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

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6
votes
2answers
306 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 ...
7
votes
1answer
1k views

Short time existence on nonlinear parabolic PDE

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact? in which book we have this fact, the number of ...
10
votes
1answer
719 views

Regularity of the Maxwell equations

As is well-known, the Maxwell equations can be phrased vectorially as, \begin{align} \nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\ \nabla \cdot \mathbf ...
6
votes
2answers
164 views

Bounded input Bounded output stability for heat equation

This is a cross-post from Computational Science. I am interested in proving or obtaining a counterexample to the following conjecture. Let $\Omega\subset\mathbb{R}^d$ be a bounded open domain. Let ...
5
votes
3answers
363 views

Structure of sign changes under the heat flow

Let $f$ be a smooth function on $R^2$, and define $N_f$ to be the set of points $p$ such that the nodal set of $f$ ($\{x\in R^2: f(x)=0\}$) divided every neighborhood of $p$ into four regions. Indeed, ...
4
votes
2answers
413 views

Analytic solution of a system of linear, hyperbolic, first order, partial differential equations

In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form ...