# Questions tagged [linear-pde]

Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.

246
questions

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32 views

### A linear first order PDE with boundary condition

I want to solve the following first order PDE
$$
(\star)\quad\begin{cases}
\nabla u\cdot \nabla\xi=f \quad\text{in}\,\Omega, \\
u\mid_{\partial \Omega}=0
\end{cases}
$$
where $\xi\in C^2(\overline{\...

**2**

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59 views

### Solution of equation on vector field

I have a vector field function $\vec{J}: {\bf R}^3\to {\bf R}^3$ looking like:
$$ \vec{J}(\vec{r}) = (\vec{B} \times \vec{v}(\vec{r}))\rho(\vec{r}) $$
with a (very well behaved) real, positive, ...

**0**

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49 views

### Linear system formulation for a PDE with Neumann boundary condition

PDE with Dirichlet boundary condition can be written as a linear system:
$Au(x)=f(x); \ \forall x \in \Omega$,
s.t. $u(x)=g(x); \ \forall x \in \Gamma$.
This can be solved for instance using the ...

**5**

votes

**1**answer

367 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 ...

**2**

votes

**1**answer

71 views

### Reference request: Schauder estimates for parabolic equations

Where can I find Schauder estimates for second order linear parabolic equations (in divergence form with potential)?
Any reference would be highly appreciated.

**5**

votes

**2**answers

145 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**

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115 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 ...

**1**

vote

**1**answer

105 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+\...

**3**

votes

**1**answer

168 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

61 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, ...

**1**

vote

**1**answer

92 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 ...

**0**

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**0**answers

25 views

### approach to perturb a linear operator

My question is related to how one normally would perturb a linear operator.
Let $B_1$ denote the open unit ball in $ R^N$ and suppose $\gamma>0$ is such that the operator
$$L(\phi):=\Delta \phi(...

**2**

votes

**0**answers

34 views

### Wave equation regularity

I have an equation of the type
$$\hat{\rho} u_{tt}-\hat{E}u_{xx}=f(x,t)$$
for $x\in (0,1)$ and $t>0$, where $\hat{\rho}$ and $\hat{E}$ are constants, $u(0,t)=u(1,t)=0$, $u(x,0)=p(x)$, $u_t(x,0)=q(...

**0**

votes

**0**answers

94 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**

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158 views

### Replacing the initial conditions for a PDE

The problem
The PDE I am working with is given by $\left(\partial_a^b \leftrightarrow\frac{\partial^b}{\partial a^b}\right)$
$$\partial_t \psi = i \partial_x^2 \psi$$
$$\psi(x,t=0) = \psi_0(x)$$
$$\...