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.
4,466 questions
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Gluing of two solutions to the same parabolic equation
Consider the domain $[0,1] \times [0,T]$ and the uniformly parabolic operator $L -\partial_t$ with smooth coefficient. Suppose I have $u_1(x,t) \in C^\infty([0,1] \times [0,T])$ solving
\begin{...
- 5,380
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0
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150
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What role do semiclassical methods play in the study of Ginzburg--Landau-type equations?
As far as I understand, semiclassical limits are used in quantum mechanics to analyse equations that depend on a small parameter $\hbar$. Apparently studying properties of the PDE as $\hbar \to 0$ ...
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436
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Rewriting PDE as "push-forward"
Suppose that we have the following PDE
$$\partial_t \mu_t = \nabla\cdot \left(\nabla \mu_t - (b*\mu_t)\mu_t\right), \tag{1}$$
with $\mu_0$ being a (smooth) probability measure/density on $\mathbb{R}^d$...
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126
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Reference for global theory of Schrödinger operators
Question. What is a good reference to learn about the spectral properties of Schrödinger operators in $\mathbf{R}^n$? I am specifically interested in references that discuss examples where the ...
- 5,048
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1
answer
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Compactness for initial-to-final map for heat equation
Let $M$ be a compact smooth manifold without boundary. Let $T>0$ and let $g$ be a smooth Riemannian metric on $M$. Given any $f \in L^2(M)$ let $u$ be the unique solution to the equation
$$\...
- 5,048
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0
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70
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Measure and other properties of nodal domains of Laplacian
Let $(\phi_k,\lambda_k)$ be the couple of eigenfunctions and eigenvalues of the the Laplacian operator on $\Omega \subset \mathbb R^n$.
The nodal set of $\phi_k$ is the set $$\mathcal N_k = \{x \in \...
- 161
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1
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Continuation (extension) of harmonic functions
Suppose $(M,g)$ is a simply connected smooth Riemannian manifold with smooth boundary and suppose that $U \subset \partial M$ is a smooth open connected subset of the boundary. Now my question is the ...
- 63.9k
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Iterated integrations by parts using the fractional Laplacian
Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that
$$\...
- 17.2k
2
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1
answer
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Viscosity solutions of $(-\Delta)^s u = 0$ in $\Omega $ with non-homogeneous data $u = 1$ in $\mathbb R^n \setminus \Omega$
Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem
$$
(1) \quad \begin{cases}
(-\Delta)^s u +\lambda u= 0 & x \in \Omega \\
u = 1 & x \in \mathbb R^n \...
- 161
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Biharmonic operator and maximum principle (PPP)
I have a question related to the Positivity Preserving Principle (PPP) for $ \Delta^2$ and related topics. Recall if $u$ solves
$$\Delta^2 u = f(x) \mbox{ in } \Omega, \quad u=\partial_\nu u =0 \...
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Solving second order hyperbolic pdes arising from combinatorics
Question. Given the following second order hyperbolic pde
$$u_{tt} - a(x)u_{t} = u_{xx} + b(x) u_{x}+ c(x)u + d(x) \label{1}\tag{$\dagger$}$$
where $a(x)$, $b(x)$, $c(x)$ and $d(x)$ are smooth, is ...
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Points where harmonic functions fail to give a coordinates system
Let $\Omega$ be a bounded domain in $\mathbb R^n$ with a smooth boundary and let $g$ be a smooth Riemannian metric on $\Omega$. Let $f_1,f_2,\ldots,f_n$ be non-constant smooth functions on $\partial \...
- 4,115
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Second order estimates for Dirichlet problem for complex Monge-Ampere equation
Let $\Omega\subset \mathbb{C}^n$ be a bounded pseudo-convex domain. Let $0<f\in C^{\infty}(\bar\Omega)$, $\phi\in C^\infty(\partial \Omega)$. Consider the Dirichlet problem for the complex Monge -...
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On the Schrödinger equation and the eigenvalue problem
Li-Yau 1983_Article
The second part of above paper used the discrete eigenvalues of $\frac{-\Delta}{q}$ where $q>0$ to proof the the number of non-positive eigenvalues of
Schrödinger operator $-\...
- 6,035
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1
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Energy estimates for nonlinear wave type equation
Consider the following wave-type equation,
$$u_{tt}-\frac{2}{t}u_t-\Delta u=g(t,x)$$
where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u_{t})=(0,0)...
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Bounding trace operator from below
In a paper, I've read the following thing. Here $\Omega$ is a smooth domain
From the standard trace theorem we know there exists a bounded linear operator $$\gamma: H^1(\Omega) \rightarrow H^{\frac{1}...
