All Questions
Tagged with ap.analysis-of-pdes linear-pde
126 questions with no upvoted or accepted answers
2
votes
0
answers
683
views
Laplace problem with Robin boundary condition on a wedge
I'm trying to understand what the essential differences between Dirichlet/Neumann and Robin boundary conditions are. Therefore, let $\omega \in \left(0, 2\pi\right)$ and let
\begin{equation*}
\Omega = ...
2
votes
0
answers
102
views
Elliptic equation with Laplace-Beltrami boundary condition
For my research, I've come across the following type of equation (under variational form).
Assume $\Omega\subset\mathbb{R}^d$ is a Lipschitz domain, $\phi \in L^2(\partial \Omega)$ and $\nabla_{\...
2
votes
0
answers
178
views
are these norms equivalent?
If it is known that $\sum_{i,j=1}^{n}a_{ij}\xi_i\xi_j\geq \alpha^2|\xi|^2$, where $\xi=(\xi_1,\xi_2,...,\xi_n)\in\mathbb{R}^n$ then can it be said that $\sum_{i,j=1}^{n}a_{ij}\frac{\partial u}{\...
- 157
2
votes
0
answers
110
views
Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions
I asked this question on Mathematics network but it didn't receive any answers. So I assume it is just beyond the classic things in PDEs and I decided to ask it here too.
Consider the following ...
- 271
2
votes
0
answers
211
views
Regularity on Neumann problem on polygonal domain
I asked a similar question before but didn't get any responses. So I will attempt again (the prior question was regarding Holder continuity).
Let $ \Omega$ denote a cube in $ R^n$ and consider ...
- 1,385
2
votes
0
answers
142
views
elliptic regularity for Neumann BVP on square
I am interested in the regularity of ellitpic equations like
$$ -\Delta u(x) +a(x) \cdot \nabla u(x) + C(x) u(x) =f(x) \quad \Omega$$ with $ \partial_\nu u =0$ on $ \partial \Omega$ where $ \Omega=(...
- 1,385
2
votes
0
answers
114
views
biharmonic equation with L^1 data and Navier Condition
I am reading an article that, a section of it is mentioned below . I have some question about this section. I will ask my question after the section below. I am thanksed if some one could help me , ...
- 371
2
votes
0
answers
142
views
Uniform bounds for a coupled parabolic system of PDE (linear)
Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$.
Consider the system, with $u^\epsilon, v^\epsilon \...
- 43
2
votes
0
answers
115
views
Solve a PDE related to free boundary problem
I would like to solve the following system for my problem:
$$\max\Big(\frac{1}{2}u_{ss}+u_l\delta(s-s_0), F(l)-\lambda(s)-u(s,l)\Big)=0.$$
where $u=u(s,l): R\times R_+\to R$ is the unknown function ...
- 1,835
2
votes
0
answers
104
views
Analyticity of one-dimensional PDE solutions with respect to the space variable
Let $n>1$ and $u$ be a solution of a linear PDE with constant coefficients
$$
u_t-\sum_{k=0}^n a_k \partial_x^k u=0,\quad a_k\in \mathbb C,\quad a_n\ne0,
$$
in some neighborhood of a point $(x_0,...
- 2,715
2
votes
0
answers
329
views
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\...
- 281
1
vote
0
answers
23
views
Uniform bound on the first moment for a perturbed advection-diffusion equation
I am studying the solution $u = u(t,x)$ to the following problem on the positive half-line:
$$
\begin{cases}
u_t = u_{xx} + u_x - \frac{1}{1+t}(xu)_x, & \quad t > 0, \, x > 0, \\[2mm]
-u_x = ...
- 43
1
vote
0
answers
39
views
Hyperbolic equation without initial state
Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$:
$$
a^2 u_{xx} - b^2 u_{yy} = f(x, y),
$$
with Dirichlet boundary conditions on $u$.
By using the ...
- 617
1
vote
0
answers
120
views
Well-posedness result for a linear parabolic equation on the torus
Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$
$$ \partial_t u- ...
- 185
1
vote
0
answers
40
views
Is the average of two viscosity sub-solutions to linear elliptic equations is also a sub-solution?
