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questions :Elliptic Partial Differential Equations
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[Visitor (23.83.*.*)]answers [Chinese ]Time :2020-07-16
Elliptic variable differential equation

Its typical representatives are Laplace equation and Poisson equation (called Δu as Laplace operator)

Δu=-4πρ(x, y, z)(2)

The quadratic continuous differentiable solution of the Laplace equation is called the harmonic function, and equation (1) has the form

Special solution, where S is a curved surface, μ is a continuous function defined on S, (3) the function defined outside S satisfies (1), non-homogeneous equation (ie Poisson's equation) (2) There is an important special solution, which is the body potential with ρ as the density

When ρ is continuously differentiable within Ω, the function u determined by (4) satisfies (2) within Ω and (1) outside Ω. Applying Green's formula

This shows that the value of the harmonic function at any point in the area can be represented by the value of this function on the area interface and the normal derivative.
In the Dirichlet problem on the unit sphere, for a point with spherical coordinates (ρ, θ, j)

Where (θ0, j0) is the argument of integration, which is the spherical coordinate. cosυ is the cosine of the angle between the directions (θ, j) and (θ0, j0). The theory of elliptic equations is quite complete.

Elliptic partial differential equations, numerical methods

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