# Questions tagged [elliptic-operators]

For questions about elliptic differential operators.

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### Existence and uniqueness of solutions to an elliptic system of linear 2nd order PDEs with Neumann boundary conditions

I am considering the following elliptic system of PDEs. I would like to establish existence of solutions in the Holder space $C^{2,\alpha}$ for some $0<\alpha<1$. Let $\Omega\subset\mathbb R^n$ ...
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### A coordinate-free criterion for ellipticity of a linear differential operator

In Chapter 6 of Warner's Foundations of Differentiable Manifolds and Lie Groups, he develops a self-contained theory of local elliptic operators to establish the Hodge theorem. I got a bit stuck on a ...
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### estimation use Hodge theory

I'm confused by how we get the following estimation from Hodge theory It seems to me that it is using the fundamental inequality of elliptic operators, but I could not see what this operator is.
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### Are solutions to this PDE bounded uniformly in $\mathbb{R}^n$

Let $u\in \dot{H}^1(\mathbb{R}^n)$ be a weak solution in dimension $n\geq 3$ to the following PDE, $$-\Delta u = \lambda \rho u$$ where $\rho\in L^{n/2}(\mathbb{R}^n)\cap L^{\infty}(\mathbb{R}^n)$ and ...
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i would to like know what fact in the pde theory is used in the following article , pg 15 "Since $Dw$ is Hölder continuous, we can see this equation as a linear ellip- tic equation with $C^{\... 3 votes 2 answers 353 views ### Inverting the vorticity via Biot-Savart in Navier Stokes Given the Navier Stokes equation$\partial_t u+u\cdot \nabla u+\nabla p=\nu \Delta u$in$\mathbb{R}^3$with$u$divergence-free, one is often interested in the vorticity$\omega=\text{curl} \ u$. In ... 2 votes 1 answer 67 views ### Two Definition of Elliptic Symbols There two definition of elliptic symbol. A smooth matrix function$p(x,\xi)$is a elliptic symbol of order$m\in\mathbb{R}$if exist a constant$c>0$such that for all$|\xi|>c$we have$p(x,\xi)...
Consider the Hilbert space $H_0^1(\mathbb R^n)$ defined as the closure of $C^\infty_c(\mathbb R^n)$ in $H^1(\mathbb R^n)$. I am interested in the following convex subspace: \mathscr C \,:= \left\{\,...