# Tagged Questions

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### Fundamental solution of the Laplacian on the surface of a cylinder

Does the Laplace operator have a fundamental solution on the surface of a cylinder in $\mathbb{R}^3$? Intuitively, I can visualize a function that diverges to $\infty$ at a point, decreases to a ...
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### Tensor differential equations

I am reading Ringstrom's book The Cauchy problem in General Relativity, But I don't really understand Chapter 12 associating to tensor equations. I want to read some other material about this. Could ...
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### Are there any results linking hyperbolic operators and Pseudo-Riemaniann geometry?

In the case of a $n$-dimensional pseudo-riemannian manifold one has a symmetric bilinear form, $h$, with signature $(1,n)$ and the analogous operator to the Laplace -Beltrami operator,$\Delta$, is the ...
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### non-smooth minmal surfaces and differenteial equations

This is the equation for a function $u(x,y)$ whose graph is a minimal surface (its mean curvature is $0$): $$(1+u_x^2)u_{yy}-2u_xu_yu_{xy}+(1+u_y^2)u_{xx}=0$$ My question is if there are non-smooth ...
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### Existence of positive solutions of a linear PDE on closed manifolds

I was wondering is there a sufficient condition (or sufficient and necessary condition) for the existence of positive solutions of the following linear PDE on a closed manifold $(M, g)$, ...
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### Relation between $L^1(\partial\Omega)$ and the surface integral on $C^1$ domains

Let $\Omega \subset \mathbb{R}^n$ be a $C^{1}$ bounded domain. It is possible to define the space $L^1(\partial\Omega)$ as the set of functions $u\colon \partial\Omega \to \mathbb{R}$ with the finite ...
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### $C^1$ domains vs. Lipschitz domains in PDEs, do we need transformation of coordinates?

I am getting a bit confused. In the definition of $C^1$ manifold in Renardy and Rogers, they say that $\partial\Omega$ is of class $C^1$ if every point on $\partial\Omega$ has a neighbourhood within ...
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### Two definitions of $H^1(\partial\Omega)$, one using charts and one use tangential gradients

Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$. There are two ways to define a space $H^1(\partial\Omega)$: By using charts, we can define $H^1(\partial\Omega)$ to ...
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### Differentiating the determinant of the Jacobian of a diffeomorphism (don't understand a proof)

For each $t$, let $A_t:\Omega_0 \to \Omega_t$ be a bi-Lipschitz map between open sets in $\mathbb{R}^n$. The map is also invertible. It satisfies $$\frac{d}{dt}A_t(y) = w(A_t(y),t)$$ where $w$ is a ...
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### Change of Variables from Sphere to Plane

Say we are in the space $\mathbb{R}^{n}$. Consider the $n-1$ dimensional surface measure $dS$ on the boundary of the upper half sphere. We can define the coordinates on this sphere by ...
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### Definition of weak solutions from geometrical point of view

Why are weak solutions defined like: A function $u \in H^1(\Omega)$ is a weak solution of $$Lu:=\operatorname{div}(A\nabla u)+b\cdot\nabla u+cu=f+\operatorname{div}F, \;\text{in } \Omega$$ if ...
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### Isometric embeddings with prescribed second fundamental form

I'm looking for some non-rigidity result for isometric embeddings in euclidean space (codimension 2). For example, any isometric embedding of the round $S^2$ into $\mathbb R^3$ is unique up to rigid ...
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Let $\Omega$ be a domain in $\mathbb{R}^2$ with a Riemannian metric $g$, and let $\Delta$ be the laplace-beltrami operator induced by the metric, with dirichlet boundary conditions. For $x,y ... 2answers 48 views ### Problem with simple laplacian equation I would like to solve the following PDE: $$\partial_x^2 u + \partial_y^2 u = -\frac{2 x^2 (x^2-y^2)}{\left(x^2+y^2\right)^2}$$ The right side comes from$ x^2 \partial_x^2 \log(x^2 +y^2) $. ... 1answer 167 views ### How to obtain this Pohozaev identity for the Gross-Pitaevskii equation? The Gross-Pitaveskii equation (after plugging in the traveling wave ansatz and writing in moving frame coordinates) reads ic\partial_1 v +\Delta v +v(1-\vert v \vert^2)=0. ... 1answer 112 views ### Differential forms: The authors of a paper define$d(u\times du)$, but what is$u \times du$supposed to mean? I'm reading [1] recently and have another question about a remark in this paper. I tried to solve it myself (see below) but did not succeed. It could be just a notation problem. The Setup: Let$u ...
Let $(M,g)$ be a conformally compact surface. An example situation is a hyperbolic surface of infinite area like the quotient $\Gamma\backslash \mathbb{H}$, where $\mathbb{H}$ is the hyperbolic plane ...
Let $U\subseteq \mathbb{R}^n$ be an open set and let $f\in\mathcal{C}^k(U)$ for some positive integer $k$. Are the following definitions of $\mathcal{C}^k$ regularity "up to boundary" equivalent? ...