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.
4,468 questions
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Physical interpretation of Robin boundary conditions
In a (bounded) domain $\Omega \subset \mathbb{R}^n$, if we're studying the Laplace equation or heat equation or such PDE's we can impose the Dirichlet
$u|_{\partial\Omega} \equiv 0$,
Neumann
$D_{\nu} ...
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3
answers
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History of fundamental solutions
I have a few questions on the history of PDE.
Who first wrote down the formula for the solution of the Cauchy problem for the heat equation involving the heat kernel? I have seen it called Poisson's ...
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2
answers
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What's the idea behind Carleman estimates?
A standard Carleman-type estimate is of the form
$$
\sum_{|\alpha|<m}{\tau^{2(m-|\alpha|-1)}\int{|D^{\alpha}u|^{2}e^{2\tau\phi}}dx}\leq K\int{|Pu|^{2}e^{2\tau\phi}dx},\quad u\in C_{0}^{\infty},
$$
...
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answer
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Image of the trace operator
It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset L^...
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Can two drums almost sound the same?
Let $D\subset \mathbb R^2$ be a region and let $\Lambda=\{\lambda_1,\lambda_2,\dots\}$ be the set of eigenvalues of the Laplacian $-\Delta$ (with boundary condition $\psi=0$ on $\partial D$).
Mark Kac,...
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Characterising critical points of $E(f)=\int_{M}| \bigwedge^2 df|^2 \text{Vol}_{M}$
$\newcommand{\id}{\operatorname{Id}}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\TM}{\operatorname{TM}}$
$\newcommand{\Hom}{\operatorname{Hom}}$
$\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\...
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A Hölder continuous function which does not belong to any Sobolev space
I'm seeking a function which is Hölder continuous but does not belong to any Sobolev space.
Question: More precisely, I'm searching for a function $u$ which is in $C^{0,\gamma}(\Omega)$ for $\gamma \...
- 2,641
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votes
2
answers
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Elliptic regularity for the Neumann problem
I'm trying to understand how to establish regularity for elliptic equations on bounded domains with Neumann data.
For simplicity, let's presume we are focusing on $-\Delta u = f$ in $\Omega$ and $\...
- 2,641
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votes
1
answer
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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 ...
- 6,035
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answer
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Algebraic microlocal analysis and nonlinear PDE
Though originating in the study of linear partial differential equations, microlocal analysis has become an invaluable tool in the study of nonlinear pde. Of particular importance has been the ...
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A Green's function for the Laplacian on k-forms
Let $X$ be a compact, oriented, Riemannian $n$-fold. Then we have a Laplacian operator $\Delta = d d^{\ast} + d^{\ast} d$ from $\Omega^k(X)$ to itself. We have the Hodge decomposition $\Omega^k(X) = \...
- 156k
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Why don't existence and uniqueness for the Boltzmann equation imply the same for Navier-Stokes?
As I understand it, Lions and DiPerna demonstrated existence and uniqueness for the Boltzmann equation. Moreover, this paper claims that
Appropriately scaled families of
DiPerna–Lions ...
- 15.4k
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answers
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Textbooks for PDE between Strauss and Folland
Walter A. Strauss's Partial Differential Equations: An Introduction is a classic PDE textbook for the undergraduate students. While Folland's Introduction to Partial Differential Equations, is a nice ...
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answers
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Epsilon regularity: what does it say and where does it come from?
The $\varepsilon$-regularity phenomenon shows up in several different contexts. I try to describe it focussing on the harmonic map situation, but I really would like to understand the situation in ...
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Reference request: parabolic PDE
I want to learn about parabolic PDE and it seems to me that there is no established reference as far as where one should look if one wants to learn the subject from basics.
I think I have a firm grip ...
- 1,211
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answers
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The speed of gravitational waves in general relativity
Is it possible to mathematically prove that the speed of gravitational waves in general relativity equals the speed of light, without linearizing the Einstein field equations? The approach via the ...
