# 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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### Continuity of solutions of Elliptic PDE with respect to parameters

Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy $$\Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$ where $f$ is a fast decaying smooth function. I would like to know how the ...
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### Comparison principle for Elliptic PDE with exponential nonlinearity

Suppose $\varphi$ is a radial (and radially decreasing) solution of $$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R),$$ with $R>0$, and $\psi$ is a decreasing radial function ...
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### Solving a particular delay PDE $\partial_q f(q,s-1) = -\sqrt{s(2+s)}f(q,s)$

I recently encountered a particular delay PDE in my work, the solution of which corresponds to the Laplace transform of some probability distribution. I'm having trouble to solve this equation. The ...
1 vote
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### 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 \...
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### Existence of solution to linear inhomogeneous first order PDEs systems

Maybe I am asking a triviality. If that is the case, please let me know and I will close the question. I have searched a lot but I didn't find an adequate and forceful response. For $i=1,\ldots, r$, ...
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### Pohozaev identity for linear equations

For $-\Delta u =0$, the Pohozaev identity on say $B_1$ says $$\int_{S_1} |u_T|^2 \,d\sigma = \int_{S_1} |u_N|^2 \,d\sigma + (n-2) \int_{B_1} |\nabla u|^2 \ dx$$ Here $u_T$ are the tangential ...
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### Generalizing a formula with distributions — Distributional Radon transform

I will try to describe the problem, it will necessarily be incomplete, so please if you have questions or remarks to make it more clear do not hesitate to leave them in comments. The problem Let $a$ ...
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### Surjectivity of perturbed linear operators

Consider two Banach spaces $X$ and $Y$ and two linear bounded operators $A,B:Y\rightarrow Y$. Suppose the following: (1) Y is reflexive (or even uniformly convex); (2) $X\cap Y$ is dense in $X$ and $Y$...
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### Linear PDE, analytic continuation, Green's function and boundary conditions

I'm looking at the linear PDE in 3+1 dimensions, $$\left[ -(\partial_t - \xi)^2 - \partial_k \partial_k \right] \phi(t,x) = 4\pi^2 \delta(t)\delta(x)\label{1} \tag{1}$$ Where $\xi$ is generally a ...
1 vote
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### Fundamental Solution to Biharmonic Equation in 3D

(This is a repost of a question posed in StackExchange that didn't get any replies.) Is anything known about the fundamental solution to the equation: $$\nabla^4 (Au) + \nabla^2 (Bu)+Cu=0$$ for ...
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### Numerical method for solving simple linear PDE with a grey box component

Consider the following linear PDE: $$\nabla_q V(q) - M_d(q)M^{-1}(q)\nabla_q V_d(q) = 0,$$ where $V(q)$ and $M(q)$ are known and $M_d(q)$ is a grey box function (e.x., $M_d(q)$ is fitted using a ...
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### Question about the second order linear elliptic PDE on closed manifold

Recently I see a question linear second order PDE in which user Pedro post a reference in Gilbarg's book, which said that the solvability of the linear PDE $$\Delta u +B^{i}(x)u_{i}+C(x)u=f$$ is ...
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### Riesz transform after linear transformation

I am encountering the term $\partial_x \mathcal{R}_x(f(x,y))$. I needed to do the following linear transformation $$x' = a x+ by,\,\,\,\,\, y'=ax-by,\,\,\, and \,\,f(x,y)=g(x',y')$$ I ended up with ...
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### Under which conditions does this PDE have unique solutions

Assume that $f:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ is smooth and consider the linear equation $$\mathrm{div}\, (u)(x) = f(x,u(x)),$$ where $u:\mathbb{R}^n \to \mathbb{R}^n$ is a smooth ...
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### Decay of solution for linear system with damping

Let us consider the following linear system with damping: $$\begin{cases} u_t - u_x = -\frac{1}{2} (u+v)\\ v_t + v_x = -\frac{1}{2} (u+v) \end{cases}$$ Let's write the solution as $w=(u,v)$ ...
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### Solve a coupled PDE in a rectangle

We consider a coupled PDE in a rectangle $\Omega=(-1,1)\times(-1,1)$. For the simplicity, we assume that the functions are periodic in $x_{1}$ direction. \begin{cases} \nabla\cdot u=f_{1},\ & \...
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### Regularity of solution to Cauchy problem given regular initial data

Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus) be a solution of a Cauchy problem $$\frac{d}{dt} f(t) = A f(t)$$ f(0)...
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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 ...
In a neighbourhood of $0$ in $\mathbb{R}^n$ a smooth function $h=h(x)$, $h(0)=0$, is given. Take arbitrary real numbers $w,\lambda_1,\dots,\lambda_n\in\mathbb{R}$. The problem is to find a smooth ...