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      Questions tagged [linear-pde]

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

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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{\...
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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, ...
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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 ...
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      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 ...
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      1answer
      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.
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      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 ...
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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 ...
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      1answer
      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+\...
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      1answer
      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 $c$. ...
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      1answer
      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
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      1answer
      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 ...
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      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(...
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      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(...
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      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 ...
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      158 views

      Replacing the initial conditions for a PDE

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

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