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answers
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"Physical" construction of nonconstant meromorphic functions on compact Riemann surfaces?
Miranda's book on Riemann surfaces ignores the analytical details of proving that compact Riemann surfaces admit nonconstant meromorphic functions, preferring instead to work out the algebraic ...
- 118k
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votes
4
answers
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unique continuation principle
I recently encountered a paper by Protter ("Unique continuation of elliptic equations") that starts out by saying "any solution of an elliptic equation that is defined on a domain $D$ must vanish on ...
- 1,701
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votes
4
answers
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Einstein field equations in perspectives from PDE and functional analysis
The Einstein field equations have been subject of research in theoretical physics, and differential geometry, apparently with methods from classical analysis and geometry. In particular, solutions in ...
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answers
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PDE on manifolds
I am currently in a PDE course where one of the requirements is to present a paper in PDE. I am wondering if anyone can suggest an early (read foundational, first introductory) paper talking about PDE ...
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What is an "Instanton" in classical gauge theory? (to a mathematician)
There's already a question about the same topic but I think its aim is different.
Classical (non-quantum) gauge theory is a completely rigorous mathematical theory. It can be phrased in completely ...
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Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators
EDIT: According to some comments on this post I revise the title to remove the misunderestanding.
Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated ...
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3
answers
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Poincare lemma for non-smooth differentiable forms
The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for $C^k$-...
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Why is the harmonic oscillator so important? (pure viewpoint sought). How to motivate its role in Getzler's work on Atiyah-Singer?
I'm in the process of understanding the heat equation proof of the Atiyah-Singer Index Theorem for Dirac Operators on a spin manifold using Getzler scaling. I'm attending a masters-level course on it ...
- 1,771
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votes
3
answers
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What is known about sufficient conditions for the rigidity of a convex surface?
A convex surface is a connected open subset of the boundary of a convex body in $\mathbb{R}^3$.
An "infinitesimal bending" of a convex surface $S$ is a deformation of $S$ given by a velocity field $v:...
- 437
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votes
2
answers
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Convergence of solutions to Navier-Stokes to Euler's equation for viscosity $\to$ zero
Let
$$
\partial_t u + \nabla_u u = - \nabla p
$$
be Euler's equation (Wikipedia) for an ideal incompressible fluid. Let
$$
\partial_t u + \nabla_u u - \nu \Delta u = - \nabla p
$$
be the Navier-...
- 1,544
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Can the "physical argument" for the existence of a solution to Dirichlet's problem be made into an actual proof?
Caveat: I don't really know anything about PDEs, so this question might not make sense.
In complex analysis class we've been learning about the solution to Dirichlet's problem for the Laplace ...
- 1,527
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votes
1
answer
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heat kernel on n-sphere
I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...
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Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)
Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates
\...
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0
answers
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Is there an Infinite dimensional sheaf theory for analysis on manifolds?
I apologize if this question is slightly vague but I don't know how to ask it non-vaguely. Moreover, my question is about an ideal situation. If there's a close answer which doesn't satisfy all the ...
- 7,789
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Is there a connection between representation theory and PDEs?
As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination ...
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How to generalize the various vector calculus theorems to distributions?
Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
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Where do some "energy identities" in PDE theory come from?
There are a lot of very complicated expression that helps us obtain useful estimates for PDEs. To just give one example, the following is one of the Virial identities:
$$
\frac12\frac{d}{dt} \Im \left(...
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What is the intuition behind Almgren's frequency function?
It is by now well-known that for a harmonic function $u : B_1^n(0) \to \mathbb{R}$, the ratio
$$
N(r) := \frac{r\int_{B_r(0)}|\nabla u|^2}{\int_{\partial B_r(0)} u^2}
$$
is a non-decreasing function ...
- 1,179
16
votes
1
answer
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views
Are isospectral manifolds necessarily homeomorphic?
It's known that there are pairs of closed Riemannian manifolds which are isospectral but not isometric.
