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.
4,467 questions
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Regularity properties of the derivatives of a particular function on $D \times D\to \bar{D} $ ?
This question might sound a little less rigorously formulated, but I hope the question still makes sense.
Let $h: S^1 \to S^1$ be an oriention-preserving homeomorphism and let $p(z,t) = \frac{1}{2\...
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Numerical solution
Last time, I asked this question
but after discussing with some friends, I have given up finding closed-form solutions. Now I have a simpler question.Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ ...
CommunityBot
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General solutions for HJB equations in a special case.
I am reading the book of Wendell Flemming in control theorem to learn the HJB equation
Here is the setting that interests me: Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ functions such that $0\...
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What would the best treatment of Gehring's lemma look like?
In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a ...
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Finding a dimension-free bound for a certain multiplier on Euclidean space
The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
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Endpoint Strichartz Estimates for the Schrödinger Equation
The non-endpoint Strichartz estimates for the (linear) Schrödinger equation:
$$
\|e^{i t \Delta/2} u_0 \|_{L^q_t L^r_x(\mathbb{R}\times \mathbb{R}^d)} \lesssim \|u_0\|_{L^2_x(\mathbb{R}^d)}
$$
$$
2 \...
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Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising
For a image denoising problem (below):
http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf
the author has a functional E defined
$E(u) = \int\int_\Omega F \\ d\Omega$
which he wants to ...
- 459
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kernel of the conformal Laplacian
Let $M$ be a smooth, closed manifold of dimension $n>2$. Let $L_g$ be the conformal Laplacian of the metric $g$. That is, $L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g$, where $R_g$ is the scalar ...
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How to motivate and interpret the geometric solutions of Hamilton-Jacobi equation?
Studying the Hamilton-Jacobi equation, I meet a generalization of the notion of its solutions, which is found already in the work of Sophus Lie.
For an H-J eqn, I mean a first order pde $H\circ dS=...
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Long time behavior of the heat equation on R
Let $\mu\in\mathcal{S}'(R)$ be a Schwartz distribution. The solution of a heat equation with $\mu$ as the initial data is
$$
u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y)
$$
...
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Undergraduate Derivation of Fundamental Solution to Heat Equation
It is well known that the 1-dimensional heat equation $$\frac{\partial}{\partial t} u(x,t)=a\cdot\frac{\partial^2}{\partial x^2} {u(x,t)}$$ has the fundamental solution $$K(x,t)=\frac{1}{\sqrt{4\pi a ...
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Applications of geometric evolution equations.
Hi everybody,
I'm looking for applications of geometric evolution equations such as the Ricci flow and the extrinsic flows by Gauss and mean curvature. Applications other than topological ...
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Dirichlet-to-Neumann map on $C^{k,1}$ domains
I am interested in the mapping properties of the Dirichlet-to-Neumann map (also called the Poincare-Steklov operator) for $C^{k,1}$ domains, between Sobolev spaces on the boundary. What I know is in ...
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PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field
Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases).
Let $\nabla^{2}$ be the vector Laplacian. Let $<\cdot,...
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Maximum principle for heat eq. with boundary conditions on derivatives
The Maximum principle for parabolic eq. is based on the fact that the boundary conditions are given on u.
How can this Maximum principle be used, when having boundary conditions including derivatives....
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Higher order Sobolev inequality
Let $(M,g)$ be a closed, Riemannian manifold of dimension $n>4$. Let $K$ be the best constant for the Sobolev inequality
$||u||^2_p \leq K \int_{{\Bbb{R}}^n} (\Delta u)^2 dx,$
where $p=\frac{2n}{...
- 1,701
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A more accurate method to solve hyperbolic PDE
Hi all,
I have a set of hyperbolic PDE and I have been solving this equation uisng Lax-Wendroff method (from Richtmyer). The solution is OK while I am looking for a better approach to do it. Is there ...
CommunityBot
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Does elliptic regularity guarantee analytic solutions?
Let $D$ be an elliptic operator on $\mathbb{R}^n$ with real analytic coefficients. Must its solutions also be real analytic? If not, are there any helpful supplementary assumptions? Standard ...
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A name for PDE systems which are neither under- nor overdetermined?
