# Tagged Questions

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**1**answer

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### Existence of the solution of a linear parabolic pde

Good day!
Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$.
Consider the linear parabolic equation $y' + Ay = f$ where $f \in L^q(0,T;V')$, $y \in W = \{y \in L^p(0,T;V) \colon dy/dt \in ...

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155 views

### 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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**0**answers

52 views

### 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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**1**answer

177 views

### Pseudoinverse of Neumann-Laplacian

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 ...

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votes

**2**answers

259 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 ...

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**1**answer

272 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 $\int f(x) dx =0$ . ...

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**1**answer

575 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
$$
u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y)
$$
...

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**3**answers

440 views

### How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets

$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 ...

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**1**answer

376 views

### orthonormal basis of eigenvectors for laplacian on a concave polygon

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, ...