# Questions tagged [ap.analysis-of-pdes]

Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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### Regularity of conformal maps

In order to define what it means for a map $f \colon \Omega \subseteq \mathbb R^n \to \mathbb R^n$ to be conformal, it is sufficient to require that $f$ is everywhere differentiable. Does conformality ...
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### Reference for kinetic theory on manifolds

I am looking for a reference for kinetic theory on (Riemannian) manifolds. (In particular for mean-field limits for the Vlasov equation in this setting.) In 'A review of the mean field limits for ...
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### Non-trivial examples of regular Lagrangian flow in BV case

What is a concrete example of BV vector field $v$ with $\mathrm{div}\, v = 0$ that makes Ambrosio's theory of regular Lagrangian flow relevant? With concrete I mean that we can compute the flow ...
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### Finite energy solution for Allen -Cahn equation

I am interested in the Allen-Cahn equation in $R^N$ and one can consider the related energy functional $$E(u):= \frac{1}{2}\int_{R^N}| \nabla u(x)|^2 dx + \frac{1}{4} \int_{R^N} (u^2-1)^2dx.$$ ...
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### Log-concavity of function

Consider the function $$f_{n}(x)=e^{-x^2}x^n.$$ My goal is to show that $$G(y):=\frac{(f_2*f_0)(y)}{(f_0*f_0)(y)}- \left(\frac{(f_1*f_0)(y) }{(f_0*f_0)(y)}\right)^2$$ is log-concave. Let us ...
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### Non real eigenvalues for elliptic equations

I am looking for an example of a pure second order uniformly elliptic operator $L=\sum_{i,j=1}^da_{ij}(x)D_{ij}$ in a bounded domain $\Omega$ (with Dirichlet boundary conditions, for example) having a ...
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### Some properties of fractional Dirichlet heat kernel

Let $\Omega\subset \mathbb{R}^n (n\geq 2)$ be a bounded open domain with smooth boundary $\partial\Omega$. Consider the fractional heat equation with Dirichlet boundary condition： \begin{equation} \...
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### Reference (foundamental sol. and grad estimate, etc.): a particular elliptic PDE

In $\mathbb{R}^d$, consider the following equation $$\Delta u -x\cdot \nabla u = f$$ where $f$ can be $C^\infty$ and decay like $e^{-\frac{c|x|^2}{2}}$. I would like to know fundamental sol. to this ...
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### Explanation for the energy method used here

I am reading a paper where the authors prove $$\frac{d}{dt} \Vert f \Vert_{H^4} (t)\leq C\Vert f \Vert^5_{H^4}(t)$$ Where $f=f(x,t)$ and $H^4$ stands for the usual Sobolev space. Using Gronwall's ...