| Ji XINHUA,CHERN DEQUAN.[J].数学年刊A辑,1980,1(3-4):375~386 |
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| A PARTIAL DIFFERENTIAL EQUATION WITH A DEGENERATE SURFACE Ш EXTENDED SPACE OF N-DEMENSION |
| Received:January 24, 1979 Revised:April 19, 1979 |
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| Professor Loo-keng Hua studied the partial differential equation
\[{(1 - x{x^'})^2}\sum\limits_{i = 1}^n {\frac{\partial }{{\partial x_i^2}}} + 2(n - 2)(1 - x{x^'})\sum\limits_{i = 1}^n {{x_i}} \frac{{\partial U}}{{\partial {x_i}}} = 0\] (1)
by the method of geometry. He proved that Poisson formula
\[U(x) = \frac{1}{{{\omega _{n - 1}}}}\int_{v{v^'} = 1} {...\int {(\frac{{1 - x{x^'}}}{{1 - 2x{v^'} + x{x^'}}}} } {)^{n - 1}}U(v)\mathop v\limits^ \] (2)
is the unique isolution of Diriohlet problem in the interior of the unit sphere.
In this paper we also study equation (1), the solution of which is called harmonio
funotion, too. Equation (1) is elliptic in the interior and exterior of the unit sphere,
but has a degenerate surface \[x{x^'} = 1\]. When we consider Dirichlet problem in a domain
whose interior includes a degenerate surface; the maximum modulus principle is not valid.?
In this paper, at first we prove the uniqueness theorem, and then give various
solutions of ?Ь? problem, suoii as:
i) A harmonic function on whole space (including ∞) which satisfies the known
condition on the degenerate surface.
ii) A harmonic function on whole space (including ∞) which satisfies the known
condition on a concentric sphere with the unit sphere.
iii) A solution of equation (?) of Diriohlet problem in a domain whose interior
includes a degenerate surface.
iy) A solution of equation (1) of Diriohlet problem in a domain whose boundary
is a degenerate surface. |
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