Introduction to partial differential equations

By Peter J. Olver. Boletim da Sociedade Paranaense de Matemática, 2007

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A Berry-Esseen type inequality for convex bodies with an unconditional basis (2007, background): “, e.g, [5]). We refer to λ1 as the ﬁrst non-zero Neumann eigenvalue. Suppose u∈ C∞(K). Then u∈ L2(K) and hence u= P∞ i=0uiϕi, where the sum converges in L2(K). The standard theory (see, e.g., [15] or [18]) implies that D(u,u) = Z K |∇u|2 = X∞ i=0 λi|ui| 2. (17) We say that the boundary of Kis unifor”
A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs (2008, methodology): “1) is Fréchet differentiable (it is actually infinitely differentiable) and since it is a contraction for sufficiently small values of t, we can apply the implicit functions theorem (see for example [RR04] for a Banach space version) to deduce that for every realisation of the driving noise, the map”
MICROLOCAL ANALYSIS OF GENERALIZED FUNCTIONS: PSEUDODIFFERENTIAL TECHNIQUES AND PROPAGATION OF SINGULARITIES (2004, positive): “[5]), we introduce a notion of microsupport for operators.”
On Kelvin Transformation (2006, background): “see (51) and the remark following (25).”
Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach (2007, neutral): “ΩT F (x, t)v(x, t) dx dt, ∀v ∈ H (ΩT ), (6) where H (ΩT ) is the Sobolev space with the norm [12,15] ‖u‖1 := {∫ ∫”
Cloaking via change of variables for the Helmholtz equation (2010, background): “[6] Proposition 3.”