The concepts of overdetermined and underdetermined PDE systems are well known. However, all sources I have so far looked into appear to avoid giving any name to PDE systems which are neither ...
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Positivity of Second-Order Elliptic Differential Operators
Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $\Delta = -div\nabla$ be the Laplace-Beltrami operator. Let $h$ be a smooth function on $M$. Is there a condition on $h$ weaker than non-...
- 1,701
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Separability of continuous functions with compact support [closed]
Hi,
is the space $C_0(\mathbb{R}^m)$, $m \in \mathbb{N}$ of continuous functions with compact support separable? If yes: where can I find a proof for that?
Please note: this is not a duplicate of ...
CommunityBot
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A quick and elementary question from Hubbard's Teichmuller Theory : Volume I
Hi,
On page 120, chapter 4, proposition 4.2.7 in Hubbard's Teichmuller Theory book, volume 1, he proves :
Let $U,V$ be open in $C, f:U \to V $ be a homeomorphism and the restriction of $f$ on $U \...
- 1,546
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Characterize where the Dirichlet Problem for the Laplacian is always solvable
Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner ...
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answer
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Solvable PDE Problems [closed]
I'm currently in a PDE course where one of the requirements is to find a common PDE problem and explain how to solve it.
The problems found easily on google won't help me, since every student has to ...
- 1,993
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Sobolev Embedding Theorems
I am getting a little confused about the huge number of slight variations on the Sobolev Embedding Theorem.
Let $\Omega\subseteq\mathbb{R}^n$ be a bounded Lipschitz domain and suppose that $f\in L_\...
- 13k
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Maximum principle corner
Hello,
consider a parabolic boundary value problem, for instance
$-\partial_tu+\Delta u=0$, in $\Omega$,
$\partial_\nu u=0$ on $\partial\Omega$,
in a domain $Q=(0,T)\times\Omega$, where $\Omega\...
- 13k
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1
answer
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convergence of metrics
Hi, I have the following question: take a Riemannian manifold M, with a family of smooth metrics $g(t)$ in $[0,T)$, call $D_0$ the Levi-Civita connection of $g(0)$ and assume that for every $m\geq 0$
$...
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1
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Stuck on a convergence argument in $H_0^1(\Omega)$.
I'm trying to verify that a functional I have satisfies the Palais Smale condition for appliction of the Mountain Pass lemma.
However I've encountered this step along the way which seems clear to me ...
2
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Poisson equation in the plane
Hello,
as I'm not an analyst, I'm having difficulties with the following, certainly well-known problem: one is given the PDE $\Delta u(x,y)=\sqrt{x^2+y^2}$ in the "region" $x^2+y^2\leq1$ with the ...
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metrics compatible with conformal structures
I have three related questions:
(1) How does one describe the possible Riemannian metrics that are compatible with a conformal structure on a two dimensional surface?
(2) Can all conformable ...
- 108k
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Proof of L^p Elliptic Regularity
Let $L = \sum_{i,j=1}^n -\frac{\partial}{\partial x^i} (a^{ij}(x)\frac{\partial}{\partial x^j}) + \sum_{i=1}^n b^i(x) \frac{\partial}{\partial x^i} + c(x)$ be a second order elliptic operator with ...
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Can the solution manifold for an exterior differential system be represented using alternating multivectors?
Differential equations can be written as an ideal of n-forms. Solutions are manifolds where the forms pull back to zero. Is it possible, or useful, to represent the solution by multivectors? For ...
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Removable Singularities for Elliptic Equations
The following fact is quite standard and does not have a very long proof:
$(\*)$ If $u$ is harmonic on $B_1(0)\setminus \{0\}$ and uniformly bounded, then $u$ in fact extends to a harmonic function ...
3
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2
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Localization of Laplacian eigenfunction on the unit square?
Let A be the unit square, $\{u_k\}$ is the set of all L2-normalized Laplacian eigenfunctions with Dirichlet boundary condition. Is it true that for any open subset V, $C_V = \inf\limits_k \int\...
