# Tagged Questions

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### Tempered distribution solution to a simple PDE

Let's consider the following PDE in $\mathbb R^d$ : $$\frac{\partial^d u}{\partial x_1...\partial x_d}=f$$ where $f$ is a tempered distribution with support in $\mathbb R^d_+$. There is a result by ...
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### Parabolic partial differential equation, initial conditions

Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$. Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
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Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that $$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0.$$ Further assume a solvability condition $$\int_\Omega f ... 2answers 260 views ### Uniform bound on the eigenfunctions of the Laplacian Is it possibly to have L_\infty bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty ... 1answer 273 views ### Upper bounds for the solution of an elliptic PDE depending on a parameter. Suppose I have the following PDE on [0,1]^n$$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$with periodic boundary conditions and \int f(x) dx =0 . ... 1answer 576 views ### Long time behavior of the heat equation on R Let \mu\in\mathcal{S}'(R) be a Schwartz distribution. The solution of a heat equation with \mu as the initial data is$$ u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y)  ...
$Qn#1$ : Let $f:U\to V$ be a $K$ quasiconformal homeomorphism ( NOT diffeomorphism ) of plane open subsets of $C$. By my definition of quasiconformality, I mean 1)$f$ is continuous, 2)the weak ...
I am interested in the Laplace operator $\Delta$ on a concave polygon. When the polygon is convex, it is known that $\Delta: H^2(\Omega) \rightarrow L^2(\Omega)$ is boundedly invertible. In addition, ...