All Questions
Tagged with ap.analysis-of-pdes linear-pde
303 questions
2
votes
0
answers
162
views
Explicit computation of a norm in context of operator-semigroups and differential equations
I am interested in the explicit calculation of the following norm $\vert \cdot \vert$.
Let $X$ a Banach space with norm $\Vert \cdot \Vert$ and $(T(t))_{t \geq 0}$ a strongly continuous one-parameter ...
- 185
2
votes
1
answer
164
views
The only rotation fields satisfying this PDE are constant
$\newcommand{\div}{\operatorname{div}}$$\newcommand{\SO}{\operatorname{SO(2)}}$$\newcommand{\R}{\operatorname{\mathbb{R}}}$$\newcommand{\bdx}{\partial_x}$$\newcommand{\bdy}{\partial_y}$$\newcommand{\...
- 6,741
3
votes
1
answer
404
views
Neumann/Robin Laplacian semigroup well-known estimate
Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:
$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \...
- 1,759
1
vote
0
answers
282
views
Fourier Transform; half space baby problem (new)
This question is related to a prior question i asked, see Fourier Transform ; half space elliptic baby problem.
Essentially I am asking the same question now but taking a lot more care.
So lets ...
- 1,385
4
votes
2
answers
226
views
Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight
Let $\zeta, u_0\in L^2(\Omega)$, with $\zeta \geq 0$ and $\Omega\subset \Bbb R^d$ open and bounded.
\begin{equation}\label{Star-3.7}
\begin{cases}
\partial_t u -\Delta u + \zeta u=0 &\mbox{ in }\...
- 1,101
1
vote
0
answers
70
views
Solutions of constant coefficients differential operator on $\mathbb{R}^n$
Let $D$ be a constant coefficients linear differential operator on complex valued functions on $\mathbb{R}^n$. Let $\tilde D=\mathbb{F}^{-1}\circ D\circ \mathbb{F}$ be its conjugation with the Fourier ...
- 21.8k
1
vote
0
answers
400
views
Calculating frequency of sound of ringing metal coin
I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...
- 119
3
votes
1
answer
230
views
Is there a metric on Euclidean space that turns the Helmholtz equation into the Laplace equation?
Is there a Riemannian metric $\tilde g$ on $\mathbb R^d$ such that
$$\tag{1}
\Delta_{\tilde g}=e^f(\Delta +1),$$
for some $f\in C^\infty(\mathbb R^d)$? Here $\Delta=\partial_{x_1}^2+\ldots+\partial_{...
- 692
3
votes
0
answers
98
views
Semiclassical analysis and reflection law
I was curious about the following vague question. In pseudodifferential calculus and semi-classical analysis there are various theorems that relate geodesics of the underlying space to the behavior of ...
- 445
2
votes
1
answer
305
views
Reconstructing the metric on $CP^2$ with special one forms
I know that $(z_1,z_2)$ are the affine\inhomogeneous coordinates on the complex projective space $CP^2$. Now I have four one forms $(Y_1,Y_2, Y_3, Y_4)$. I want to rewrite the Fubini Study metric on $...
- 69
1
vote
0
answers
123
views
Perturbation in the equation $u_t=\epsilon Pu$, where $P$ is an elliptic partial differential operator
Let $Pu=\sum_{ij} \partial_j(a_{ij}(x) \partial_{i} u)$ be an elliptic operator. Consider the equation
$$
(u=u_\epsilon)\\
\partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\
u(0,x)=u_0(...
- 1,755
2
votes
0
answers
100
views
$L^\infty$ Transport Equation Estimate: Characteristics of the Milne Problem on a Finite Slab
Cross-posted from MSE here.
I'm trying to justify equation (3.43) on page 18 of this paper by Lei Wu and Yan Guo on diffusion approximation of the radiative transport equation.
Consider the following ...
- 123
3
votes
1
answer
136
views
Maximum principle for an elliptic like operator
I am trying to prove some monotonicity of a solution of a given pde; after considering a quantity like $ \phi(x) = x \cdot \nabla v(x)$ ($v$ is the solution of a given pde) I arrive at something ...
- 1,385
4
votes
0
answers
165
views
Continuity of weak solutions to wave equation with time-dependent coefficients
Consider the following second-order wave equation
$$
u_{tt} - div( a\cdot \nabla u) = f \quad \text{ in } (0,T)\times \Omega
$$
with boundary conditions
$$
u(0)=g, \ u_t(0)=h, \ u|_{\partial \Omega}=0....
- 273
2
votes
0
answers
133
views
Isometric immersions of $\mathbb{S}^n$ into $\mathbb{S}^{2n}$
A major open problem in submanifold geometry is to determine whether there exists a nontotally geodesic isometric immersion of $\mathbb{S}^n$ into $\mathbb{S}^{2n}$. Every isometric immersion of $\...
