Communications on Pure and Applied Mathematics

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Fast wavelet transforms and numerical algorithms I
Communications on Pure and Applied Mathematics - Tập 44 Số 2 - Trang 141-183 - 1991
Gregory Beylkin, Ronald R. Coifman, Vladimir Rokhlin
AbstractA class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors. Previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices. In contrast, the methods presented here are based on the recently developed theory of wavelets and are applicable to all Calderon‐Zygmund and pseudo‐differential operators. The algorithms of this paper require order O(N) or O(N log N) operations to apply an N × N matrix to a vector (depending on the particular operator and the version of the algorithm being used), and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here.
Boundary concentration phenomena for a singularly perturbed elliptic problem
Communications on Pure and Applied Mathematics - Tập 55 Số 12 - Trang 1507-1568 - 2002
Andrea Malchiodi, Marcelo Montenegro
AbstractWe exhibit new concentration phenomena for the equation − ε2 Δu + u = up in a smooth bounded domain Ω ⊆ ℝ2 and with Neumann boundary conditions. The exponent p is greater than or equal to 2 and the parameter ε is converging to 0. For a suitable sequence εn → 0 we prove the existence of positive solutions un concentrating at the whole boundary of Ω or at some component. © 2002 Wiley Periodicals, Inc.
On the location and profile of spike‐layer solutions to singularly perturbed semilinear dirichlet problems
Communications on Pure and Applied Mathematics - Tập 48 Số 7 - Trang 731-768 - 1995
Wei‐Ming Ni, Juncheng Wei
On the shape of least‐energy solutions to a semilinear Neumann problem
Communications on Pure and Applied Mathematics - Tập 44 Số 7 - Trang 819-851 - 1991
Wei‐Ming Ni, Izumi Takagi
The partial differential equation u<sub>t</sub> + uu<sub>x</sub> = μ<sub>xx</sub>
Communications on Pure and Applied Mathematics - Tập 3 Số 3 - Trang 201-230 - 1950
Eberhard Hopf
Weak solutions of nonlinear hyperbolic equations and their numerical computation
Communications on Pure and Applied Mathematics - Tập 7 Số 1 - Trang 159-193 - 1954
Peter D. Lax
Formation and decay of shock waves
Communications on Pure and Applied Mathematics - Tập 1 Số 3 - Trang 211-245 - 1948
K. O. Friedrichs
On the exponential solution of differential equations for a linear operator
Communications on Pure and Applied Mathematics - Tập 7 Số 4 - Trang 649-673 - 1954
Wilhelm Magnus
Polar factorization and monotone rearrangement of vector‐valued functions
Communications on Pure and Applied Mathematics - Tập 44 Số 4 - Trang 375-417 - 1991
Yann Brenier
AbstractGiven a probability space (X, μ) and a bounded domain Ω in ℝd equipped with the Lebesgue measure |·| (normalized so that |Ω| = 1), it is shown (under additional technical assumptions on X and Ω) that for every vector‐valued function u ∈ Lp (X, μ; ℝd) there is a unique “polar factorization” u = ∇Ψs, where Ψ is a convex function defined on Ω and s is a measure‐preserving mapping from (X, μ) into (Ω, |·|), provided that u is nondegenerate, in the sense that μ(u−1(E)) = 0 for each Lebesgue negligible subset E of ℝd.Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real‐valued functions are unified.The Monge‐Ampère equation is involved in the polar factorization and the proof relies on the study of an appropriate “Monge‐Kantorovich” problem.
Periodic vibrations of systems governed by nonlinear partial differential equations
Communications on Pure and Applied Mathematics - Tập 19 Số 4 - Trang 371-420 - 1966
Joseph B. Keller, Lu Ting
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