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Results tagged with laplace-equation
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user 8799
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A series question related to solution of Laplace equation
Let $v,w,z$ be the functions obtained from $u$ by composing with a rotation of angle $\frac\pi4,\frac\pi2,\frac{3\pi}4$ about $(\frac12,\frac12)$. The sum $u+v+w+z$ is a harmonic function, whose value …
- 52.3k
8
votes
Convex solutions of the Poisson equation
I apologize for having posted this question too early. I realize that the answer to the first question is negative.
Actually suppose that $D=D(0;R)$ is a disk and $f=f(r)$ is a radial function. If a c …
- 52.3k
9
votes
Estimate on $C^1$-norm of solution of the Dirichlet problem for the Laplace equation
The first derivatives of $u$ are harmonic too. Therefore the Maximum Principle tells us
$$\|\nabla u\|_{C^0(\bar\Omega)}=\left\|\left.\nabla u\right|_{\partial\Omega}\right\|_{C^0(\partial\Omega)}.$$
…
- 52.3k