# Tagged Questions

Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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### Is there a true many-body green's function for interacting systems?

I've recently been trying to compute the Green's function for a non-interacting system of fermions. Since this is a site for mathematicians, for context, let me provide the following definition: ...
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### Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds

For experts in the analysis of minimal surfaces I will state the question first; then I will follow up with details. Question: Does the $\varepsilon$-regularity theorem of Choi and Schoen (http://...
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### Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions

I asked this question on Mathematics network but it didn't receive any answers. So I assume it is just beyond the classic things in PDEs and I decided to ask it here too. Consider the following ...
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### Exactly solvable examples of diffusion equation with variable diffusivity?

There are many examples of potentials $V(x)$ for which Schrodinger's equation for a single particle in one dimension is exactly solvable, in the sense that we can give "nice" expressions for the ...
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### existence of a special conformal mapping

Sorry I don't know how to give an appropriate title. In the complex plane, suppose there is a graph $x+if(x)$ separating the plane into two unbounded components, where $f(x)$ is smooth and bounded, ...
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### Recent observation of gravitational waves

It was exciting to hear that LIGO detected the merging of two black holes one billion light-years away. One of the black holes had 36 times the mass of the sun, and the other 29. After the merging the ...
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### Suggestion for books in Pertubation theory with an emphasis on the theory

As the title suggest I am looking for another good coverage of the theory of Pertubation theory. Currently I am working through Murodock's book: Pertubations: Theory and Methods. But I am rest assure ...
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### $L^\infty-L^1$ smoothing effect for the heat equation

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Let $u \in L^2(0,T;V)$ be the weak solution of the heat equation $$u_t - \Delta u = 0$$ $$u(0) = u_0$$ where $u_0$ is bounded initial data. Here ...
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### $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

I posted this question first in Math.StackExchange one week ago here, but I didn't get an answer or a helpful comment so I repost it here: Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded ...
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### Heat kernel on manifold with boundary

Let $M$ be a Riemannian manifold with boundary, let $\mathcal{V}$ be a metric vector bundle over $M$ and let $L$ be a formally self-adjoint Laplace type operator, acting on sections of $\mathcal{V}$. ...
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### The space $L^p(\partial\Omega)$ in cited references

The space $L^p(\partial\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ appears in a lot of PDE textbooks without being given any definitions, not even in those with a detailed appendix ...
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### Heat semigroup ultracontractive?

Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...
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### Has anyone studied a transport equation of this form?

Let $L\colon \mathbb{R}^2 \times \mathbb{R}^+\to \mathbb R$ satisfy $$\frac{\partial L}{\partial t} (x,t) = \max\left\{ \frac{\partial L}{\partial x_1}, \frac{\partial L}{\partial x_2} \right\}$$ ...
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### Can you hear the shape of a drum by choosing where to drum it?

I find the problem of hearing the shape of a drum fascinating. Specifically, given two connected subsets of $\mathbb R^2$ with piecewise-smooth boundaries (or a suitable generalization to a riemannian ...
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### Travelling waves for nonlinear Schrödinger equation

Consider the following nonlinear Schrödinger equation: $$-\Delta \Phi - i\frac{\partial \Phi}{\partial t} = f(|\Phi|^2)\Phi,$$ where $\Delta$ is the Laplacian on $\mathbb{R}^n$, $f$ gives the ...
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### difference between the dual space of $H^1(\Omega)$ and the dual of $H^1_0(\Omega)$

This is cross-posted on MSE: http://math.stackexchange.com/q/1596565/9464 In the Partial Differential Equations by Evans (2nd edition p299), $H^{-1}(\Omega)$ denotes the dual space to $H^1_0(\Omega)$ ...
I am reading Jost's Partial Differential Equations and meet an estimation( only stated in the book ) which I cannot verify by myself. In p.111, the book says that iteratively, we get $$|S_{n}(x_{0},y,... 0answers 66 views ### A \mathcal{C}^1 domain and Hausdorff dimension estimate Let us consider an open connected domain E\subset \mathbb{R}^N and fix a point x_0\in\partial E on its boundary. Suppose now, that there exists R>0 such that the set \partial E \cap B_{R}(... 0answers 38 views ### A \mathcal{C}^1 differentiable domain is F_\sigma? Let us consider a domain E\subset \mathbb{R}^N and fix a point x_0\in\partial E on its boundary. Suppose now, that for every R>0 the set \partial E \cap B_{R}(x_0) is \mathcal{C}^1, i.e. ... 1answer 189 views ### Is Lax-Milgram true without the separability assumption? I read the Lax-Milgram Theorem in the Navier-Stokes Equations by Temam: Let X be a separable Hilbert space (norm \|\cdot\|_X) and let$$ a:X\times X\to\Bbb{R} $$be a bilinear continuous ... 1answer 125 views ### H_0^1(\Omega) in the study of the Navier-Stokes Equations This is cross-posted on MSE: http://math.stackexchange.com/q/1584519/9464 Let \mathcal{V} be the space (without topology)$$\displaystyle \mathcal{V}=\{u\in C_0^\infty(\Omega)\mid \nabla\cdot u=0\}...
I have a PDE of the following form, from a physics problem:  y \left(\alpha \frac{\partial }{\partial y}+x \frac{\partial^2 }{\partial x \partial y} \right)f(x,y) = \left( z_1 + z_2 x^\alpha y^{-2} \...