共查询到20条相似文献,搜索用时 15 毫秒
1.
Ognjen Milatovic 《Journal of Mathematical Analysis and Applications》2004,295(2):513-526
We consider a Schrödinger-type differential expression , where ∇ is a C∞-bounded Hermitian connection on a Hermitian vector bundle E of bounded geometry over a manifold of bounded geometry (M,g) with metric g and positive C∞-bounded measure dμ, and V∈Lloc1(EndE) is a linear self-adjoint bundle map. We define the maximal operator HV,max associated to HV as an operator in L2(E) given by HV,maxu=HVu for all , where ∇∗∇u in is understood in distributional sense. We give a sufficient condition for the self-adjointness of HV,max. The proof adopts Kato's technique to our setting, but it requires a more general version of Kato's inequality for Bochner Laplacian operator as well as a result on the positivity of u∈L2(M) satisfying the equation (ΔM+b)u=ν, where ΔM is the scalar Laplacian on M, b>0 is a constant and ν?0 is a positive distribution on M. For local estimates, we use a family of cut-off functions constructed with the help of regularized distance on manifolds of bounded geometry. 相似文献
2.
Ognjen Milatovic 《Integral Equations and Operator Theory》2010,68(2):243-254
We consider a differential expression ${H=\nabla^*\nabla+V}We consider a differential expression H=?*?+V{H=\nabla^*\nabla+V}, where ?{\nabla} is a Hermitian connection on a Hermitian vector bundle E over a manifold of bounded geometry (M, g) with metric g, and V is a locally integrable section of the bundle of endomorphisms of E. We give a sufficient condition for H to have an m-accretive realization in the space L
p
(E), where 1 < p < +∞. We study the same problem for the operator Δ
M
+ V in L
p
(M), where 1 < p < ∞, Δ
M
is the scalar Laplacian on a complete Riemannian manifold M, and V is a locally integrable function on M. 相似文献
3.
Weilin Zou 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(9):3069-3082
This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L1(Ω) and f∈(Lp′(Ω))N. 相似文献
4.
Ognjen Milatovic 《Journal of Mathematical Analysis and Applications》2003,283(1):304-318
We prove self-adjointness of the Schrödinger type operator , where ∇ is a Hermitian connection on a Hermitian vector bundle E over a complete Riemannian manifold M with positive smooth measure dμ which is fixed independently of the metric, and V∈Lloc1(EndE) is a Hermitian bundle endomorphism. Self-adjointness of HV is deduced from the self-adjointness of the corresponding “localized” operator. This is an extension of a result by Cycon. The proof uses the scheme of Cycon, but requires a refined integration by parts technique as well as the use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher. 相似文献
5.
Radek Honzik 《Annals of Pure and Applied Logic》2010,161(7):895-915
We say that κ is μ-hypermeasurable (or μ-strong) for a cardinal μ≥κ+ if there is an embedding j:V→M with critical point κ such that H(μ)V is included in M and j(κ)>μ. Such a j is called a witnessing embedding.Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V∗ where F is realised on all V-regular cardinals and moreover, all F(κ)-hypermeasurable cardinals κ, where F(κ)>κ+, with a witnessing embedding j such that either j(F)(κ)=κ+ or j(F)(κ)≥F(κ), are turned into singular strong limit cardinals with cofinality ω. This provides some partial information about the possible structure of a continuum function with respect to singular cardinals with countable cofinality.As a corollary, this shows that the continuum function on a singular strong limit cardinal κ of cofinality ω is virtually independent of the behaviour of the continuum function below κ, at least for continuum functions which are simple in that 2α∈{α+,α++} for every cardinal α below κ (in this case every κ++-hypermeasurable cardinal in the ground model is witnessed by a j with either j(F)(κ)≥F(κ) or j(F)(κ)=κ+). 相似文献
6.
7.