- 89
1
vote
2
answers
598
views
Difference between semilinear and fully nonlinear
I'm confused why the Hamilton Jacobi Bellman equation:
$$\frac{\partial u}{\partial t}(t,x)+\Delta u(t,x) -\lambda||\nabla u(t,x) ||^{2}=0$$
is considered fully nonlinear, but not semilinear.
By ...
- 11
2
votes
0
answers
206
views
Laplacian on a manifold with two boundary components
I am interested in the Laplace equation on knot complements. The full complement of a knot $K$ is in $S^3$, but for compactness, we delete an open tubular neighborhood around $K$. The Laplace PDE on $...
- 217
3
votes
1
answer
89
views
Space of holomorphic functions multiplied by smooth functions taking real values
Suppose we have a fixed function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (sufficiently regular, say $C^\infty$ ). The question is: for which $f$ there exists a scalar function $g: \mathbb{R}^2 \...
- 33
1
vote
0
answers
70
views
Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)
I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it:
...
- 47
3
votes
1
answer
486
views
Ancient Heat equation and Liouville's theorem
I encounter a difficulty when proving the bounded solution of ancient heat equation implying constant function. Suppose $u(t,x)$ is the solution of ancient heat equation:
\begin{equation}
u_{t} = \...
- 139
4
votes
2
answers
1k
views
Gradient $L^p$ estimates for heat equation
I'm looking for a proof of the gradient estimate associated to the heat equation with Dirichlet boundary conditions, to see if I can express the constant $\color{red}{C}$.
$$\|e^{t\Delta_d}f\|_{W^{1,...
- 299
2
votes
1
answer
408
views
Regularity on the boundary for the heat equation with linear source
This is probably a known problem but I was not able to find exactly what I am looking for.
I have the following linear heat equation with zero-flux boundary conditions:
\begin{equation}
\begin{cases}...
- 205
2
votes
0
answers
169
views
A basic question about the Spectral Theorem
Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e.
$$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...
- 434
1
vote
1
answer
491
views
Existence for an overdetermined system of PDEs
I am interested in the existence of a vector valued solution $y = y(x, t) \in\mathbb{R}^n$ to a system of $2n$ equations: there are twice more equations than unknowns. More precisely:
Let $A$ and $...
- 115
7
votes
2
answers
634
views
System of linear pde with non constant coefficients
I'm a physicist and during my research work, I found a system of linear pde with non constant coefficients that I have to study, since I have totally no experience about systems of pde and I have even ...
- 73
2
votes
1
answer
163
views
Justification for uniqueness of solutions to dispersive PDE
For the sake of concreteness, we consider the linear Schrödinger equation
$$
\partial_t u = i\Delta u, \ \ \ \ u(0, x) = u_0(x).
$$
The solution is typically obtained by taking the Fourier transform ...
- 419
5
votes
1
answer
486
views
Lack of exponential $L^2_{t,x}$ decay for a heat equation with growing coefficients
Edit: I have changed the nature of the question, but in order to have a better idea of what I can expect for the original problem (see below).
Given $T>0$ and $n \in \bf Z$, consider the following ...
- 309
4
votes
3
answers
504
views
Solutions of $\Delta \phi + \phi =0$ on $\mathbb{R}^2$
I know a few solutions of the equation $\Delta \phi + \phi =0$ on $\mathbb{R}^2$. They can be described as the Fourier transform of any finite measure $\mu$ on $S^1$. In particular take $\mu$ a finite,...
- 399
11
votes
1
answer
1k
views
Research topics in microlocal analysis
Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution ...
- 589
5
votes
2
answers
325
views
$W^{k,1}$ regularity for elliptic equations
Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and assume $u$ is a solutions of $\nabla \cdot (a \nabla u)=f$ with $a>c>0$ in $\Omega$, where $a\in C^k(\bar{\Omega})$. Is the following ...
1
vote
0
answers
125
views
6 linear PDE for only 3 unknowns?
Let $x \in (0,L)$, $t \in (0,T)$, and let $u_0 = u_0(x) \in \mathbb{R}^3$, $g=g(t) \in \mathbb{R}^3$, $P = P(x,t) \in \mathbb{R}^3$ and $Q = Q(x,t) \in \mathbb{R}^3$ be continuously differentiable ...
- 115
1
vote
1
answer
261
views
First order partial differential equation [closed]
I know there is a solution to this pde
$$\partial_{t} f(t,x)= \partial_{x}(v(x)f(t,x))$$
$$ f(0,x)=g(x)$$
( Where $v$ and $g$ are known functions)
which is given by
$$ f(t,x)=\frac{1}{v(x)} h(t+\...
user135936
3
votes
1
answer
205
views
Eigenfunctions of elliptic equations
Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $a_1, a_2$ be smooth positive functions such that $a_1-a_2$ is compactly supported in $\Omega$, and $a_i>c>0$, for some constant $c$. ...
