共查询到20条相似文献,搜索用时 28 毫秒
1.
Let \({\Omega}\) a bounded domain in \({\mathbb{R} ^N }\), and let \({u\in C^1 (\overline{\Omega})}\) a weak solution of the following overdetermined BVP: \({-\nabla (g(|\nabla u|)|\nabla u|^{-1}\nabla u)=f(|x|,u)}\), \({ u > 0 }\) in \({\Omega }\) and \({u=0, \ |\nabla u(x)|=\lambda (|x|)}\) on \({\partial \Omega }\), where \({g\in C([0,+\infty)\cap C^1 ((0,+\infty ) ) }\) with \({g(0)=0}\), \({g'(t) > 0}\) for \({t > 0}\), \({f\in C([0,+\infty ) \times [0, +\infty ) )}\), f is nonincreasing in \({|x|}\), \({\lambda \in C([0, +\infty )) }\) and \({\lambda }\) is positive and nondecreasing. We show that \({\Omega }\) is a ball and u satisfies some “local” kind of symmetry. The proof is based on the method of continuous Steiner symmetrization. 相似文献
2.
Let \({\mathcal{M}}\) be a semifinite von Neumann algebra with a faithful, normal, semifinite trace \({\tau}\) and E be a strongly symmetric Banach function space on \({[0,\tau({\bf 1}))}\) . We show that an operator x in the unit sphere of \({E(\mathcal{M}, \tau)}\) is k-extreme, \({k \in {\mathbb{N}}}\) , whenever its singular value function \({\mu(x)}\) is k-extreme and one of the following conditions hold (i) \({\mu(\infty, x) = \lim_{t\to\infty}\mu(t, x) = 0}\) or (ii) \({n(x)\mathcal{M}n(x^*) = 0}\) and \({|x| \geq \mu(\infty, x)s(x)}\) , where n(x) and s(x) are null and support projections of x, respectively. The converse is true whenever \({\mathcal{M}}\) is non-atomic. The global k-rotundity property follows, that is if \({\mathcal{M}}\) is non-atomic then E is k-rotund if and only if \(E(\mathcal{M}, \tau)\) is k-rotund. As a consequence of the noncommutative results we obtain that f is a k-extreme point of the unit ball of the strongly symmetric function space E if and only if its decreasing rearrangement \({\mu(f)}\) is k-extreme and \({|f| \geq \mu(\infty,f)}\) . We conclude with the corollary on orbits Ω(g) and Ω′(g). We get that f is a k-extreme point of the orbit \({\Omega(g),\,g \in L_1 + L_{\infty}}\) , or \({\Omega'(g),\,g \in L_1[0, \alpha),\,\alpha < \infty}\) , if and only if \({\mu(f) = \mu(g)}\) and \({|f| \geq \mu(\infty, f)}\) . From this we obtain a characterization of k-extreme points in Marcinkiewicz spaces. 相似文献
3.
This work deals with the solvability near the characteristic set Σ = {0} × S 1 of operators of the form \({L=\partial/\partial t + (x^na(x) + ix^mb(x))\partial/\partial x}\), \({b\not\equiv0}\) and a(0) ≠ 0, defined on \({\Omega_\epsilon=(-\epsilon,\epsilon)\times S^1}\), \({\epsilon >0 }\), where a and b are real-valued smooth functions in \({(-\epsilon,\epsilon)}\) and m ≥ 2n. It is shown that given f belonging to a subspace of finite codimension of \({C^\infty(\Omega_\epsilon)}\) there is a solution \({u\in L^\infty}\) of the equation Lu = f in a neighborhood of Σ; moreover, the L ∞ regularity is sharp. 相似文献
4.
Let \({\mathcal{B}^\omega(p, q, B_d)}\) denote the \({\omega}\)-weighted Hardy–Bloch space on the unit ball B d of \({\mathbb{C}^d}\), \({d\ge 1}\). For \({2< p,q < \infty}\) and \({f\in \mathcal{B}^\omega(p, q, B_d)}\), we obtain sharp estimates on the growth of the p-integral means M p (f, r) as \({r\to 1-}\). 相似文献
5.
Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of up, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\). 相似文献
6.
Yanheng Ding Andrzej Szulkin 《Calculus of Variations and Partial Differential Equations》2007,29(3):397-419
We study the existence and the number of decaying solutions for the semilinear Schrödinger equations \({-\varepsilon^{2}\Delta u + V(x)u = g(x,u)}\), \({\varepsilon > 0}\) small, and \({-\Delta u + \lambda V(x)u = g(x,u)}\), \({\lambda > 0}\) large. The potential V may change sign and g is either asymptotically linear or superlinear (but subcritical) in u as \({|u| \to \infty}\) . 相似文献
7.