For an oriented graph D, let ID[u,v] denote the set of all vertices lying on a u-v geodesic or a v-u geodesic. For S⊆V(D), let ID[S] denote the union of all ID[u,v] for all u,v∈S. Let [S]D denote the smallest convex set containing S. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set S with ID[S]=V(D) and the hull number h(D) of an oriented graph D is the minimum cardinality of a set S with [S]D=V(D). For a connected graph G, let O(G) be the set of all orientations of G, define g−(G)=min{g(D):D∈O(G)}, g+(G)=max{g(D):D∈O(G)}, h−(G)=min{h(D):D∈O(G)}, and h+(G)=max{h(D):D∈O(G)}. By the above definitions, h−(G)≤g−(G) and h+(G)≤g+(G). In the paper, we prove that g−(G)<h+(G) for a connected graph G of order at least 3, and for any nonnegative integers a and b, there exists a connected graph G such that g−(G)−h−(G)=a and g+(G)−h+(G)=b. These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256-262]. 相似文献
8.
Ognjen Milatovic 《Journal of Mathematical Analysis and Applications》2009,354(1):125-133
We consider a Schrödinger differential expression P=ΔM+V on a complete Riemannian manifold (M,g) with metric g, where ΔM is the scalar Laplacian on M and V is a real-valued locally integrable function on M. We study two self-adjoint realizations of P in L2(M) and show their equality. This is an extension of a result of S. Agmon. 相似文献
9.
J.M. Calabuig E.A. Sánchez-Pérez 《Journal of Mathematical Analysis and Applications》2011,373(1):316-321
Let X be a Banach space and E an order continuous Banach function space over a finite measure μ. We prove that an operator T in the Köthe-Bochner space E(X) is a multiplication operator (by a function in L∞(μ)) if and only if the equality T(g〈f,x∗〉x)=g〈T(f),x∗〉x holds for every g∈L∞(μ), f∈E(X), x∈X and x∗∈X∗. 相似文献
10.
Carl Johan Casselgren 《European Journal of Combinatorics》2012,33(2):168-181
Let G=G(n) be a graph on n vertices with girth at least g and maximum degree bounded by some absolute constant Δ. Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all 2-subsets of a color set C of size σ(n). In this paper we determine, for each fixed g and growing n, the asymptotic probability of the existence of a proper coloring φ such that φ(v)∈L(v) for all v∈V(G). In particular, we show that if g is odd and σ(n)=ω(n1/(2g−2)), then the probability that G has a proper coloring from such a random list assignment tends to 1 as n→∞. Furthermore, we show that this is best possible in the sense that for each fixed odd g and each n≥g, there is a graph H=H(n,g) with bounded maximum degree and girth g, such that if σ(n)=o(n1/(2g−2)), then the probability that H has a proper coloring from such a random list assignment tends to 0 as n→∞. A corresponding result for graphs with bounded maximum degree and even girth is also given. Finally, by contrast, we show that for a complete graph on n vertices, the property of being colorable from random lists of size 2, where the lists are chosen uniformly at random from a color set of size σ(n), exhibits a sharp threshold at σ(n)=2n. 相似文献
11.
Janusz Brzd?k 《Journal of Mathematical Analysis and Applications》2011,381(1):299-307
Let C be a convex symmetric subset of a real Banach space F and K be a subgroup of the group (F,+). Let E be a real linear space, h:E→F, and h(x+y)−h(x)−h(y)∈K+C for x,y∈E. We prove that under some additional assumptions h can be represented in the form: h=A+γ+κ with an additive (or linear) A:E→F and some γ:E→C, κ:E→K. 相似文献
12.
Zhijun Zhang Yiming Guo Huabing Feng 《Journal of Mathematical Analysis and Applications》2009,352(1):77-915
By Karamata regular variation theory and constructing comparison functions, we derive that the boundary behaviour of the unique solution to a singular Dirichlet problem −Δu=b(x)g(u)+λq|∇u|, u>0, x∈Ω, u|∂Ω=0, which is independent of λq|∇uλ|, where Ω is a bounded domain with smooth boundary in RN, λ∈R, q∈(0,2], lims→0+g(s)=+∞, and b is non-negative on Ω, which may be vanishing on the boundary. 相似文献
13.
Given an undirected multigraph G=(V,E), a family W of sets W⊆V of vertices (areas), and a requirement function r:W→Z+ (where Z+ is the set of nonnegative integers), we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least r(W) edge-disjoint paths between v and W for every pair of a vertex v∈V and an area W∈W. So far this problem was shown to be NP-hard in the uniform case of r(W)=1 for each W∈W, and polynomially solvable in the uniform case of r(W)=r?2 for each W∈W. In this paper, we show that the problem can be solved in time, even if r(W)?2 holds for each W∈W, where n=|V|, m=|{{u,v}|(u,v)∈E}|, p=|W|, and r*=max{r(W)∣W∈W}. 相似文献
14.
