear elliptic operators started in [M.C. Cerutti, F. Ferrari, S. Salsa, Two-phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are C1,γ, Arch. Ration. Mech. Anal. 171 (2004) 329-348]. In particular we prove that Lipschitz and flat free boundaries (in a suitable sense) are smooth. As byproduct, we prove that Lipschitz free boundaries are smooth in the case of quasilinear operators of the form div(A(x,u)∇u) with Lipschitz coefficients. 相似文献
11.
Electrostatically deflected elastic systems are increasingly used in technical applications. We study a typical situation as a free boundary problem and prove, under quite general hypotheses, the existence of a local branch of solutions. Hadamard’s variational formula is crucial in the proof, which is based on the implicit function theorem in Banach spaces. 相似文献
12.
Wenrui Hao Jonathan D. Hauenstein Bei Hu Yuan Liu Yong-Tao Zhang 《Nonlinear Analysis: Real World Applications》2012,13(2):694-709
We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive numbers ρ<R, there exists a radially symmetric stationary solution with tumor boundary r=R and necrotic core boundary r=ρ. The system depends on a positive parameter μ, which describes the tumor aggressiveness. There also exists a sequence of values μ2<μ3<? for which branches of symmetry-breaking stationary solutions bifurcate from the radially symmetric solution branch. 相似文献
13.
Wenrui Hao Jonathan D. Hauenstein Bei Hu Timothy McCoy Andrew J. Sommese 《Journal of Computational and Applied Mathematics》2013
We consider a free boundary problem modeling tumor growth where the model equations include a diffusion equation for the nutrient concentration and the Stokes equation for the proliferation of tumor cells. For any positive radius R, it is known that there exists a unique radially symmetric stationary solution. The proliferation rate μ and the cell-to-cell adhesiveness γ are two parameters for characterizing “aggressiveness” of the tumor. We compute symmetry-breaking bifurcation branches of solutions by studying a polynomial discretization of the system. By tracking the discretized system, we numerically verified a sequence of μ/γ symmetry breaking bifurcation branches. Furthermore, we study the stability of both radially symmetric and radially asymmetric stationary solutions. 相似文献
14.
Fahuai Yi 《Journal of Mathematical Analysis and Applications》2003,288(2):442-461
In this paper the Muskat problem which describes a two-phase flow of two fluids, for example, oil and water, in porous media is discussed. The problem involves in seeking two time-dependent harmonic functions u1(x,y,t) and u2(x,y,t) in oil and water regions, respectively, and the interface between oil and water, i.e., the free boundary Γ:y=ρ(x,t), such that on the free boundary
15.
This paper is devoted to the study of the bifurcation of a free boundary problem modeling the growth of tumors with the effect of surface tension being considered. The existence of infinitely many branches of bifurcation solutions is proved. The method of analysis is based on reducing the problem to an operator equation in certain Hölder space with a nonlinear Fredholm operator of index 0. The desired result then follows from the Crandall-Rabinowitz bifurcation theorem. 相似文献
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17.
Avner Friedman 《Journal of Mathematical Analysis and Applications》2007,327(1):643-664
We consider a free boundary problem modeling tumor growth in fluid-like tissue. The model equations include a diffusion equation for the nutrient concentration, and the Stokes equation with a source which represents the proliferation of tumor cells. The proliferation rate μ and the cell-to-cell adhesiveness γ which keeps the tumor intact are two parameters which characterize the “aggressiveness” of the tumor. For any positive radius R there exists a unique radially symmetric stationary solution with radius r=R. For a sequence μ/γ=Mn(R) there exist symmetry-breaking bifurcation branches of solutions with free boundary r=R+εYn,0(θ)+O(ε2) (n even ?2) for small |ε|, where Yn,0 is the spherical harmonic of mode (n,0). Furthermore, the smallest Mn(R), say Mn∗(R), is such that n∗=n∗(R)→∞ as R→∞. In this paper we prove that the radially symmetric stationary solution with R=RS is linearly stable if μ/γ<N∗(RS,γ) and linearly unstable if μ/γ>N∗(RS,γ), where N∗(RS,γ)?Mn∗(RS), and we prove that strict inequality holds if γ is small or if γ is large. The biological implications of these results are discussed at the end of the paper. 相似文献
18.
Claude-Michel Brauner Josephus Hulshof Alessandra Lunardi 《Journal of Evolution Equations》2001,1(1):85-113
We study the stability of the planar travelling wave solution to a free boundary problem for the heat equation in the whole
. We turn the problem into a fully nonlinear parabolic system and establish a stability result which is the proper generalization
of the one-dimensional case. The curvature terms contribute a gradient squared corresponding to critical growth. The latter
is eliminated by means of the Hopf-Cole transformation.
Received August 18, 2000, accepted September 27, 2000. 相似文献
19.
Borys Bazaliy 《Journal of Mathematical Analysis and Applications》2007,328(1):84-100
Some bacteria move inside cells by recruiting the actin filaments of the host cells. The filaments are polymerized at the back surface of the bacteria, and they move away, forming a “comet” tail behind the bacterium, which consists of gel network. We develop a one-dimensional mathematical model of the gel based on partial differential equations which involve the number of filaments, the density and velocity of the gel, and the pressure. The two end-points of the gel form two free boundaries. The resulting free boundary problem is rather non-standard. We prove local existence and uniqueness. 相似文献
20.
Guy Bayada 《Journal of Mathematical Analysis and Applications》2003,282(1):212-231
The asymptotic behaviour of a Stokes flow with Tresca free boundary friction conditions when one dimension of the fluid domain tends to zero is studied. A specific Reynolds equation associated with variational inequalities is obtained and uniqueness is proved. 相似文献