共查询到20条相似文献,搜索用时 31 毫秒
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In this paper we prove a quantitative form of Landis’ conjecture in the plane. Precisely, let W(z) be a measurable real vector-valued function and V(z) ≥0 be a real measurable scalar function, satisfying ‖W‖ L ∞(R 2) ≤ 1 and ‖V‖ L ∞(R 2) ≤ 1. Let u be a real solution of Δu ? ?(Wu) ? Vu = 0 in R 2. Assume that u(0) = 1 and |u(z)| ≤exp (C 0|z|). Then u satisfies inf |z 0| =R sup |z?z 0| <1|u(z)| ≥exp (?CRlog R), where C depends on C 0. In addition to the case of the whole plane, we also establish a quantitative form of Landis’ conjecture defined in an exterior domain. 相似文献
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Luis A. Caffarelli 《偏微分方程通讯》2013,38(3):391-414
Consider a function u defined on n , except, perhaps, on a closed set of potential singularities . Suppose that u solves the eikonal equation ‖Du‖ = 1 in the pointwise sense on n \, where Du denotes the gradient of u and ‖·‖ is a norm on n with the dual norm ‖·‖?. For a class of norms which includes the standard p-norms on n , 1 < p < ∞, we show that if has Hausdorff 1-measure zero and n ≥ 2, then u is either affine or a “cone function,” that is, a function of the form u(x) = a ± ‖x ? z‖?. 相似文献
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Justin Holmer 《偏微分方程通讯》2013,38(5):878-905
We consider solutions u(t) to the 3d NLS equation i? t u + Δu + |u|2 u = 0 such that ‖xu(t)‖ L 2 = ∞ and u(t) is nonradial. Denoting by M[u] and E[u], the mass and energy, respectively, of a solution u, and by Q(x) the ground state solution to ?Q + ΔQ + |Q|2 Q = 0, we prove the following: if M[u]E[u] < M[Q]E[Q] and ‖u 0‖ L 2 ‖?u 0‖ L 2 > ‖Q‖ L 2 ‖?Q‖ L 2 , then either u(t) blows-up in finite positive time or u(t) exists globally for all positive time and there exists a sequence of times t n → + ∞ such that ‖?u(t n )‖ L 2 → ∞. Similar statements hold for negative time. 相似文献
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ABSTRACT The variational problem in L ∞ considered is to minimize F(u) = ‖Du‖ L ∞(Ω) subject to ∈ t Ω |Du|2 dx ≤ E for given E > 0. It is proven that a constrained minimizer exists and satisfies an Aronsson-Euler equation in the viscosity sense which depends on a parameter Λ∞ ≥ 0. This parameter splits Ω into two parts. In one part the minimizer satisfies the infinity laplace equation and in the remaining part the minimizer is the solution of the elasto-plastic torsion problem with constraint ‖Du‖ L ∞ ≤ Λ∞. 相似文献
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We study the approximation properties of a harmonic function u ∈ H1?k(Ω), k > 0, on a relatively compact subset A of Ω, using the generalized finite element method (GFEM). If Ω = ??, for a smooth, bounded domain ??, we obtain that the GFEM‐approximation uS ∈ S of u satisfies ‖u ? uS‖ ≤ Chγ‖u‖, where h is the typical size of the “elements” defining the GFEM‐space S and γ ≥ 0 is such that the local approximation spaces contain all polynomials of degree k + γ. The main technical ingredient is an extension of the classical super‐approximation results of Nitsche and Schatz (Applicable Analysis 2 (1972), 161–168; Math Comput 28 (1974), 937–958). In addition to the usual “energy” Sobolev spaces H1(??), we need also the duals of the Sobolev spaces Hm(??), m ∈ ?+. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 相似文献
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Yang Zhijian 《Mathematical Methods in the Applied Sciences》2009,32(9):1082-1104
The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in elasto‐plastic flow utt?div{|?u|m?1?u}?λΔut+Δ2u+g(u)=f(x). It proves that under rather mild conditions, the dynamical system associated with above‐mentioned IBVP possesses a global attractor, which is connected and has finite Hausdorff and fractal dimension in the phase spaces X1=H(Ω) × L2(Ω) and X=(H3(Ω)∩H(Ω)) × H(Ω), respectively. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
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Nikos Katzourakis 《偏微分方程通讯》2013,38(11):2091-2124
Let H ∈ C 2(? N×n ), H ≥ 0. The PDE system arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E ∞(u, Ω) = ‖H(Du)‖ L ∞(Ω) defined on maps u: Ω ? ? n → ? N . (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on ? N×n , which we call the “∞-Laplacian”. By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non existence of zeros of |Du| and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u′) depending on all the arguments. 相似文献
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《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 ≤ C‖f‖ 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
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The Cheeger problem for a bounded domain Ω⊂RN, N>1 consists in minimizing the quotients |∂E|/|E| among all smooth subdomains E⊂Ω and the Cheeger constant h(Ω) is the minimum of these quotients. Let be the p-torsion function, that is, the solution of torsional creep problem −Δp?p=1 in Ω, ?p=0 on ∂Ω, where Δpu:=div(|∇u|p−2∇u) is the p-Laplacian operator, p>1. The paper emphasizes the connection between these problems. We prove that . Moreover, we deduce the relation limp→1+‖?p‖L1(Ω)?CNlimp→1+‖?p‖L∞(Ω) where CN is a constant depending only of N and h(Ω), explicitely given in the paper. An eigenfunction u∈BV(Ω)∩L∞(Ω) of the Dirichlet 1-Laplacian is obtained as the strong L1 limit, as p→1+, of a subsequence of the family {?p/‖?p‖L1(Ω)}p>1. Almost all t-level sets Et of u are Cheeger sets and our estimates of u on the Cheeger set |E0| yield |B1|hN(B1)?|E0|hN(Ω), where B1 is the unit ball in RN. For Ω convex we obtain u=|E0|−1χE0. 相似文献
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We consider an inverse boundary value problem for the heat equation ? t u = div (γ? x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| t=0 = u 0, in a bounded domain Ω ? ? n , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k 2 (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?ν u(t, x)|(0, T)×?Ω. The knowledge of the initial data u 0 is not used in the proof. If we know min0≤t≤T (sup x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking. 相似文献
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Luan Thach Hoang 《Journal of Differential Equations》2008,245(9):2585-2594
We show that ‖Au+Δu‖L2(Ωε)?C(ε‖∇u‖L2(Ωε)+‖u‖L2(Ωε)), where Ωε is a thin domain in R3 of depth ε, the vector field u belongs to the domain of A, which is the Stokes operator for divergence-free vector fields on Ωε satisfying the Navier boundary condition. 相似文献