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1.
We consider the class of semistable solutions to semilinear equations ?Δu = f(u) in a bounded smooth domain Ω of \input amssym $\Bbb R^n$ (with Ω convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions n ≤ 4, we establish an a priori L‐bound that holds for every positive semistable solution and every nonlinearity f. This estimate leads to the boundedness of all extremal solutions when n = 4 and Ω is convex. This result was previously known only in dimensions n ≤ 3 by a result of G. Nedev. In dimensions 5 ≤ n ≤ 9 the boundedness of all extremal solutions remains an open question. It is only known to hold in the radial case Ω = BR by a result of A. Capella and the author. © 2010 Wiley Periodicals, Inc.  相似文献   

2.
On a three–dimensional exterior domain Ω we consider the Dirichlet problem for the stationary Navier–Stokes system. We construct an approximation problem on the domain ΩR, which is the intersection of Ω with a sufficiently large ball, while we create nonlinear, but local artificial boundary conditions on the truncation boundary. We prove existence and uniqueness of the solutions to the approximating problem together with asymptotically precise pointwise error estimates as R tends to infinity.  相似文献   

3.
We study the large-time asymptotics for solutions u( x , t) of the wave equation with Dirichlet boundary data, generated by a time-harmonic force distribution of frequency ω, in a class of domains with non-compact boundaries and show that the results obtained in [11] for a special class of local perturbations of Ω0 ? ?2 × (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω0. In particular, we prove that u is bounded as t → ∞ if Ω does not allow admissible standing waves of frequency ω in the sense of [8]. This implies in connection with [8]. Theorem 3.1 that the logarithmic resonances of the unperturbed domain Ω0 at the frequencies ω = πk (k = 1, 2,…) observed in [14] can be simultaneously removed by small perturbations of the boundary. As a main step of our analysis, the determination of admissible solutions of the boundary value problem ΔU + κ2U = ? f in Ω, U = 0 on ?Ω is reduced to a compact operator equation.  相似文献   

4.
The structure of nontrivial nonnegative solutions to singularly perturbed quasilinear Dirichlet problems of the form –?Δpu = f(u) in Ω, u = 0 on ?Ω, Ω ? R N a bounded smooth domain, is studied as ? → 0+, for a class of nonlinearities f(u) satisfying f(0) = f(z1) = f(z2) = 0 with 0 < z1 < z2, f < 0 in (0, z1), f > 0 in (z1, z2) and f(u)/up–1 = –∞. It is shown that there are many nontrivial nonnegative solutions with spike‐layers. Moreover, the measure of each spike‐layer is estimated as ? → 0+. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0,). Uniqueness of a solution with a boundary‐layer and many positive intermediate solutions with spike‐layers are obtained for ? sufficiently small. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

5.
The structure of positive solutions to the quasilinear elliptic problems –div(|Du|p–2Du = λf(u) in Ω, u = 0 on ∂Ω, p > 1, Ω ⊂ RNa bounded smooth domain, is precisely studied when λ is sufficiently large, for a class of logistic‐type nonlinearities f(u) satisfying that f(0) = f(a) = 0, a > 0, f(u) > 0 for u ∈ (0,a), , while u = a is a zero point of f with order ω. It is shown that if ωp – 1, the problem has a unique positive solution uλ with sup Ω uλ < a, which develops a boundary layer near ∂Ω. It is shown that if 0 < ω < p – 1, the problem also has a unique positive solution u λ, but the flat core {x ∈ Ω : uλ(x) = a} ≠ ∅︁ exists. Moreover, the asymptotic behaviour of the flat core is studied as λ → ∞.  相似文献   

6.
7.
Given a bounded regular domain Ω in ℝN, we study existence and asymptotic behaviour of the solutions of the equation Δu + |Du|q = f(u) in Ω, which diverge on ∂Ω. We extend and complete some results contained in [4].  相似文献   

8.
We consider here solutions of a nonlinear Neumann elliptic equation Δu +?f (x, u) =?0 in Ω, ?u/?ν =?0 on ?Ω, where Ω is a bounded open smooth domain in ${\mathbb{R}^N, N\geq2}$ and f satisfies super-linear and subcritical growth conditions. We prove that L ?bounds on solutions are equivalent to bounds on their Morse indices.  相似文献   

9.
In this paper we study the following problem: ut−Δu=−f(u) in Ω×(0, T)≡QT, ∂u ∂n=g(u) on ∂Ω×(0, T)≡ST, u(x, 0)=u0(x) in Ω , where Ω⊂ℝN is a smooth bounded domain, f and g are smooth functions which are positive when the argument is positive, and u0(x)>0 satisfies some smooth and compatibility conditions to guarantee the classical solution u(x, t) exists. We first obtain some existence and non-existence results for the corresponding elliptic problems. Then, we establish certain conditions for a finite time blow-up and global boundedness of the solutions of the time-dependent problem. Further, we analyse systems with same kind of boundary conditions and find some blow-up results. In the last section, we study the corresponding elliptic problems in one-dimensional domain. Our main method is the comparison principle and the construction of special forms of upper–lower solutions using related equations. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.  相似文献   

10.
Given a bounded domain Ω we consider local weak blow-up solutions to the equation Δpu=g(x)f(u) on Ω. The non-linearity f is a non-negative non-decreasing function and the weight g is a non-negative continuous function on Ω which is allowed to be unbounded on Ω. We show that if Δpw=−g(x) in the weak sense for some and f satisfies a generalized Keller-Osserman condition, then the equation Δpu=g(x)f(u) admits a non-negative local weak solution such that u(x)→∞ as x→∂Ω. Asymptotic boundary estimates of such blow-up solutions will also be investigated.  相似文献   