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Does the pointwise mean value property imply harmonicity?
Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property:
for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that
$$
u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(...
- 12.9k
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2
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Reference Request: Elliptic differential operators in the Fréchet setting
Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...
- 5,824
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A singular differential equation
In a neighbourhood of $0$ in $\mathbb{R}^n$ a smooth function $h=h(x)$, $h(0)=0$, is given. Take arbitrary real numbers $w,\lambda_1,\dots,\lambda_n\in\mathbb{R}$.
The problem is to find a smooth ...
- 17.2k
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Techniques for showing non-degeneracy results (PDE)
Motivation:
Consider the equation,
$$-\Delta u = u^p$$
in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...
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Almost(?) elliptic operators
I would like some references concerning the following subject.
Suppose that $\Omega$ is a bounded subset in $\mathbb{R}^n$ with smooth boundary and consider the following PDE there stated
$$L(f)(x) = ...
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85
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Does such a vector field exist?
Does there exist a velocity field $\mathbf{u}(x,t)\in \mathbb{R}^3$ such that $$\text{Div}\begin{bmatrix}
\mathbf{u}\cdot \nabla w_1\\
\mathbf{u}\cdot \nabla w_2\\
\mathbf{u}\cdot \nabla w_3\\
\end{...
- 63.9k
5
votes
1
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Seeking for references on some PDEs
This is not a technical mathematical question. I came across some PDEs with no references nor their names.
$$-\Delta u + \int_\Omega udx = f\qquad \hbox{in $\Omega$} \label{1}\tag{Eq1}$$
The above ...
- 313
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1
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Expected properties for a PDE whose solution is supposed to be something that doesn't exist
My understanding of Lecture #33, 34: The Characteristic Function for a Diffusion:
As an alternative to directly computing the characteristic function of a random variable $X_t$ in a stochastic ...
- 247
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1
answer
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Estimate on $C^1$-norm of solution of the Dirichlet problem for the Laplace equation
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^\infty$-smooth boundary. Let $\phi\in C^\infty(\partial \Omega)$. Let $u$ be the solution of the Dirichlet problem of the Laplace equation
\...
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Has this form of the heat equation been solved for the radiation boundary condition
Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ...
3
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Non-linear, hyperbolic, 2nd order system of PDEs
This is a cross-post.
In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system
\begin{...
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do hyperfunction solutions always exist?
I have two questions---the second question (which is what I'm really interested in) is a generalization of the first, but I think the first may be more likely to get an answer. I'll be happy with an ...
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Is the time of solution shorter as the initial data increases?
I'm reading the book superlinear parabolic problems and I came across the following situation twice: given two initial data $u_0$ and $\underline{u_0}$ with $u_0\geq \underline{u_0}$, $u_0\neq \...
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Continuity of the entropy of the solution of a parabolic PDE at $t=0$
Consider the following initial value problem for a parabolic PDE :
$$\begin{cases}
\textrm{div}\big(A\,\nabla u(t,x) + b(x)\, u(t,x)\big) \,=\,\partial_t u(t,x) \quad x\in\mathbb R^d\,,\ t>0 \\[4pt]...
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Elliptic equations in asymptotically hyperbolic manifolds
I am interested in reading about existence and regularity theorems for elliptic equations on manifolds with negative (constant) curvature outside a compact subset. I am aware of some results in this ...
- 1,263
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1
answer
141
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Averaging and fractional Laplacian
Let $u,\phi:\mathbb R \to \mathbb R$ be smooth functions and $\Omega_\epsilon$ be a bounded domain in $\mathbb R$ with diameter $\epsilon>0$ (consider for exaple the ball $B_{\epsilon/2}(0)$). Is ...
CommunityBot
- 1
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votes
2
answers
921
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Exotic spectrum of Laplace operator
Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator,
it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
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0
answers
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Results on the eigenspace of weighted elliptic eigenvalue problems
I am considering the following eigenvalue problem in $\Omega=\mathbb{R}^n_+$
$$-\operatorname{div}(a(x)\nabla \varphi) = \lambda a(x)w(x)\varphi$$
where the weights $a>0$ and $w\in L^{\infty}$ (and ...
- 547
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votes
1
answer
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Semi-linear elliptic problem, energy functionals, Fréchet derivatives and the Newton method in Banach spaces
Suppose $\Omega\subset\mathbb{R}^n$ is a regular open set, $f\in L^2(\Omega)$ and consider the following elliptic problem.
$$-\Delta u + u=f'(u) , \;\;u_{|\partial \Omega}=0,$$
where $f'$ is the ...
