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Consider the supremal functional applied to \(W^{1,\infty }\) maps \(u:\Omega \subseteq \mathbb {R}\longrightarrow \mathbb {R}^N\), \(N\ge 1\). Under certain assumptions on \(\mathscr {L}\), we prove for any given boundary data the existence of a map which is: Our method is based on \(L^p\) approximations and stable a priori partial regularity estimates. For item ii) we utilise the recently proposed by the author notion of \(\mathcal {D}\)-solutions in order to characterise the limit as a generalised solution. Our results are motivated from and apply to Data Assimilation in Meteorology.
相似文献
$$\begin{aligned} E_\infty (u,A) := \Vert \mathscr {L}(\cdot ,u,\mathrm {D}u)\Vert _{L^\infty (A)},\quad A\subseteq \Omega , \end{aligned}$$
(1)
- (i)a vectorial Absolute Minimiser of (1) in the sense of Aronsson,
- (ii)a generalised solution to the ODE system associated to (1) as the analogue of the Euler-Lagrange equations,
- (iii)a limit of minimisers of the respective \(L^p\) functionals as \(p\rightarrow \infty \) for any \(q\ge 1\) in the strong \(W^{1,q}\) topology and
- (iv)partially \(C^2\) on \(\Omega \) off an exceptional compact nowhere dense set.
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We construct identity-based encryption and inner product encryption schemes under the decision linear assumption. Their private user keys are leakage-resilient in several scenarios. In particular, In addition, we prove that our IBE schemes are anonymous under the DLIN assumption, so that ciphertexts leaks no information on the corresponding identities. Similarly, attributes in IPE are proved computationally hidden in the corresponding ciphertexts.
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- In the bounded memory leakage model (Akavia et al., TCC, vol. 5444, pp. 474–495, 2009), our basic schemes reach the maximum-possible leakage rate \(1-o(1)\).
- In the continual memory leakage model (Brakerski et al., Overcoming the hole in the bucket: public-key cryptography resilient to continual memory leakage, 2010; Dodis et al., Cryptography against continuous memory attacks, 2010), variants of the above schemes enjoy leakage rate at least \(\frac{1}{2} -o(1)\). Among the results, we improve upon the work of Brakerski et al. by presenting adaptively secure IBE schemes.
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Let \(\Omega \subset \mathbb {R}^n\), \(n\ge 2\), be a bounded domain satisfying the separation property. We show that the following conditions are equivalent: For domains satisfying the separation property, in particular, for finitely connected domains in the plane, our result provides a geometric characterization of the Korn inequality, and gives positive answers to a question raised by Costabel and Dauge (Arch Ration Mech Anal 217(3):873–898, 2015) and a question raised by Russ (Vietnam J Math 41:369–381, 2013). For the plane, our result is best possible in the sense that, there exist infinitely connected domains which are not John but support Korn’s inequality.
相似文献
- (i)\(\Omega \) is a John domain;
- (ii)for a fixed \(p\in (1,\infty )\), the Korn inequality holds for each \(\mathbf {u}\in W^{1,p}(\Omega ,\mathbb {R}^n)\) satisfying \(\int _\Omega \frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\,dx=0\), \(1\le i,j\le n\),$$\begin{aligned} \Vert D\mathbf {u}\Vert _{L^p(\Omega )}\le C_K(\Omega , p)\Vert \epsilon (\mathbf {u})\Vert _{L^p(\Omega )}; \qquad (K_{p}) \end{aligned}$$
- (ii’)for all \(p\in (1,\infty )\), \((K_p)\) holds on \(\Omega \);
- (iii)for a fixed \(p\in (1,\infty )\), for each \(f\in L^p(\Omega )\) with vanishing mean value on \(\Omega \), there exists a solution \(\mathbf {v}\in W^{1,p}_0(\Omega ,\mathbb {R}^n)\) to the equation \(\mathrm {div}\,\mathbf {v}=f\) with$$\begin{aligned} \Vert \mathbf {v}\Vert _{W^{1,p}(\Omega ,\mathbb {R}^n)}\le C(\Omega , p)\Vert f\Vert _{L^p(\Omega )};\qquad (DE_p) \end{aligned}$$
- (iii’)for all \(p\in (1,\infty )\), \((DE_p)\) holds on \(\Omega \).
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In this paper, we consider the generation of strongly continuous analytic semigroups on \(L^p((0,\omega ),\mu _{p}\, dx)\) and \(L^p((0,\omega ), dx), 1<p<\infty \), by a family of second order elliptic operators of the form As in [24], we shall prove the generation results on \(L^2\)-spaces using the sesquilinear forms. More general results are obtained by using interpolation procedure and Neuberger’s theorem.