Let $b\in C_b(R;R)$. Consider the following LINEAR equation on $R^2$:
\begin{equation}
u-\partial_{xx}^2 u + (b(x+y)-b(x)) \partial_y u=f\in C^\infty_c(R^2). \tag{1}
\end{equation}
Assume that $...
- 515
1
vote
0
answers
211
views
Understanding the effect of PDE solution on critical strip?
I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...
- 383
1
vote
0
answers
131
views
Regularity of elliptic equation with Neumann boundary conditions
In the context of the regularity of the free boundary of the one-phase (a.k.a. Bernoulli) problem we want to show $C^{1,\alpha}$ regularity of the free boundary implies smoothness of the free boundary ...
- 11
1
vote
0
answers
32
views
Reference for an IBVP for the linear homogeneous 1-D Schrödinger equation
I asked the following question on MSE some time ago, but got no answer (sorry if the question is not appropriate for MO).
Consider the following initial boundary value problem for the linear ...
- 4,034
1
vote
0
answers
106
views
Solution to hyperbolic linear second order PDE
I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution ...
- 11
1
vote
0
answers
38
views
Studying the evolution of laplacian in NS equation
The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided by:
\begin{equation}\label{Eq1}
\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...
- 317
1
vote
0
answers
43
views
Mixed boundary condition of parabolic equations
Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that
$$
\partial\Omega=\partial\Omega_D\cup\partial\Omega_N,
$$
where $ \partial\Omega_D $ and $ \partial\Omega_N $ are ...
- 1,219
1
vote
0
answers
315
views
Maximal regularity heat equation
Considering the heat equation on the flat torus $\mathbf{T}^d$, we have the maximal regularity estimate
\begin{align*}
\forall \varphi\in\mathscr{D}(\mathbf{R}\times\mathbf{T}^d),\quad \|\Delta \...
- 3,425
1
vote
0
answers
252
views
Has anyone studied the PDE generalization of Teichmüller Space?
We begin by recalling the definition of Teichmüller space but stated a little more convolutedly (which will make it easy to generalize).
Given a surface $S$ we can define Teichmüller space $T(S)$ to ...
- 2,689
1
vote
0
answers
39
views
Parabolic theory for singular coefficients on bounded domains (Reference Request)
In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems.
Is ...
- 111
1
vote
0
answers
159
views
Generalized functional for solution of PDEs
Asked this on Math Stack Exchange awhile ago but it got ignored then deleted.
To solve a differential equation of one variable, you need constraints equal to the number of derivatives.
For a partial ...
- 131
1
vote
0
answers
120
views
Liouville theorem for an elliptic equation with gradient perturbation
How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation?
Let $u \in L^2(\mathbb R^n;\mathbb R)$ be a smooth solution of
$$ -\Delta u + v \cdot \nabla u = 0 ...
- 217
1
vote
0
answers
47
views
Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter
Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example
$$
\left\{\begin{aligned}
\Delta \Phi &=\sigma f(x, y) \...
- 809
1
vote
0
answers
126
views
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 ...
- 5,038
1
vote
0
answers
133
views
How can you make a PDE solution stabilize over time?
this is a problem I have been thinking about lately. I tried asking on stack exchange as well but did not find an answer.
Suppose I have a simple linear first order PDE of the form:
$$au_x+bu_y=0$$
I ...
- 251
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
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
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
$$
(u=u_\epsilon)\\
\partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\
u(0,x)=u_0(...
- 1,755
1
vote
0
answers
70
views
Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)
I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it:
...
- 47
1
vote
0
answers
125
views
6 linear PDE for only 3 unknowns?
Let $x \in (0,L)$, $t \in (0,T)$, and let $u_0 = u_0(x) \in \mathbb{R}^3$, $g=g(t) \in \mathbb{R}^3$, $P = P(x,t) \in \mathbb{R}^3$ and $Q = Q(x,t) \in \mathbb{R}^3$ be continuously differentiable ...
