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We study aspects of the analytic foundations of integration and closely related problems for functions of infinitely many variables x1,x2,…∈Dx1,x2,D. The setting is based on a reproducing kernel kk for functions on DD, a family of non-negative weights γuγu, where uu varies over all finite subsets of NN, and a probability measure ρρ on DD. We consider the weighted superposition K=uγukuK=uγuku of finite tensor products kuku of kk. Under mild assumptions we show that KK is a reproducing kernel on a properly chosen domain in the sequence space DNDN, and that the reproducing kernel Hilbert space H(K)H(K) is the orthogonal sum of the spaces H(γuku)H(γuku). Integration on H(K)H(K) can be defined in two ways, via a canonical representer or with respect to the product measure ρNρN on DNDN. We relate both approaches and provide sufficient conditions for the two approaches to coincide.  相似文献   

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We study viscous shock waves that are associated with a simple mode (λ,r)(λ,r) of a system ut+f(u)x=uxxut+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=0rλ=0 and (r⋅∇)2λ≠0(r)2λ0. Such loss of genuine nonlinearity, the simplest example of which is the cubic scalar conservation law ut+(u3)x=uxxut+(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 r,s∈]1,2[r,s]1,2[ and λ,μ∈]0,+∞[λ,μ]0,+[. In this paper, we deal with the existence and multiplicity of nonnegative and nonzero solutions of the Dirichlet problem with 00 boundary data for the semilinear elliptic equation −Δu=λus−1−ur−1Δu=λus1ur1 in Ω⊂RNΩRN, where N≥2N2. We prove that there exists a positive constant ΛΛ such that the above problem has at least two solutions, at least one solution or no solution according to whether λ>Λλ>Λ, λ=Λλ=Λ or λ<Λλ<Λ. In particular, a result by Hernandéz, Macebo and Vega is improved and, for the semilinear case, a result by Díaz and Hernandéz is partially extended to higher dimensions. Finally, an answer to a conjecture, recently stated by the author, is also given.  相似文献   

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Let XX be a finite graph. Let |V||V| be the number of its vertices and dd be its degree. Denote by F1(X)F1(X) its first spectral density function which counts the number of eigenvalues ≤λ2λ2 of the associated Laplace operator. We provide an elementary proof for the estimate F1(X)(λ)−F1(X)(0)≤2⋅(|V|−1)⋅d⋅λF1(X)(λ)F1(X)(0)2(|V|1)dλ for 0≤λ<10λ<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 L2L2-torsion.  相似文献   

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In this paper we establish the boundedness of the extremal solution uu in dimension N=4N=4 of the semilinear elliptic equation −Δu=λf(u)Δu=λf(u), in a general smooth bounded domain Ω⊂RNΩRN, with Dirichlet data u|Ω=0u|Ω=0, where ff is a C1C1 positive, nondecreasing and convex function in [0,∞)[0,) such that f(s)/s→∞f(s)/s as s→∞s.  相似文献   

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We study boundary value problems for semilinear elliptic equations of the form −Δu+g°u=μΔu+g°u=μ in a smooth bounded domain Ω⊂RNΩRN. Let {μn}{μn} and {νn}{νn} be sequences of measure in Ω and ∂Ω   respectively. Assume that there exists a solution unun with data (μn,νn)(μn,νn), i.e., unun satisfies the equation with μ=μnμ=μn and has boundary trace νnνn. Further assume that the sequences of measures converge in a weak sense to μ and ν   respectively while {un}{un} converges to u   in L1(Ω)L1(Ω). In general u   is not a solution of the boundary value problem with data (μ,ν)(μ,ν). However there exists a pair of measures (μ??)(μ?,ν?) such that u   is a solution of the boundary value problem with this data. The pair (μ??)(μ?,ν?) is called the reduced limit of the sequence {(μn,νn)}{(μn,νn)}. We investigate the relation between the weak limit and the reduced limit and the dependence of the latter on the sequence. A closely related problem was studied by Marcus and Ponce [3].  相似文献   

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We study the problem (−Δ)su=λeu(Δ)su=λeu in a bounded domain Ω⊂RnΩRn, where λ   is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7n7 for all s∈(0,1)s(0,1) whenever Ω   is, for every i=1,...,ni=1,...,n, convex in the xixi-direction and symmetric with respect to {xi=0}{xi=0}. The same holds if n=8n=8 and s?0.28206...s?0.28206..., or if n=9n=9 and s?0.63237...s?0.63237.... These results are new even in the unit ball Ω=B1Ω=B1.  相似文献   

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We study a multi-dimensional nonlocal active scalar equation of the form ut+v⋅∇u=0ut+vu=0 in R+×RdR+×Rd, where v=Λ−2+α∇uv=Λ2+αu with Λ=(−Δ)1/2Λ=(Δ)1/2. We show that when α∈(0,2]α(0,2] certain radial solutions develop gradient blowup in finite time. In the case when α=0α=0, the equations are globally well-posed with arbitrary initial data in suitable Sobolev spaces.  相似文献   

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By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term hh affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)u=b(x)g(u)+λh(x), u>0u>0 in ΩΩ, u|Ω=∞u|Ω=, where ΩΩ is a bounded domain with smooth boundary in RNRN, λ>0λ>0, g∈C1[0,∞)gC1[0,) is increasing on [0,∞)[0,), g(0)=0g(0)=0, gg is regularly varying at infinity with positive index ρρ, the weight bb, which is non-trivial and non-negative in ΩΩ, may be vanishing on the boundary, and the inhomogeneous term hh is non-negative in ΩΩ and may be singular on the boundary.  相似文献   

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We exhibit balance conditions between a Young function A and a Young function B   for a Korn type inequality to hold between the LBLB norm of the gradient of vector-valued functions and the LALA norm of its symmetric part. In particular, we extend a standard form of the Korn inequality in LpLp, with 1<p<∞1<p<, and an Orlicz version involving a Young function A   satisfying both the Δ2Δ2 and the 22 condition.  相似文献   

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We give a Sobolev inequality with the weight K(x)K(x) belonging to the class A2GnA2Gn for the function |u|t|u|t and the weight K(x)−1K(x)1 for |∇u|2|u|2. The constant in the relevant inequality is seen to depend on the GnGn and A2A2 constants of the weight.  相似文献   

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