共查询到20条相似文献,搜索用时 718 毫秒
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We show the interpolation inequalities for derivatives in variable exponent Lebesgue–Sobolev spaces by applying the boundedness of the Hardy–Littlewood maximal operator on Lp(⋅). 相似文献
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We study viscous shock waves that are associated with a simple mode (λ,r) of a system ut+f(u)x=uxx of conservation laws and that connect states on either side of an ‘inflection’ hypersurface Σ in state space at whose points r⋅∇λ=0 and (r⋅∇)2λ≠0. Such loss of genuine nonlinearity, the simplest example of which is the cubic scalar conservation law ut+(u3)x=uxx, occurs in many physical systems. We show that such shock waves are spectrally stable if their amplitude is sufficiently small. The proof is based on a direct analysis of the eigenvalue problem by means of geometric singular perturbation theory. Well-chosen rescalings are crucial for resolving degeneracies. By results of Zumbrun the spectral stability shown here implies nonlinear stability of these shock waves. 相似文献
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Let X be a finite graph. Let |V| be the number of its vertices and d be its degree. Denote by F1(X) its first spectral density function which counts the number of eigenvalues ≤λ2 of the associated Laplace operator. We provide an elementary proof for the estimate F1(X)(λ)−F1(X)(0)≤2⋅(|V|−1)⋅d⋅λ for 0≤λ<1 which has already been proved by Friedman (1996) [3] before. We explain how this gives evidence for conjectures about approximating Fuglede–Kadison determinants and L2-torsion. 相似文献
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We study the problem of propagation of analytic regularity for semi-linear symmetric hyperbolic systems. We adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius (= width) ε0, the same happens for the solution u(t,⋅) for a certain radius ε(t), as long as the solution exists. Our focus is on precise lower bounds on the spatial radius of analyticity ε(t) as t grows. 相似文献
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The self-adjointness of H+V is studied, where H=−iα⋅∇+mβ is the free Dirac operator in R3 and V is a measure-valued potential. The potentials V under consideration are given by singular measures with respect to the Lebesgue measure, with special attention to surface measures of bounded regular domains. The existence of non-trivial eigenfunctions with zero eigenvalue naturally appears in our approach, which is based on well known estimates for the trace operator defined on classical Sobolev spaces and some algebraic identities of the Cauchy operator associated to H. 相似文献
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Susana Merchán Luigi Montoro Ireneo Peral Berardino Sciunzi 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2014
In this work we deal with the existence and qualitative properties of the solutions to a supercritical problem involving the −Δp(⋅) operator and the Hardy–Leray potential. Assuming 0∈Ω, we study the regularizing effect due to the addition of a first order nonlinear term, which provides the existence of solutions with a breaking of resonance. Once we have proved the existence of a solution, we study the qualitative properties of the solutions such as regularity, monotonicity and symmetry. 相似文献
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We consider the Cauchy problem in Rn for strongly damped wave equations. We derive asymptotic profiles of these solutions with weighted L1,1(Rn) data by using a method introduced in [9] and/or [10]. 相似文献
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We consider the minimum problem for the functional in three dimensional space, where λ>0 is a constant. We will establish a Liouville type theorem for this variational problem: if u∈C(R3) is a nonnegative and nonzero global minimizer, then u(x)=λ((x−x0)⋅ν)+ for some point x0 and some unit vector ν. 相似文献
EΩ(u)=∫Ω(|Du|2+λ2χ{u>0})
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We investigate the Cauchy problem and the initial-boundary value problem for multi-dimensional conservation laws with degenerate viscosity in the whole space and in the half-space respectively. We give the optimal decay estimates in the W1,p(1≤p≤∞) norm for the perturbation from the planar viscous rarefaction wave. The analysis based on the new Lp-energy method and L1-estimates. 相似文献
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This note deals with the strongly damped nonlinear wave equation with Dirichlet boundary conditions, where both the nonlinearities f and g exhibit a critical growth, while h is a time-independent forcing term. The existence of an exponential attractor of optimal regularity is proven. As a corollary, a regular global attractor of finite fractal dimension is obtained. 相似文献
utt−Δut−Δu+f(ut)+g(u)=h
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We prove existence and uniqueness of a renormalized solution to nonlinear elliptic equations with variable exponents and L1 data. The functional setting involves Lebesgue–Sobolev space with variable exponents W1,p(⋅)(Ω). 相似文献