# 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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### Elliptic operators over noncompact manifold

We know for two vector bundles $E$ and $F$ over compact manifold $M$,an elliptic operator $D:\Gamma(\mathrm{E})\to \Gamma(\mathrm{F})$ is automatically Fredholm. And for the case $M$ is noncompact, in ...
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### Elliptic regularity for Dirichlet problem

Let $\overline{M}=M \cup \partial M$ be a compact manifold with boundary, where $\partial M$ is the boundary of $\overline{M}$ and $M$ is the interior of $\overline{M}$. Let $P$ be an injective ...
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### Any theory on the elliptic operator $Lu=\Delta u + b_iu_i + cu$ when $c>0$

I wonder if there are theories on elliptic operator $$Lu=\Delta u + b_iu_i + cu$$ when $c>0$, when $c<0$, we are glad to have maximum principle, so the bijectivity can be easily analyzed, but I ...
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### Elliptic regularity on manifolds: Is this true?

Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
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### Elliptic PDEs in BSDEs and in optimal control

This soft/reference question is related to this MO post of a similar nature. What are some examples of elliptic PDEs appearing in control and BSDEs?
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1 vote
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### Regularity of Feynman-Kac formula for a simple diffusion

Let consider the diffusion process given by: $$dX_t = \alpha(X_t) dW_t$$ where $\alpha(x) = \alpha_1\mathbf{1}_{x\geq 0} + \alpha_2\mathbf{1}_{x< 0}$ ($\alpha_1,\alpha_2>0$) and $W$ a Wiener ...
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### Examples of symmetry-breaking solitons which retain a subgroup symmetry

There are many works on spontaneous symmetry breaking in the Nonlinear Schrödinger equation with asymmetric soliton solutions. However, all symmetry breaking soliton examples I have seen go from the ...
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### Elliptic PDEs in Finance

In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary ...
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### A question on the maximum principle of second order elliptic equations

Let $Lu=a^{ij}u_{ij} + b^i u_i$ be an elliptic operator of second order in a bounded domain $\Omega$. Assume that $a^{ij}$ is uniformly elliptic. Then it's well known that the following maximum ...
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