Titlebook: Essentials of Partial Differential Equations; With Applications Marin Marin,Andreas Öchsner Book 2019 Springer International Publishing AG,

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The Theory of PotentialThe Newtonian potential, or the potential of volume, associated to the Laplace equation . is, by definition, the following improper integral: .where ., and . is the area of the unit sphere from the .-dimensional space ..

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Boundary Value Problems for Elliptic OperatorsConsider the bounded domain . with boundary . and the linear operator of second order

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Parabolic EquationsThe prototype of a parabolic equation is given by the equation of propagation of heat in a body. Let . be a bounded domain from . having boundary . and .. For a constant of time ., arbitrarily fixed, consider the interval of time . given by

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Linear Partial Differential Equations of Second OrderLet us consider the differential operator .in which the coefficients . and . are regular functions which depend only on one variable . and which are defined on an open set . from ., where . is not necessarily bounded.