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Seon-Hong Kim Sung Yoon Kim Tae Hyung Kim Sangheon Lee 《Proceedings Mathematical Sciences》2018,128(2):23
It is known that no two roots of the polynomial equation where \(0 < r_1 \le r_2 \le \cdots \le r_n\), can be equal and the gaps between the roots of (1) in the upper half-plane strictly increase as one proceeds upward, and for \(0< h< r_k\), the roots of and the roots of (1) in the upper half-plane lie alternatively on the imaginary axis. In this paper, we study how the roots and the critical points of (1) and (2) are located.
相似文献
$$\begin{aligned} \begin{aligned} \prod _{j=1}^n (x-r_j) + \prod _{j=1}^n (x+r_j) =0, \end{aligned} \end{aligned}$$
(1)
$$\begin{aligned} (x-r_k-h)\prod _{\begin{array}{c} j=1\\ j\ne k \end{array}}^n(x-r_j) + (x+r_k+h)\prod _{\begin{array}{c} j=1\\ j\ne k \end{array} }^n (x+r_j) = 0 \end{aligned}$$
(2)
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Yonatan Gutman Elon Lindenstrauss Masaki Tsukamoto 《Geometric And Functional Analysis》2016,26(3):778-817
Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a \({\mathbb{Z}^k}\)-action on a compact metric space X, we study the following three problems closely related to mean dimension.
These were investigated for \({\mathbb{Z}}\)-actions in Lindenstrauss (Inst Hautes Études Sci Publ Math 89:227–262, 1999), but the generalization to \({\mathbb{Z}^k}\) remained an open problem. When X has the marker property, in particular when X has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3).A key ingredient is a new method to continuously partition every orbit into good pieces. 相似文献
- (1)When is X isomorphic to the inverse limit of finite entropy systems?
- (2)Suppose the topological entropy \({h_{\rm top}(X)}\) is infinite. How much topological entropy can be detected if one considers X only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer?
- (3)When can we embed X into the \({\mathbb{Z}^k}\)-shift on the infinite dimensional cube \({([0,1]^D)^{\mathbb{Z}^k}}\)?
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Nikos Katzourakis Tristan Pryer 《NoDEA : Nonlinear Differential Equations and Applications》2016,23(6):61
A map \(u : \Omega \subseteq \mathbb {R}^n \longrightarrow \mathbb {R}^N\), is said to be \(\infty \)-harmonic if it satisfies The system (1) is the model of vector-valued Calculus of Variations in \(L^\infty \) and arises as the “Euler-Lagrange” equation in relation to the supremal functional In this work we provide numerical approximations of solutions to the Dirichlet problem when \(n=2\) and in the vector valued case of \(N=2,3\) for certain carefully selected boundary data on the unit square. Our experiments demonstrate interesting and unexpected phenomena occurring in the vector valued case and provide insights on the structure of general solutions and the natural separation to phases they present.
相似文献
$$\begin{aligned} E_\infty (u,\Omega )\, :=\, \Vert \text {D}u \Vert _{L^\infty (\Omega )}. \end{aligned}$$
(2)
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Hiroshi Sakai 《Archive for Mathematical Logic》2018,57(3-4):317-327
Minami–Sakai (Arch Math Logic 55(7–8):883–898, 2016) investigated the cofinal types of the Katětov and the Katětov–Blass orders on the family of all \(F_\sigma \) ideals. In this paper we discuss these orders on analytic P-ideals and Borel ideals. We prove the following: In the course of the proof of the latter result, we also prove that for any analytic ideal \(\mathcal {I}\) there is a Borel ideal \(\mathcal {J}\) with \(\mathcal {I} \subseteq \mathcal {J}\).
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- The family of all analytic P-ideals has the largest element with respect to the Katětov and the Katětov–Blass orders.
- The family of all Borel ideals is countably upward directed with respect to the Katětov and the Katětov–Blass orders.
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The construct M of metered spaces and contractions is known to be a superconstruct in which all metrically generated constructs can be fully embedded. We show that M has one point extensions and that quotients in M are productive. We construct a Cartesian closed topological extension of M and characterize the canonical function spaces with underlying sets Hom(X,Y) for metered spaces X and Y. Finally we obtain an internal characterization of the objects in the Cartesian closed topological hull of M. 相似文献
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Zongming Guo Linfeng Mei Fangshu Wan 《NoDEA : Nonlinear Differential Equations and Applications》2016,23(6):59
We obtain non-radial bifurcation from radial solutions of a semilinear elliptic equation in expanding annuli of \(\mathbb {R}^N\). To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equation The main results of this paper can be seen as applications of the properties of regular entire solutions of (0.1).
相似文献
$$\begin{aligned} -\Delta u=\lambda u^p \quad \text {in}\quad \mathbb {R}^N. \end{aligned}$$
(0.1)