Let L be the divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on (0,∞) of strictly critical lower type pω∈(0,1] and ρ(t)=t−1/ω−1(t−1) for t∈(0,∞). In this paper, the authors study the Orlicz-Hardy space Hω,L(Rn) and its dual space BMOρ,L*(Rn), where L* denotes the adjoint operator of L in L2(Rn). Several characterizations of Hω,L(Rn), including the molecular characterization, the Lusin-area function characterization and the maximal function characterization, are established. The ρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L(Rn) are also given. As applications, the authors show that the Riesz transform ∇L−1/2 and the Littlewood-Paley g-function gL map Hω,L(Rn) continuously into L(ω). The authors further show that the Riesz transform ∇L−1/2 maps Hω,L(Rn) into the classical Orlicz-Hardy space Hω(Rn) for and the corresponding fractional integral L−γ for certain γ>0 maps Hω,L(Rn) continuously into , where is determined by ω and γ, and satisfies the same property as ω. All these results are new even when ω(t)=tp for all t∈(0,∞) and p∈(0,1). 相似文献
15.
We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0,⋅)=u0 on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any Φ∈C(R;RN), u0∈L∞(Ω), f∈L∞(Q), a∈L∞((0,T)×∂Ω). In the L1-setting, a corresponding result is proved for the more general notion of renormalised entropy solution. 相似文献
16.
We consider equations (E) −Δu+g(u)=μ in smooth bounded domains Ω⊂RN, where g is a continuous nondecreasing function and μ is a finite measure in Ω. Given a bounded sequence of measures (μk), assume that for each k?1 there exists a solution uk of (E) with datum μk and zero boundary data. We show that if uk→u# in L1(Ω), then u# is a solution of (E) relative to some finite measure μ#. We call μ# the reduced limit of (μk). This reduced limit has the remarkable property that it does not depend on the boundary data, but only on (μk) and on g. For power nonlinearities g(t)=|t|q−1t, ∀t∈R, we show that if (μk) is nonnegative and bounded in W−2,q(Ω), then μ and μ# are absolutely continuous with respect to each other; we then produce an example where μ#≠μ. 相似文献
17.
F. Charro 《Journal of Differential Equations》2011,251(6):1562-1579
In this paper we study the monotonicity of positive (or non-negative) viscosity solutions to uniformly elliptic equations F(∇u,D2u)=f(u) in the half plane, where f is locally Lipschitz continuous (with f(0)?0) and zero Dirichlet boundary conditions are imposed. The result is obtained without assuming the u or |∇u| are bounded. 相似文献
18.
H. Hamada 《Journal of Mathematical Analysis and Applications》2011,381(1):179-186
We determine the form of polynomially bounded solutions to the Loewner differential equation that is satisfied by univalent subordination chains of the form f(z,t)=etAz+?, where A∈L(Cn,Cn) has the property m(A)>0. Here m(A)=min{R〈A(z),z〉:‖z‖=1}. We also give sufficient conditions for g(z,t)=L(f(z,t)) to be polynomially bounded, where f(z,t) is an A-normalized polynomially bounded Loewner chain solution to the Loewner differential equation. 相似文献
19.
We study the nonlinear problem −Δu+V(x)=f(x,u), x∈RN, lim|x|→∞u(x)=0, where the Schrödinger operator −Δ+V is semibounded and the nonlinearity f is linearly bounded. We establish compactness of Palais-Smale sequences and Cerami sequences for the associated energy functional under general spectral-theoretic assumptions. Applying these results, we obtain existence of three nontrivial solutions if the energy functional has a mountain-pass geometry. 相似文献
20.
Vladimir Umanskiy 《Advances in Mathematics》2003,180(1):176-186
Given p≠0 and a positive continuous function g, with g(x+T)=g(x), for some 0<T<1 and all real x, it is shown that for suitable choice of a constant C>0 the functional has a minimizer in the class of positive functions u∈C1(R) for which u(x+T)=u(x) for all x∈R. This minimizer is used to prove the existence of a positive periodic solution y∈C2(R) of two-dimensional Lp-Minkowski problem y1−p(x)(y″(x)+y(x))=g(x), where p∉{0,2}. 相似文献