4
votes
1answer
28 views

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 ...
3
votes
0answers
40 views

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 ...
2
votes
0answers
45 views

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 ...
2
votes
0answers
19 views

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 ...
4
votes
1answer
44 views

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 $(M, g)$, ...
0
votes
0answers
31 views

laplace-beltrami and angular momentum operator

Im trying to understand this formula: In $\mathbb{R}^n$ we say that the laplacian can be seen as (radial part and angular part). : $$\triangle=(\frac{\partial^2}{\partial ...
0
votes
0answers
39 views

Existence and uniqueness on this semi-linear parabolic PDE

I want to know whether the existence and uniqueness of a classical solution can be found about this question: Find a classical solution $u : [0,T]\times [0,\infty] \rightarrow {\mathbb R}$, such ...
2
votes
1answer
33 views

Wave equation on a compact Riemannian surface without boundary: no mass conservation?

Consider a compact, smooth Riemmanian surface $\mathcal{S} \subset \mathbb{R}^3$ without boundary. I would like to solve the wave equation: $$u_{tt} + \Delta_{\mathcal{S}} u = 0$$ under the ...
1
vote
0answers
29 views

PDEs along parametrized curves in $\mathbb R^2$

For Riemannian surfaces in $\mathbb R^n,n\geq 3$, the Laplace-Beltrami operator can be expressed in terms of the Riemannian metric and one can consider evolution equations (e.g. heat diffusion, wave ...
0
votes
1answer
31 views

Differential equations of a plate with spherical mass load?

I would like to know the equations describing a plate surface being curved and stressed by a mass (you know, like a ball on a stretched sheet). I'm just curious :). I feel a bit confused about tensor ...
2
votes
1answer
85 views

Why should the diffusivity matrix (of elliptic operator) map tangent space to itself?

I have seen that an elliptic operator $A$ on a hypersurface $\Gamma$, written as $$Au=-\nabla_\Gamma \cdot (M(x)\nabla_\Gamma u)$$ (where $\nabla_\Gamma$ is the tangential or surface gradient) is ...
2
votes
1answer
27 views

The localization of smooth boundary

Let $\Omega$ be an open set in $\mathbb R^n$ with smooth boundary and $p \in \partial\Omega$ a fixed point. For any $0<r<R$, can we find an open set $\Omega_1$ with smooth boundary such that ...
2
votes
0answers
27 views

On the relation between PDEs and Distributions on Manifolds

Given a distribution $\Delta$ of dimension $n$ (continuous or smooth) in a $n+m$ dimensional manifold, can one always find a basis $\{X_j\}$ such that in local coordinates $(x^1,...x^m,y^1,...y^n)$: ...
1
vote
1answer
51 views

Lie derivative of a scalar and PDE

I posted this on the physics stackexchange, but they told me to post here, as it may be more relevant. I am reading about differential geometry, and in particular the Lie derivative and its relation ...
0
votes
1answer
20 views

About $C_c^\infty((0,T)\times \Omega)$

Let $\Omega = \Omega_1 \cup \Omega_2 \cup \Gamma$ where $\Omega_1, \Omega_2$ are open domains in $\mathbb{R}^n$ and $\Gamma$ has measure zero. $\Gamma$ is the interface between $\Omega_1$ and ...
21
votes
1answer
291 views

Why is the Laplacian important in Riemannian geometry?

As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why. ...
0
votes
1answer
53 views

Finding the Boundary Conditions for a Laplace's Equation in Polar Coordinates

I have solved Laplace's equation in Polar Coordinates for the scalar electric potential in a circle of radius R and have the solution $$ \phi(r,\varphi) = \phi_{0} + ...
2
votes
1answer
60 views

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 ...
0
votes
1answer
55 views

$C^1$ 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 ...
6
votes
0answers
85 views

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 ...
2
votes
1answer
78 views

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 ...
1
vote
1answer
34 views

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 ...
1
vote
1answer
36 views

Bounding an integral on boundary of Lipschitz domain

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain with bounded boundary $\Gamma.$ So $\Gamma$ is a hypersurface of dimension $(n-1)$. I want to show that $$\int_\Gamma ...
1
vote
0answers
24 views

comformal surface parameterization of simply-connected Riemann surface with boundary

$M$ and $S$ are two simply-connected surfaces with boundaries $\partial M =\partial S \neq \emptyset$, respectively. Both $M$ and $S$ are disk-like and smooth enough. Here, we assume $M$ is planar ...
3
votes
1answer
107 views

Laplace's Equation with Neumann BC

Hi fellow math enthusiasts, I am currently working on some research to do with the electric field induced within the brain via magnetic stimulation. I am trying to solve the partial differential ...
20
votes
2answers
383 views

Why do odd dimensions and even dimensions behave differently?