M.-J. Huang 《Acta Mathematica Hungarica》2018,155(2):355-361
The Dirichlet eigenvalues \({\{\lambda_{n}\}_{n=1}^{\infty}}\) and Neumann eigenvalues \({\{\mu_{n}\}_{n=1}^{\infty}}\) of the string equation \({\varphi'' (x) +\lambda \rho (x) \varphi(x) =0}\) are considered. It is known that \({ \mu_{n} < \lambda_{n} < \mu_{n+2}}\) for all n. The purpose of this paper is to provide conditions on the mass density \({\rho(x)}\) under which \({\lambda_{n} < \mu_{n+1}}\) or \({\mu_{n+1} < \lambda_{n}.}\) 相似文献
8.
For a new class of g(t, x), the existence, uniqueness and stability of \({2\pi}\)-periodic solution of Duffing equation \({x'' + cx' + g(t, x) = h(t)}\) are presented. Moreover, the unique \({2\pi}\)-periodic solution is (exponentially asymptotically stable) and its rate of exponential decay c/2 is sharp. The new criterion characterizes \({g_{x}^{\prime}(t, x) - c^2/4}\) with L p -norms \({(p \in [1, \infty])}\), and the classical criterion employs the \({L^{\infty}}\)-norm. The advantage is that we can deal with the case that \({g_{x}^{\prime}(t, x) - c^2/4}\) is beyond the optimal bounds of the \({L^{\infty}}\)-norm, because of the difference between the L p -norm and the \({L^{\infty}}\)-norm. 相似文献
9.
It is shown that Gelfand transforms of elements \({f\in L^{\infty} (\mu)}\) are almost constant at almost every fiber \({\Pi^{-1}(\{x\})}\) of the spectrum of L ∞(μ) in the following sense: for each \({f\in L^{\infty} (\mu)}\) there is an open dense subset U = U(f) of this spectrum having full measure and such that the Gelfand transform of f is constant on the intersection \({\Pi^{-1}(\{x\})\cap U}\). As an application a new approach to disintegration of measures is presented, allowing one to drop the usually taken separability assumption. 相似文献
10.
Leonid Zelenko 《Integral Equations and Operator Theory》2016,85(3):307-345
In the space \({L_{2}(\mathbf{R}^{d}) (d \le 3)}\) we consider the Schrödinger operator \({H_{\gamma}=-{\Delta}+ V(\mathbf{x})\cdot+\gamma W(\mathbf{x})\cdot}\), where \({V(\mathbf{x})=V(x_{1}, x_{2}, \dots, x_{d})}\) is a periodic function with respect to all the variables, \({\gamma}\) is a small real coupling constant and the perturbation \({W(\mathbf{x})}\) tends to zero sufficiently fast as \({|\mathbf{x}|\rightarrow\infty}\). We study so called virtual bound levels of the operator \({H_\gamma}\), i.e., those eigenvalues of \({H_\gamma}\) which are born at the moment \({\gamma=0}\) in a gap \({(\lambda_-,\,\lambda_+)}\) of the spectrum of the unperturbed operator \({H_0=-\Delta+ V(\mathbf{x})\cdot}\) from an edge of this gap while \({\gamma}\) increases or decreases. We assume that the dispersion function of H0, branching from an edge of \({(\lambda_-,\lambda_+)}\), is non-degenerate in the Morse sense at its extremal set. For a definite perturbation \({(W(\mathbf{x})\ge 0)}\) we show that if d ≤ 2, then in the gap there exist virtual eigenvalues which are born from this edge. We investigate their number and an asymptotic behavior of them and of the corresponding eigenfunctions as \({\gamma\rightarrow 0}\). For an indefinite perturbation we estimate the multiplicity of virtual bound levels. In particular, we show that if d = 3 and both edges of the gap \({(\lambda_-,\,\lambda_+)}\) are non-degenerate, then under additional conditions there is a threshold for the birth of the impurity spectrum in the gap, i.e., \({\sigma(H_\gamma)\cap(\lambda_-,\,\lambda_+)=\emptyset}\) for a small enough \({|\gamma|}\). 相似文献
11.
