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Let be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ?Ω. We show that the solution to the linear first-order system:(1) vanishes if and . In particular, square-integrable solutions ζ of (1) with vanish. As a consequence, we prove that: is a norm if with , for some with as well as . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let , , satisfy for some with . Then there exists a constant translation vector and a constant skew-symmetric matrix , such that . 相似文献
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We study computably enumerable equivalence relations (or, ceers), under computable reducibility ≤, and the halting jump operation on ceers. We show that every jump is uniform join-irreducible, and thus join-irreducible. Therefore, the uniform join of two incomparable ceers is not equivalent to any jump. On the other hand there exist ceers that are not equivalent to jumps, but are uniform join-irreducible: in fact above any non-universal ceer there is a ceer which is not equivalent to a jump, and is uniform join-irreducible. We also study transfinite iterations of the jump operation. If a is an ordinal notation, and E is a ceer, then let denote the ceer obtained by transfinitely iterating the jump on E along the path of ordinal notations up to a. In contrast with what happens for the Turing jump and Turing reducibility, where if a set X is an upper bound for the A-arithmetical sets then computes , we show that there is a ceer R such that , for every finite ordinal n, but, for all k, (here Id is the identity equivalence relation). We show that if are notations of the same ordinal less than , then , but there are notations of such that and are incomparable. Moreover, there is no non-universal ceer which is an upper bound for all the ceers of the form where a is a notation for . 相似文献
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Roberta Filippucci Patrizia Pucci Frédéric Robert 《Journal de Mathématiques Pures et Appliquées》2009,91(2):156-177
Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that admits a positive weak solution in of class , whenever , and . The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if is a weak solution in of , then when either , or and u is also of class . 相似文献
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V.S. Guliyev M.N. Omarova M.A. Ragusa A. Scapellato 《Journal of Mathematical Analysis and Applications》2018,457(2):1388-1402
In this paper we study the behavior of Hardy–Littlewood maximal operator and the action of commutators in generalized local Morrey spaces and generalized Morrey spaces . 相似文献
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As a consequence of integral bounds for three classes of quaternionic spherical harmonics, we prove some bounds from below for the norm of quaternionic harmonic projectors, for . 相似文献
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Gerd Grubb 《Journal of Functional Analysis》2018,274(9):2634-2660
This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations with a nonlocal operator P:
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1) For strongly elliptic pseudodifferential operators (ψdo's) P on of order , a symbol calculus on is introduced that allows showing optimal regularity results, globally over and locally over : for , . The are anisotropic Sobolev spaces of Bessel-potential type, and there is a similar result for Besov spaces.2) Let Ω be smooth bounded, and let P equal (), or its generalizations to singular integral operators with regular kernels, generating stable Lévy processes. With the Dirichlet condition , the initial condition , and , (*) has a unique solution with . Here if , and is contained in if , but contains nontrivial elements from if (where ). The interior regularity of u is lifted when f is more smooth. 相似文献
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Carme Cascante Joan Fàbrega Joaquín M. Ortega 《Journal of Mathematical Analysis and Applications》2018,457(1):722-750
In this paper we characterize the boundedness of the bilinear form defined by in the product of homogeneous Sobolev spaces , . We deduce a characterization of the space of pointwise multipliers from to its dual in terms of trace measures. 相似文献
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In this paper, we apply the variational method with Structural Prescribed Boundary Conditions (SPBC) to prove the existence of periodic and quasi-periodic solutions for the planar four-body problem with two pairs of equal masses and . A path on satisfies the SPBC if the boundaries and , where A and B are two structural configuration spaces in and they depend on a rotation angle and the mass ratio .We show that there is a region such that there exists at least one local minimizer of the Lagrangian action functional on the path space satisfying the SPBC for any . The corresponding minimizing path of the minimizer can be extended to a non-homographic periodic solution if θ is commensurable with π or a quasi-periodic solution if θ is not commensurable with π. In the variational method with the SPBC, we only impose constraints on the boundary and we do not impose any symmetry constraint on solutions. Instead, we prove that our solutions that are extended from the initial minimizing paths possess certain symmetries.The periodic solutions can be further classified as simple choreographic solutions, double choreographic solutions and non-choreographic solutions. Among the many stable simple choreographic orbits, the most extraordinary one is the stable star pentagon choreographic solution when . Remarkably the unequal-mass variants of the stable star pentagon are just as stable as the equal mass choreographies. 相似文献
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Henri Martikainen Mihalis Mourgoglou Xavier Tolsa 《Journal of Functional Analysis》2018,274(5):1255-1275
In the context of local Tb theorems with testing conditions we prove an enhanced Cotlar's inequality. This is related to the problem of removing the so called buffer assumption of Hytönen–Nazarov, which is the final barrier for the full solution of S. Hofmann's problem. We also investigate the problem of extending the Hytönen–Nazarov result to non-homogeneous measures. We work not just with the Lebesgue measure but with measures μ in satisfying , . The range of exponents in the Cotlar type inequality depend on n. Without assuming buffer we get the full range of exponents for measures with , and in general we get , . Consequences for (non-homogeneous) local Tb theorems are discussed. 相似文献