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Dokl. Akad. Nauk SSSR 146, 1275–1278 (1962). ˇ 15. : General boundary-value problems for elliptic equations with discontinuous coefficients. Dokl. Akad. Nauk SSSR 148, 1034–1037 (1963). 16. : Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95(4), 707–734 (2003) 17. : Error control in multi-element generalized polynomial chaos method for elliptic problems with random coefficients. Commun. Comput. Phys. 5(2-4), 793–820 (2009) Galerkin FEM for Fractional Order Parabolic Equations with Initial Data in H −s , 0 ≤ s ≤ 1 Bangti Jin, Raytcho Lazarov, Joseph Pasciak, and Zhi Zhou Department of Mathematics, Texas A&M University, College Station, TX 77843, USA Abstract.

See [13]) For α > 0, α = n + 1/2, n ∈ N, the spaces H˙ + (a, b), α α α ˙ ˙ ˙ H− (a, b), Hc (a, b) and H (a, b) are equal and their seminorms as well as norms are equivalent. S. G. Vulkov, and A. Deli´c For the functions of two variables, x and t, defined in the rectangle Q = (0, 1) × (0, T ), we introduce anisotropic Sobolev spaces H α,β (Q), α, β ≥ 0, in the usual manner [14]: H α,β (Q) = L2 ((0, T ), H α (0, 1)) ∩ H β ((0, T ), L2 (0, 1)) . Analogously we define α,β β H± (Q) = L2 ((0, T ), H α (0, 1)) ∩ H± ((0, T ), L2 (0, 1)) .

Li, V. Nistor, and Y. Qiao Step 5. We finally reduce to the case of a half-space or a full space using a partition of unity as in the classical case, as follows. We choose a smooth partition of unity (φj ) on Ω consisting of functions with small supports. The supports should be small enough so that if the support of φj intersects the boundary of Ω or the interface Γ , then the boundary or the interface can be straightened in a small neighborhood of the support of φj . We arrange that the resulting operators are positive and we complete the proof as in [13].

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Bounded positive critical points of some multiple integrals of the calculus of variations by Boccardo L., Pellacci B.


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