- 115
1
vote
0
answers
173
views
Replacing the initial conditions for a PDE
The problem
The PDE I am working with is given by $\left(\partial_a^b \leftrightarrow\frac{\partial^b}{\partial a^b}\right)$
$$\partial_t \psi = i \partial_x^2 \psi$$
$$\psi(x,t=0) = \psi_0(x)$$
$$\...
- 69
1
vote
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
$$\partial_t \vec u(x,t)=\sum_{j=1}^...
- 21.8k
1
vote
0
answers
71
views
An existence result for solutions of elliptic equations with a mixed boundary problem
Assume that $\Omega$ is a bounded domain such that
$\partial\Omega=\Gamma_1\cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint and closed. Let us consider the following elliptic equations.
...
- 181
1
vote
0
answers
70
views
Smoothing in linear hyperbolic equations
This is a bit fuzzy, but I've somewhere read or heard something like:
"For linear hyperbolic equations smoothing in time leads to smoothing in space"
Is this in any sense true?
References, ...
- 43
1
vote
0
answers
116
views
Eigenvalues of elliptic operator analytic with respect to a parameter
I am interested when one can say the eigenvalues of an elliptic operator
are real analytic with respect to a parameter. In particular I have seen many people say the first eigenvalue is analytic but ...
- 1,385
1
vote
0
answers
74
views
Poisson boundary estimates
I asked this question Poisson equation estimates near boundary a few days ago but haven't gotten any response. So I will ask a related question. Suppose $-\Delta u(x)=f(x)$ in $B_1^+$ in the (...
- 1,385
1
vote
0
answers
122
views
Poisson equation estimates near boundary
Fix $ \Omega$ a bounded smooth domain in $\mathbb R^N$ (take $N$ big) and let $ \frac{N+1}{2}<p<N$. We now consider nonnegative smooth functions $f$ such that $-\Delta u(x)=f(x) $ in $ \Omega$ ...
- 1,385
1
vote
0
answers
231
views
Cauchy–Kowalevski Theorem for PDEs
I hope to extend the PDEs across the boundary using Cauchy–Kowalevski Theorem.
Given a real analytic manifold $M$ with boundary $\partial M$, the solution $u$ to the PDE $$\triangle u+b(x)\nabla u+c(...
- 103
1
vote
0
answers
105
views
Positivity of solution of Poisson equation
Let $B$ denote the unit ball centered at the origin in $R^N$ and take $N \ge 3$. Let $( \phi_k(\theta), \lambda_k)$ for $ k \ge 0$ denote the Eigenpairs of $ -\Delta_\theta$ on $S^{N-1}$ which are $L^...
- 1,385
1
vote
0
answers
788
views
$C^{1,2}$ regularity of (weak) solutions to the heat equation
Let $\Omega$ be a bounded Lipschitz domain (smoother if needed), and consider the heat equation
$$u_t - \Delta u = 0$$
$$\frac{\partial u(t,x)}{\partial \nu(x)} = a(t,x) - b(t,x)u(t,x)$$
$$u(0) = u_0$$...
1
vote
0
answers
81
views
About the "method of lines": when are such solutions good approximations for **all** future time?
This question is about approximate solutions to some classes of PDEs obtained using the "method of lines".
For example, for an initial-value problem given by a PDE on a circle, one can choose $n$ ...
- 111
1
vote
0
answers
84
views
Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions
I am looking for a potential theory representation for the following equation in $2$D:
$$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$
$$u = g \,...
- 151
1
vote
0
answers
74
views
Fundamental gap for Neumann BVP with potential
I am sure this is extremely well known but I have been digging a bit and I can't find what I need. Consider $B$ to be the unit ball in $ \mathbb R^N$ and consider the eigenvalue problem
\begin{cases}...
- 1,385
1
vote
0
answers
37
views
Regularity of a flux induced by a potential
Take
$\Omega\subset R^n$ with smooth boundary (take a ball for example)
a function $f\in L^{\infty}(\Omega)$ with support strictly contained in $\Omega$ and with $\int _{\Omega} f \; dx=0$
a scalar ...
- 71
1
vote
0
answers
95
views
Construct a PDE solution from a net of approximations
Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$.
Let $...
- 5,407