# Questions tagged [elliptic-equations]

For questions about elliptic partial differential equations. If your question is specific to the Laplace equation, see (harmonic-functions).

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### Confusion on definition of fundamental solution for the heat equation

As mentioned on Wikipedia, a fundamental solution for a linear differential operator $L$ is a function (or distribution) $G$ such that $$LG = \delta$$ which by linearity of $L$ gives the following ...
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### On Schauder estimates and harmonic coordinates in Petersen's book

In Peter Petersen's Riemannian Geometry(3rd ed), Section 11.2, following form of local Schauder estimate is stated without proof (Theorem 11.2.2): For elliptic 2nd order differential operator $L$ on ...
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### Type of solution (as the critical point of the energy functional) after doing scaling to the PDE.

Now I have a nonlinear elliptic PDE, if I do scaling such as $u=\lambda v$ and then get a new PDE, then I turn to study the new equation since it has a simpler form. My question is easy to understand: ...
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The problem is Suppose $\phi\in\mathcal{S}(\mathbb{R}^2)$, $u$ on $\mathbb{R}^3_-=\mathbb{R}^2\times(-\infty,0)$ satisfies \begin{cases} \Delta u(x,y,z)=0\\ \lim\limits_{z\to 0^-} u(x,y,z)=\phi(x,y)\\ ...