Abstract
We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations of second order. A linear partial differential operator L defined by
is elliptic on Ω ⊂ ℝn if the symmetric matrix [a ij ] is positive definite for each x ∈ Ω. We have used the notation D i u, D ij u for partial derivatives with respect to x i and x i , x j and the summation convention on repeated indices is used. A nonlinear operator Q,
[D u = (D 1 u, ..., D n u)], is elliptic on a subset of ℝn × ℝ × ℝn] if [a ij (x, u, p)] is positive definite for all (x, u, p) in this set. Operators of this form are called quasilinear. In all of our examples the domain of the coefficients of the operator Q will be Ω × ℝ × ℝn for Ω a domain in ℝn. The function u will be in C 2(Ω) unless explicitly stated otherwise.
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Bassanini, P., Elcrat, A.R. (1997). Elliptic Partial Differential Equations of Second Order. In: Theory and Applications of Partial Differential Equations. Mathematical Concepts and Methods in Science and Engineering, vol 46. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1875-8_5
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