For each \({\alpha\in[0,2)}\) we consider the eigenvalue problem \({-{\rm div}(|x|^\alpha \nabla u)=\lambda u}\) in a bounded domain \({\Omega\subset \mathbb{R}^N}\) (\({N\geq 2}\)) with smooth boundary and \({0\in \Omega}\) subject to the homogeneous Dirichlet boundary condition. Denote by \({\lambda_1(\alpha)}\) the first eigenvalue of this problem. Using \({\Gamma}\)-convergence arguments we prove the continuity of the function \({\lambda_1}\) with respect to \({\alpha}\) on the interval \({[0,2)}\). 相似文献
12.
Lauren B. M. Sager 《Integral Equations and Operator Theory》2016,86(3):377-407
We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\). 相似文献
13.
Volker Heiermann 《manuscripta mathematica》2016,150(3-4):571-583
Let F be a non-Archimedean local field of characteristic 0, let G be the group of F-rational points of a connected reductive group defined over F and let \({G\prime}\) be the group of F-rational points of its quasi-split inner form. Given standard modules \({I(\tau, \nu )}\) and \({I(\tau\prime, \nu\prime)}\) for G and \({G\prime}\) respectively with \({\tau\prime}\) a generic tempered representation, such that the Harish-Chandra \({\mu}\)-function of a representation in the supercuspidal support of \({\tau}\) agrees with the one of a generic essentially square-integral representation in some Jacquet module of \({\tau\prime}\) (after a suitable identification of the underlying spaces under which \({\nu = \nu\prime}\)), we show that \({I(\tau, \nu)}\) is irreducible whenever \({I(\tau\prime, \nu\prime)}\) is. The conditions are satisfied if the Langlands quotients \({J(\tau, \nu})\) and \({J(\tau\prime, \nu\prime)}\) of respectively \({I(\tau, \nu)}\) and \({I(\tau\prime, \nu\prime)}\) lie in the same Vogan L-packet (whenever this Vogan L-packet is defined), proving that, for any Vogan L-packet, all the standard modules with Langlands quotient in a given Vogan L-packet are irreducible, if and only if this Vogan L-packet contains a generic representation. This result for generic Vogan L-packets was proven for quasi-split orthogonal and symplectic groups by Moeglin-Waldspurger and used in their proof of the general case of the local Gan-Gross-Prasad conjectures for these groups. 相似文献
14.
We use the variational concept of \({\Gamma}\)-convergence to prove existence, stability and exhibit the geometric structure of four families of stationary solutions to the singularly perturbed parabolic equation \({u_t=\epsilon^2 {\rm div}(k\nabla u)+f(u,x)}\), for \({(t,x)\in \mathbb{R}^+\times\Omega}\), where \({\Omega\subset\mathbb{R}^n}\), \({n\geq 1}\), supplied with no-flux boundary condition. The novelty here lies in the fact that the roots of the bistable function f are not isolated, meaning that the graphs of its roots are allowed to have contact or intersect each other along a Lipschitz-continuous (n ? 1)-dimensional hypersurface \({\gamma \subset \Omega}\); across this hypersurface, the stable equilibria may have corners. The case of intersecting roots stems from the phenomenon known as exchange of stability which is characterized by \({f(\cdot,x)}\) having only two roots. 相似文献
15.
The main purpose of this paper is to prove the following result. Let R be a 2-torsion free semiprime ring with symmetric Martindale ring of quotients Q s and let \({\theta}\) and \({\phi}\) be automorphisms of R. Suppose \({T:R\rightarrow R}\) is an additive mapping satisfying the relation \({T(xyx)=T(x)\theta (y)\theta (x)-\phi (x)T(y)\theta (x)+\phi (x)\phi (y)T(x)}\), for all pairs \({x,y\in R}\). In this case T is of the form \({2T(x)=q\theta (x)+\phi (x)q}\), for all \({x\in R}\) and some fixed element \({q\in Q_{s}}\). 相似文献
16.
Let \({p \in (1,\infty)}\), \({s \in (0,1)}\) and \({\Omega \subset {\mathbb{R}^{N}}}\) a bounded open set with boundary \({\partial\Omega}\) of class C 1,1. In the first part of the article we prove an integration by parts formula for the fractional p-Laplace operator \({(-\Delta)_{p}^{s}}\) defined on \({\Omega \subset {\mathbb{R}^{N}}}\) and acting on functions that do not necessarily vanish at the boundary \({\partial\Omega}\). In the second part of the article we use the above mentioned integration by parts formula to clarify the fractional Neumann and Robin boundary conditions associated with the fractional p-Laplacian on open sets. 相似文献
17.