11.
Finite element tearing and interconnecting (FETI) methods and boundary element tearing and interconnecting (BETI) methods are special iterative substructuring methods with Lagrange multipliers. For elliptic boundary value problems on bounded domains, the condition number of these methods can be rigorously bounded by C(1+log(H/h))2, where H is the subdomain diameter and h the mesh size. The constant C is independent of H, h and possible jumps in the coefficients of the partial differential equation.In certain situations, e.g., in electromagnetic field computations, instead of imposing artificial boundary conditions one may be interested in modelling the real physical behaviour in an exterior domain with a radiation condition. In this work we analyze one-level BETI methods for such unbounded domains and show explicit condition number estimates similar to the one above. Our theoretical results are confirmed in numerical experiments.  相似文献   

12.
Let Ω denote an unbounded domain in ?n having the form Ω=?l×D with bounded cross‐section D??n?l, and let m∈? be fixed. This article considers solutions u to the scalar wave equation ?u(t,x) +(?Δ)mu(t,x) = f(x)e?iωt satisfying the homogeneous Dirichlet boundary condition. The asymptotic behaviour of u as t→∞ is investigated. Depending on the choice of f ,ω and Ω, two cases occur: Either u shows resonance, which means that ∣u(t,x)∣→∞ as t→∞ for almost every x ∈ Ω, or u satisfies the principle of limiting amplitude. Furthermore, the resolvent of the spatial operators and the validity of the principle of limiting absorption are studied. Copyright © 2003 John Wiley & Sons, Ltd.  相似文献   

13.
《偏微分方程通讯》2013,38(1-2):91-109
Abstract

Let Ω be a bounded Lipschitz domain in ? n , n ≥ 3 with connected boundary. We study the Robin boundary condition ?u/?N + bu = f ∈ L p (?Ω) on ?Ω for Laplace's equation Δu = 0 in Ω, where b is a non-negative function on ?Ω. For 1 < p < 2 + ?, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ‖(?u)*‖ p  ≤ Cf p , as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.  相似文献   

14.
15.
We consider a kinematic dynamo model in a bounded interior simply connected region Ω and in an insulating exterior region . In the so‐called direct problem, the magnetic field B and the electric field E are unknown and are driven by a given incompressible flow field w . After eliminating E , a vector and a scalar potential ansatz for B in the interior and exterior domains, respectively, are applied, leading to a coupled interface problem. We apply a finite element approach in the bounded interior domain Ω, whereas a symmetric boundary element approach in the unbounded exterior domain Ωc is used. We present results on the well‐posedness of the continuous coupled variational formulation, prove the well‐posedness and stability of the semi‐discretized and fully discretized schemes, and provide quasi‐optimal error estimates for the fully discretized scheme. Copyright © 2013 John Wiley & Sons, Ltd.  相似文献   

16.
In this paper, we study the local discontinuous Galerkin (LDG) methods for two‐dimensional nonlinear second‐order elliptic problems of the type uxx + uyy = f(x, y, u, ux, uy) , in a rectangular region Ω with classical boundary conditions on the boundary of Ω . Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L2 norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss–Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1 , and for both mixed Dirichlet–Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal.  相似文献   

17.
This paper deals with the problem ? Δ p u + α(x)|u| p?2 u = β(x)f(|u|) in Ω, subjected to the zero Neumann boundary condition, where p > 1, Ω ? ? N is bounded with smooth boundary, α, β ? L (Ω), essinfΩβ > 0, and f:[0,+ ∞) → ? is a not necessarily continuous nonlinearity that oscillates either at the origin or at the infinity. By using nonsmooth variational methods, we establish in both cases the existence of infinitely many distinct non-negative solutions of the Neumann problem. In our framework, α:Ω → ? may be a sign-changing or even a nonpositive potential, which is not permitted usually in earlier works.  相似文献   

18.
We consider a material that occupies a convex polygonal bounded domain Ω ⊂ ℝn, with regular boundary Γ = Γ1 ∪ Γ2 (with Γ ∩ Γ = ∅︁) with meas (Γ1) = |Γ1| > 0 and |Γ2| > 0. We assume, without loss of generality, that the melting temperature is 0°C. We consider the following steady‐state heat conduction problem in Ω: with α, q, B = Const > 0, and q and α represent the heat flux on Γ2 and the heat transfer coefficient on Γ1, respectively. In a previous article (Tabacman‐ Tarzia, J Diff Eq 77 (1989), 16– 37) sufficient and/or necessary conditions on data α, q, B, Ω, Γ1, Γ2 to obtain a temperature u of nonconstant sign in Ω (that is, a multidimensional steady‐state, two‐phase, Stefan problem) were studied. In this article, we consider a regular triangulation by finite element method of the domain Ω with Lagrange triangles of the type 1, with h > 0 the parameter of the discretization. We study sufficient (and/or necessary) conditions on data α, q, B, Ω, Γ1, and Γ2 to obtain a change of phase (steady‐state, two‐phase, discretized Stefan problem) in corresponding discretized domain, that is, a discrete temperature of nonconstant sign in Ω. Moreover, error bounds as a function of the parameter h, are also obtained. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq. 15: 355–369, 1999  相似文献   

19.
Qing Miao 《Applicable analysis》2013,92(12):1893-1905
For a given bounded domain Ω in R N with smooth boundary ?Ω, we give sufficient conditions on f so that the m-Laplacian equation △ m u = f(x, u, ?u) admits a boundary blow-up solution uW 1,p (Ω). Our main results are new and extend the results in J.V. Concalves and Angelo Roncalli [Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput. 182 (2006), pp. 13–23]. Our approach employs the method of lower–upper solution theorem, fixed point theory and weak comparison principle.  相似文献   

20.
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