不出名
发表于 2025-3-30 08:20:23
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Abjure
发表于 2025-3-30 12:44:30
Uniqueness and Energy IntegralsWe shall extend the method of energy integrals to more general second-order (hyperbolic) operators. This “energy” method is a basic technique in the modern theory of partial differential operators, and in the course of our development, we shall establish some interesting and important classical inequalities.
多骨
发表于 2025-3-30 18:27:25
An Initial-Value Problem for a Hyperbolic EquationWe consider the equation for the homogeneous operator .: . with initial data . We assume that each . is constant, and that the hyperplane . = 0 is non-characteristic with respect to .. Since the normal vector to the surface . = 0 is (1, 0, …, 0), this means . ≠ 0; thus we assume .. = 1.
Amorous
发表于 2025-3-30 23:26:43
Second-Order Linear Elliptic EquationsSolutions of elliptic equations represent .; i.e., solutions which do not vary with time. They often describe the asymptotic states achieved by solutions of time-dependent problems, as . → ∞. Physically speaking, all the “rough spots” smooth out by the time this steady state is achieved.
dissolution
发表于 2025-3-31 02:40:10
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服从
发表于 2025-3-31 08:24:25
Holmgren’s Uniqueness Theoremanalytic solution to this problem. Holmgren’s uniqueness theorem denies this possibility. We shall also find this result useful in Chapter 6 where we shall apply it to determine qualitative information on domains of dependence. For this reason, we shall prove a rather general version of the theorem.
审问,审讯
发表于 2025-3-31 09:24:39
Bifurcation Theorywhen a solution (., .) of (13.1) lies on a “curve” of solutions (., .(.)), at least locally; i.e., for |. — .| < ε. We may also inquire as to when (., .) lies on several solution curves, (., .(.)), (., .(.)),….
Rebate
发表于 2025-3-31 17:24:21
Comparison Theorems and Monotonicity Methodsabolic equations as . → + ∞. As a second application of the maximum principle, we shall show how it can be used to prove existence theorems. This is the method of “upper” and “lower” solutions, the solution being the limit of a monotone iteration scheme, where the monotonicity is a consequence of the maximum principle.
MONY
发表于 2025-3-31 20:04:02
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EXUDE
发表于 2025-4-1 01:44:17
Characteristics and Initial-Value Problemsoses the existence of a distinguished coordinate, ξ, where the equation ξ = 0 defines the “initial” surface. Of course, as we have seen in the last chapter, one needs some kind of compatibility between the equation and the initial surface. The notion of characteristic serves to classify and make more precise these intuitive ideas.