- 13k
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Spectral Galerkin method for a semi-linear parabolic PDE
I'm trying to understand how to apply the Galerkin method to $u_t - \Delta u = u^3$. I understand how to obtain all of the a-priori estimates using Sobolev embeddings and such but my question concerns ...
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another solution to PDE possible?
hi there,
i have the following pde:
$$(\partial_x x^4 \partial_x - \partial_t^2)y(x,t)=0$$ and found the solution $$y=a+t^2-1/x^2$$, with a a constant.
Is this solution unique? Does anyone know of any ...
- 52.3k
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Is this (interpolation) inequality right?
Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^3$, $F$ is bounded in $L^\infty (\Omega \times (0,T))\cap (\cap_{k=1}^\infty L^{5/3}(0,T;C^k(\bar{\Omega})))$.
Question: Can we say that $F$ ...
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Uniqueness of weak solution L[u]=0
Suppose L is a partial differential operator of arbitrary order with constant coefficients.
If u is in $L^p(\mathbb{R}^n)$ and Lu=0 in distributions, is it necessarily the case that u=0? Does the ...
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smoothness of solution for second order elliptic problem
Hello all,
could someone point me to a reference that ties the smoothness of the solution $u$ to the classical elliptic problem
$\nabla \cdot ( q \nabla u ) = f \;,\; x \in \Omega$
$u = g \;,\; x \...
CommunityBot
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PDEs, boundary conditions, and unique solvability
I'm interested in a criterion that determines whether a linear scalar PDE (arbitrary order) has a unique solution given vanishing boundary conditions at spatial infinity. I'll try to formulate the ...
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Sobolev norms of eigenfunctions
Let D be a domain in R^n, and let f be an eigenfunction of the Laplacian with Dirichlet boundary condition with eigenvalue $\lambda$. Assume that f has L^2 norm 1. I want to know if I can say anything ...
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K.Uhlenbeck's preprint "A priori estimates for Yang-Mills fields"
Does anyone have a copy of the unpublished preprint of Karen Uhlenbeck A priori estimates for Yang-Mills fields from around 1986?
It appears to have circulated for some time, and it is quoted in ...
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Rate of convergence of smooth mollifiers
How does one figure out/prove the rate of convergence (in some norm) of mollifiers given a function bounded in some other norm (say Sobolev space, Besov space)? Also, is there a dimensional analysis ...
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Is the derivative of a Lipschitz function better than L^\infty
How smooth is the first derivative (in the distribution sense) of a Lipschitz function? Taking difference quotients and testing against an $L^1$ function, we see that $Df$ is in $L^\infty$. In ${\...
- 1,771
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Vanishing solution of the Poisson equation at infinity
Hi, I am interested in finding some vanish bahavior at infinity of the solutions of this kind of equations:
$-\Delta\phi+a(x)\phi=b(x)$
where $a(x), b(x)\in L^{p}$ with $1\leq p\leq 3$. Besides $\...
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Any similar inequality in literature?
I got the following inequality:
$B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary.
$(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$,
...
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question about mixed spectrum of a linear operator $\mathcal{L}$
Suppose $\mathcal{L}$ is a bounded linear operator and I have the solution to Eigenvalue problem
$\mathcal{L} \phi + \lambda \phi = 0$
wish to solve the following PDE
$\left(-\partial_t + \mathcal{...
- 4,729
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3
answers
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Laplace equation with mixed boundary conditions
Does the Laplace equation on a rectangle with Dirichlet boundary conditions at two opposing sides and Neumann boundary conditions at the other two, always have a solution? If it does, is it unique? Is ...
- 758
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320
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Poisson problem with a "scaled" Laplacian.
Let $d_1$ and $d_2$ be positive constants. I'm considering a 2D Poisson-like problem of the form
$$ d_1\frac{\partial^2 u}{\partial x_1^2} + d_2\frac{\partial^2 u}{\partial x_2^2} = f$$ in the ...
- 617
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Controlling the Second Eigenvalue of a Schrödinger Operator
Consider a bounded domain $\Omega$ (with smooth boundary) in some Riemannian $n$-manifold $M^n$.
Let $L$ be the operator
$$
L=\Delta+V
$$
where $\Delta$ is the Laplace-beltrami operator on $M$ (so is ...
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