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Poisson kernel for fractional Laplacian
Does the Poisson kernel for fractional Laplacian $(-\Delta)^s$ exist for a smooth bounded domain with two boundaries. If so, is there a upper and lower bound for the Poisson kernel.
- 63.9k
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Degenerate Beltrami equation
Question:
Let $\mu:\mathbb C\to \mathbb C$ be a $C^\infty$ function satisfying $|\mu|\le 1$.
Let us furthermore assume that the function $\mu$ never takes the value $-1$.
Does there exist a $C^\infty$ ...
- 43.2k
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On the physics background of p-Laplacian equation
Could you tell me the physics background of p-Laplacian equation? Thank you!
Actually, I know nothing about this. But I am curious about the original of these PDEs or where they come from. Could you ...
CommunityBot
- 1
4
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0
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Minimal regularity for domains in Green's formula
The Green formula is well-known for smooth bounded domains of $\mathbb R^d$. My question is:
What is the minimal regularity known for domains where Green's formula still holds?
- 395
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2
answers
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Density of traces of solutions to an elliptic equation
Let $D_1$ be a domain with smooth boundary and assume that $D_1$ is a proper subset of $D_2$ which is itself a bounded domain in $\mathbb R^n$ with a smooth boundary. Assume also that $D_2\setminus ...
- 684
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Proving an estimate for the Neumann problem on $\mathbb{R}^3 \setminus B_1$ in Weighted Sobolev spaces
Let $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
It is known that for every $g \in H^{\frac{1}{2}} (\partial M)$, and for an appropriate $\delta$, there exists a unique solution $u$ ...
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Wave equation with porous medium term
The classical porous media equation is
$$u_t - \Delta(u^m) = 0 \quad m>1.$$
Has the (degenerate) wave equation
$$u_{tt} - \Delta(u^m) = 0$$
been subject of studies? What would the physical ...
- 839
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1
answer
691
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Regularity of weak solutions to semi-linear elliptic PDEs
Suppose that $f:\Bbb R^2\to\Bbb R$ is a continuous non-linearity and consider the following semi-linear elliptic PDE given by:
$$-\Delta u=f(x,u),\;\;x\in\Omega\subset\Bbb R^n,\tag{1}\label{1}$$
To ...
- 597
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0
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Method of characteristics and explicit formula for an IBVP for the transport equation
Let us assume $c:(0,1) \to (\epsilon,+\infty)$ and consider the PDE
$$
\begin{cases}
u_t+c(x)u_x = 0, \\
u(0,x) = g(x) \\
u(t,0) = f(t)
\end{cases} \qquad (t,x) \in (0,T)\times(0,1) \label{1}\tag{$\...
- 6,367
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2
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spaces of smooth functions for linear hyperbolic PDE
Which classes of (scalar or systems of) linear first or second order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:...
- 21.6k
2
votes
1
answer
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On a heat equation-like PDE
For $f: \mathbb R^n \to \mathbb R$ a locally integrable function, $\varepsilon \in (0, \infty)$, and $x \in \mathbb R^n$, define $I(f, \varepsilon, x)$ to be the averaged integral of $f$ over $B_{\...
- 6,323
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0
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111
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Two identical objects circling the center of mass periodically in general relativity
In Newton's gravity we can have two identical objects circle the center of mass periodically (assuming the surroundings are vacuum).
Is something like this possible in general relativity? Is there an ...
- 31
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0
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102
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Is there a non-degenerate solution for this PDE on $\mathbb{R}^3$?
$\newcommand{\tr}{\operatorname{tr}}$
$\newcommand{\R}{\mathbb{R}}$
Does there exist a smooth map $f:\mathbb{R}^3 \to \mathbb{R}^3$, which satisfies
$$\tr \big( df \otimes \delta(df \wedge df) \big)=0,...
- 6,741
2
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0
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229
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Weighted Sobolev norm in terms of Spherical harmonics coefficients
Let $M = [1,\infty) \times S^2$.
Consider the weighted Sobolev space $H^k_{\delta}(M)$ with the Sobolev norm:
$$\lVert u \rVert_{k,\delta}^2 := \sum_{n=0}^k \int_M |D^nu \,r^{n-\delta}|^2 r^{-3} dV $$
...
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3
votes
3
answers
219
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Asymptotic behavior of an integral transform
Given $g\in L^2(\mathbb{R}^3)$, consider the following function ( defined for $r>0$ ):
$$c(r):=\int_{\mathbb{R}^3}\frac{g(x)}{|x|^2+r}dx$$
I'm interested in the behavior of $c(r)$ for large $r$. A ...
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