It is well known that odd and even dimensions work differently. Waves propagation in odd dimensions is unlike propagation in even dimensions. A parity operator is a rotation in even dimensions, ...
4
votes
1answer
87 views

Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition ...
4
votes
0answers
63 views

An imcomprehensible proof on Arnold's Lectures on PDE, contact diffeomorphism

In Lectures on Partial Differential Equations, Arnold gives a theorem in the chapter Huygens' Principle in the Theory of Wave Propagation: Theorem 1 (the theory of support functions). The manifold ...
2
votes
1answer
69 views

Differential forms on the torus correspond to periodic forms on $\Bbb{R}^n$?

Let $T^n=\Bbb{R^n/Z^n}$ be the torus. Is it possible say that forms on the torus bijectively correspond to forms on $\Bbb{R}^n$ invariant under translations by integers?
1
vote
1answer
87 views

Problem with notation: Laplacian on a manifold

In the Aubin's book "Nonlinear analysis on manifolds" the Laplacian operator on functions on some smooth manifold is defined by the formula $$ \Delta = -\nabla^\gamma\nabla_\gamma, $$ where ...
0
votes
0answers
18 views

Existence of an integratiag factor (in the proof of isothermal parameters in analytic case)

Suppose $f(x,y),g(x,y)$ are analytic functions, does there exist a function $\lambda(x,y)$, such that $\lambda(fdx+gdy)=dh$ for some $h(x,y)$?(locally)
9
votes
1answer
279 views

Effect of pullback of differential forms on an ideal

Say that the exterior differential system (EDS) corresponding to a PDE system is: $$df-f_x\,dx-f_y\,dy-f_w\,dw-f_z\,dz=0,\\ a_1\,f_x+a_2\,f_y=0,\tag{sys}$$ Of course we also require the independence ...
2
votes
1answer
318 views

Studying Navier-Stokes equations using differential geometry

I study Navier-Stokes in $\mathbb{R}$. But I am interested in applying Differential Geometry for these equations. If I extend my domain to a torus, would this enable me to use DG?
3
votes
2answers
188 views

Show that the curves are circle.

For example, this question is also similar to the previous question I have asked, which is the following link; How to show the curves are conics. Question: Solve the equation ...
4
votes
0answers
145 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
1
vote
1answer
139 views

Comprehensive references on partial differential equations

How do the three volumes by Taylor's "Partial differential equations" compare with the two volumes with the same title by Friedrich Sauvigny's as a reference for study? What are the good and bad ...
0
votes
0answers
69 views

Observation on normalized Ricci flow on two sphere

Note that two sphere with nonnegative curvature converges to canonical sphere along normalized Ricci flow. So assume that if curvature is bounded below by $-\epsilon$ then the sphere meets a ...
3
votes
0answers
58 views

bounds on eigenvalues of elliptic operators on functions on riemannian manifolds

Well I have little experience with pde's and analysis, I mostly study topic related to geometric topology and I would like to see if someone can please explain to me why is it important to find bounds ...
5
votes
0answers
68 views

How are boundary conditions formally captured by the jet bundle approach to differential equations?

In the jet bundle approach to differential equations https://en.wikipedia.org/wiki/Jet_bundle#Partial_differential_equations one identifies the equation with the set of a solution of the ...
2
votes
0answers
141 views

A probable inspiring proof to Poincare lemma

Poincare lemma says if a smooth $p$-form $\omega$ is closed, then $\omega$ must be exact. Let's put it in another way, it says the solution of $d\omega=0$ is $\omega=d\eta$ for some $(p-1)$-form ...
4
votes
2answers
152 views

Why do all solutions to this equation have the same form?

In this paper on page 45, the authors state that Let's assume we know that $$ w \times dw = d\varphi + \sum_{j=1}^3\alpha_jdx_j, \tag{1}$$ where $\varphi \in H^1(\mathbb{R}^3, \mathbb{R})$, ...
3
votes
2answers
75 views

How to integrate this differential form on the boundary of the cube

The setup. Assume $u = u_1+iu_2: \mathbb{R}^3 \to \mathbb{C}$ and we have the differential 1-forms $$ \star\xi=-x_2 dx_3 + x_3 dx_2 $$ and $$ u \times du = \sum_{i=1}^3 (u \times \partial_i u) dx_i = ...
7
votes
2answers
170 views

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 ...
0
votes
1answer
54 views

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 ...
2
votes
0answers
71 views

Asymptotics of Green's function of laplacian

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 ...
3
votes
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) $. ...
3
votes
1answer
166 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 \begin{equation} ic\partial_1 v +\Delta v +v(1-\vert v \vert^2)=0. ...
8
votes
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 ...
1
vote
1answer
71 views

projection onto the nullspace of the Laplacian on a conformally compact surface

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 ...
4
votes
1answer
140 views

Definition of smoothness “up to boundary”

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? ...