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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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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 $\xi\in C^2(\overline{\... 0answers 59 views ### Solution of equation on vector field I have a vector field function$\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, ... 0answers 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 ... 1answer 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 ... 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. 2answers 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 ... 0answers 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 ... 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 $$f(t,x)=\frac{1}{v(x)} h(t+\... 1answer 168 views ### Eigenfunctions of elliptic equations Let \Omega be a bounded region in \mathbb{R}^n and a_1, a_2 be smooth positive functions such that a_1-a_2 is compactly supported in \Omega, and a_i>c>0, for some constant c. ... 1answer 61 views ### Global solutions of the wave equation with bounded initial condition Let f,g be bounded compactly supported smooth functions, and assume u is the solutions of the wave equation$$u_{tt}-c^2(x)\Delta u=0 \ \ \hbox{on} \ \ \mathbb{R}^n \times (0,\infty)u(x,0)=f, ... 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 ... 0answers 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 $$L(\phi):=\Delta \phi(... 0answers 34 views ### Wave equation regularity I have an equation of the type$$\hat{\rho} u_{tt}-\hat{E}u_{xx}=f(x,t)$$for x\in (0,1) and t>0, where \hat{\rho} and \hat{E} are constants, u(0,t)=u(1,t)=0, u(x,0)=p(x), u_t(x,0)=q(... 0answers 94 views ### Harnack Inequality for uniformly elliptic PDE via constructing a singularity I am trying to prove a Harnack inequality for a nonnegative subsolution u \in H^1(B_2) to the PDE \text{div}(A Du) \ge 0, where A = A(x) is uniformly elliptic. The proof outline I am following ... 0answers 158 views ### Replacing the initial conditions for a PDE The problem The PDE I am working with is given by \left(\partial_a^b \leftrightarrow\frac{\partial^b}{\partial a^b}\right)$$\partial_t \psi = i \partial_x^2 \psi\psi(x,t=0) = \psi_0(x)\...

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