Is it known if there are closed Riemannian manifolds which are isospectral but not homeomorphic?...
- 545
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votes
1
answer
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The determinant as a differential operator
According to Gårding, the determinant is a hyperbolic polynomial over the space $\mathbf{Sym}_n$ of real symmetric $n\times n$ matrices. More precisely, it is hyperbolic in the direction of the ...
- 52.3k
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votes
2
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Surjectivity of curl
Let: $\mathbb R^3\ni x\mapsto v(x)\in\mathbb R^3$ be a vector field with null divergence belonging to the Schwartz class such that
$$
\int_{\mathbb R^3} v(x) dx=0.
$$
Is it true that there exists a ...
- 16.2k
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votes
1
answer
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views
Hearing the 17 planar symmetry groups
Though I'm sure it's not really hard to work out for myself, does anyone know a reference for the spectra of the Laplacian on the 17 flat compact orbifolds that underlie the 17 planar symmetry groups. ...
- 17.6k
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Reference request: the theory of currents
I am a graduate student and want to study the theory of currents. What is a good reference for a beginner?
I should be familiar with the theory of distributions or generalized functions on $\mathbb R^...
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Version of Banach-Steinhaus theorem
I am wondering about the following version of the Banach-Steinhaus theorem.
Let $A$ be a closed convex subset contained in the unit ball of a Banach space $X$ and consider bounded operators $T_n \in \...
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How does one motivates the method of separation of variables when teaching PDE's?
I'm not sure if this question is appropriate for MO. Add comments if it is not. Thanks.
How to explain/motivate the method of separation of variables for PDEs to undergraduates? What's the real math ...
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votes
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Basis for the space of Harmonic homogeneous polynomial in N variables.
Hello,
Does someone know an explicit basis of the space of harmonic homogeneous polynomial in N variables.
When $N=3$, if I'm not mistaking Legendre polynomial allow to write an explicit basis.
Is ...
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Does the Legendre-Hadamard condition imply a generalized Gårding inequality?
For simplicity, we restrict to constant coefficients. Let $A^{ij}_{ab} \in \mathbb{R}$, $1 \le i, j \le n$ and $1 \le a, b\le m$, satisfy the Legendre-Hadamard condition:
$$
A^{ij}_{ab}\xi_i\xi_jv^av^...
- 27.5k
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votes
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Eliminating 1st order terms in elliptic partial differential equation
Under what conditions is it possible, using a suitable change of variables, to eliminate 1st order terms in an elliptic partial differential equation, so that the equation involves the 2nd derivatives,...
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votes
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Symbols of elliptic operators
First let me state the problem, then I'll explain its origin and finally, I'll ask the main question..
Problem S. Fix a positive integer $n$. Find all the pairs $(V, S)$, whith the following ...
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Is there a Seiberg-Witten version of Donaldson-Thomas theory?
Donaldson invariants are a count of instantons (the solutions to a particular elliptic PDE) on 4-manifolds. One thing which makes the theory difficult is a lack of compactness for the moduli spaces of ...
- 7,005
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votes
2
answers
785
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Is there a spectral theory approach to non-explicit Plancherel-type theorems?
Teaching graduate analysis has inspired me to think about the completeness theorem for Fourier series and the more difficult Plancherel theorem for the Fourier transform on $\mathbb{R}$. There are ...
- 56.6k
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votes
6
answers
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partial differential equation for ruled surfaces
We say that a surface $f(x,y,z)=0$ is ruled if for each point $p$ in the surface there is a line that passes through $p$ and is contained in the surface. See http://en.wikipedia.org/wiki/Ruled_surface ...
- 1,359
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votes
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When is a given matrix of two forms a curvature form?
Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of 1-forms $...
- 3,383
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votes
2
answers
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Who is Petrov of the Petrov-Galerkin method?
I was not able to find the origin of the name Petrov in the Petrov-Galerkin method for the numerical approximation of PDEs.
Wikipedia refers to a certain Alexander G. Petrov, but it is still not ...
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