1
vote
1
answer
438
views
Global solutions of the wave equation with bounded initial condition
Let $f,g$ be bounded compactly supported smooth functions, and assume $u$ is the solutions of the wave equation
$$u_{tt}-c^2(x)\Delta u=0 \ \ \hbox{on} \ \ \mathbb{R}^n \times (0,\infty)$$
$$u(x,0)=f, ...
2
votes
1
answer
150
views
Two PDE for one unknown?
Let $x \in (0,L)$, $t \in (0,T)$, and let $f_1 = f_1(x,t) \in \mathbb{R}$, $f_2 = f_2(x,t) \in \mathbb{R}$, $u^0 = u^0(x) \in \mathbb{R}$ and $g= g(t) \in \mathbb{R}$ be continuous functions.
My ...
- 115
0
votes
0
answers
117
views
Harnack Inequality for uniformly elliptic PDE via constructing a singularity
I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
- 1
1
vote
0
answers
173
views
Replacing the initial conditions for a PDE
The problem
The PDE I am working with is given by $\left(\partial_a^b \leftrightarrow\frac{\partial^b}{\partial a^b}\right)$
$$\partial_t \psi = i \partial_x^2 \psi$$
$$\psi(x,t=0) = \psi_0(x)$$
$$\...
- 69
1
vote
1
answer
225
views
Exact solution of two coupled transport equations
I want to solve the following system
$$\eqalign{
& y_t = -y_x + z\text{ in }(0,\text{T}) \times (0,1) \cr
& z_t = z_x + y\text{ in }(0,\text{T}) \times (0,1) \cr
& y(0,x) = y_0,\,\,z(...
- 617
11
votes
3
answers
3k
views
Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix
I am looking for the fundamental solution of the following PDE
$$\partial_i (a^{ij}\partial_j u)=f$$
where $a^{ij}(x)$ is a non-symmetric matrix with possibly non-constant coefficients.
I could find a ...
- 1,303
4
votes
1
answer
331
views
Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$
Consider the following PDE:
\begin{equation}
p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$}
\end{equation}
where $g$ is a flat function at the point (...
- 307
2
votes
0
answers
95
views
Wave equation with 'spring' integral boundary condition
I am really stuck with this small toy problem.
I am trying to solve a wave equation on unit interval $x \in [0,1]$ with a specific boundary condition which I can't see how to tackle. The problem is:
...
- 21
3
votes
3
answers
258
views
ODE with Bessel decay
This is related to my previous question, but it is probably less scary and an expert in using Mathematica could figure out an answer easily.
I would like to estimate the asymptotic behaviour of the ...
- 38
2
votes
0
answers
74
views
Bessel decay for nonhomogeneous PDE
I'm interested in the following nonhomogeneous PDE
$$ (\Delta-k^{2})u=-g $$
on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...
- 53
13
votes
1
answer
994
views
Moduli space of linear partial differential equations
Is there a way to view "the space of all possible linear PDE's" as an algebraic variety with singularities?
This is in connection with a quote from someone on the web that I saw a long time ago. At ...
- 281
1
vote
0
answers
90
views
Uniqueness of solution of linear PDE of first order
Let $\vec u\colon \mathbb{R}^n\times \mathbb{R}\to \mathbb{R}^k$
be at least $C^1$- smooth vector valued function. Assume they satisfy a first order linear equation
$$\partial_t \vec u(x,t)=\sum_{j=1}^...
- 21.8k
1
vote
1
answer
234
views
Existence of unique critical points to second order elliptic PDEs
Let $\Omega\subset\mathbb{R}^2$ is bounded and convex and $\partial\Omega$ be smooth. Consider the second order elliptic PDE (1)
$$
\begin{cases}
Lu = f &\text{ on } \Omega\,,\\ u=0 &\text{ ...
- 173
4
votes
1
answer
168
views
Method of characteristics beyond the Lipschitz setting
I have come across the following easy-looking problem that is driving me mad.
I have a family of measures (on the real line $\mathbb R$) $\{\mu_t\}_{t>0}$ which is uniformly bounded (the measures ...
- 608
0
votes
1
answer
326
views
Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)
I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting.
I want to verify and compare different Discretizations of the ...
- 11
1
vote
2
answers
899
views
Solving the Poisson equation using a random walk on $\mathbb Z ^d$
How do I solve the Poisson equation with the help of a discrete random walk on $\mathbb Z ^d$?
- 11
2
votes
0
answers
145
views
Observability inequality for the heat equation
I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions:
\begin{array}{c}
\varphi _{t}+\...
- 617