# Questions tagged [elliptic-pde]

Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

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### On a compact operator in the plane

Let $\Omega \subset \mathbb R^2$ be a bounded domain with a smooth boundary. Let $$\bar{\partial}= \frac{1}{2} ( \partial_{x^1} + i \,\partial_{x^2}),$$ and let $G: L^2(\Omega)\to H^2(\Omega)$ be the ...
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### Compactly-supported harmonic tensors

Let $({M},g)$ be a connected and non-compact Riemannian manifold without boundary. If $L:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a linear second order elliptic operator on some smooth $\mathbb{R}$...
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### Gehring lemma for fractional maximal functions

Given a function $f\in L^p(\mathbb{R}^n)$, the Gehring lemma states that if there exists $p>1$, a constant $C_0>1$ and a cube $Q \subset \mathbb{R}^n$ such that for almost every $x\in Q$, it ...
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### For the solvability of the poisson equation $\Delta u = f$ on manifold with boundary

For poisson equation $\Delta u = f$ in bounded domain in $\mathbb{R}^n$, we can directly get the solution by Green function. For poisson equation $\Delta u = f$ on closed Riemannian manifold, the ...
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### Regularity of elliptic partial differential equation with mixed Dirichlet-Robin boundary condition, to prove $u\in H^{2}(\Omega)$

I have posted this problem on Math Stackexchange but got no reply. When I deal with the wave equation with dynamical boundary condition, I am confused by the regularity of the following elliptic ...
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### A mapping property for fractional Laplace--Beltrami operator

Suppose that $g$ is a smooth Riemannian metric on $\mathbb R^3$ that is equal to the Euclidean metric outside the unit ball. Let us define the fractional Laplace--Beltarmi operator on $\mathbb R^3$ of ...
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### Given composition rules, determining whether a continuous map between smooth functions is a pseudodifferential operator

Let $M$ be a closed manifold, and let $P:C^{\infty}(M)\rightarrow C^{\infty}(M)$ be a continuous linear map in the smooth Fréchet topologies. In what is to come, if it helps, one can assume further ...
Let $A(x)$ be a smooth, uniformly elliptic, symmetric matrix, i.e., $\lambda |\xi|^2 \leq \langle A(x)\xi,\xi\rangle \leq \Lambda |\xi|^2$ for some two fixed constants $\lambda, \Lambda \in (0,\infty)$...
Let $\Omega \subset \mathbf{R}^n$ be a smooth, bounded domain. The Dirichlet problem for the minimal surface equation \begin{equation} (1 + \lvert Du \rvert^2) \Delta u - D_i u D_j u D_{ij} u = 0 \end{...