Let G be a group. We denote by \({\nu(G)}\) an extension of the non-abelian tensor square \({G \otimes G}\) by \({G \times G}\). We prove that if G is finite-by-nilpotent, then the non-abelian tensor square \({G \otimes G}\) is finite-by-nilpotent. Moreover, \({\nu(G)}\) is nilpotent-by-finite (Theorem A). Also we characterize BFC-groups in terms of \({\nu(G)}\) among the groups G in which the derived subgroup is finitely generated (Theorem B). 相似文献
18.
Shuaibing Luo 《Integral Equations and Operator Theory》2016,85(4):539-554
In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product \({\phi}\) on the Dirichlet space D. We prove that any two distinct nontrivial minimal reducing subspaces of \({M_\phi}\) are orthogonal. When the order n of \({\phi}\) is 2 or 3, we show that \({M_\phi}\) is reducible on D if and only if \({\phi}\) is equivalent to \({z^n}\). When the order of \({\phi}\) is 4, we determine the reducing subspaces for \({M_\phi}\), and we see that in this case \({M_\phi}\) can be reducible on D when \({\phi}\) is not equivalent to \({z^4}\). The same phenomenon happens when the order n of \({\phi}\) is not a prime number. Furthermore, we show that \({M_\phi}\) is unitarily equivalent to \({M_{z^n} (n > 1)}\) on D if and only if \({\phi = az^n}\) for some unimodular constant a. 相似文献
19.
Harald Woracek 《Monatshefte für Mathematik》2012,167(1):105-149
A string is a pair \({(L, \mathfrak{m})}\) where \({L \in[0, \infty]}\) and \({\mathfrak{m}}\) is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \({\mathfrak{m}}\) as its mass density. To each string a differential operator acting in the space \({L^2(\mathfrak{m})}\) is associated. Namely, the Kre?n–Feller differential operator \({-D_{\mathfrak{m}}D_x}\) ; its eigenvalue equation can be written, e.g., asA positive Borel measure τ on \({\mathbb R}\) is called a (canonical) spectral measure of the string \({\textsc S[L, \mathfrak{m}]}\) , if there exists an appropriately normalized Fourier transform of \({L^2(\mathfrak{m})}\) onto L 2(τ). In order that a given positive Borel measure τ is a spectral measure of some string, it is necessary that: (1) \({\int_{\mathbb R} \frac{d\tau(\lambda)}{1+|\lambda|} < \infty}\) . (2) Either \({{\rm supp} \tau \subseteq [0, \infty)}\) , or τ is discrete and has exactly one point mass in (?∞, 0). It is a deep result, going back to Kre?n in the 1950’s, that each measure with \({\int_{\mathbb R}\frac{d\tau(\lambda)}{1+|\lambda|} < \infty}\) and \({{\rm supp} \tau \subseteq [0, \infty)}\) is a spectral measure of some string, and that this string is uniquely determined by τ. The question remained open, which conditions characterize whether a measure τ with \({{\rm supp} \tau \not\subseteq [0, \infty)}\) is a spectral measure of some string. In the present paper, we answer this question. Interestingly, the solution is much more involved than the first guess might suggest.
相似文献
$$f^{\prime}(x) + z \int_0^L f(y)\,d\mathfrak{m}(y) = 0,\quad x \in\mathbb R,\ f^{\prime}(0-) = 0.$$
20.
Haibo Lin 《Archiv der Mathematik》2016,106(3):275-284
Let \({\varphi}\) be a Musielak–Orlicz function satisfying that, for any \({(x,\,t)\in{\mathbb R}^n \times [0, \infty)}\), \({\varphi(\cdot,\,t)}\) belongs to the Muckenhoupt weight class \({A_\infty({\mathbb R}^n)}\) with the critical weight exponent \({q(\varphi) \in [1,\,\infty)}\) and \({\varphi(x,\,\cdot)}\) is an Orlicz function with uniformly lower type \({p^{-}_{\varphi}}\) and uniformly upper type \({p^+_\varphi}\) satisfying \({q(\varphi) < p^{-}_{\varphi}\le p^{+}_{\varphi} < \infty}\). In this paper, the author obtains a sharp weighted bound involving \({A_\infty}\) constant for the Hardy–Littlewood maximal operator on the Musielak–Orlicz space \({L^{\varphi}}\). This result recovers the known sharp weighted estimate established by Hytönen et al. in [J. Funct. Anal. 263:3883–3899, 2012]. 相似文献