# Tag Info

### Basic Solution to the Heat Equation

As of your final answer, it is indeed a correct one. In more general terms, you would need to find a so-called fundamental solution $\Phi(x,t)$, which is the solution to the problem:  \left\{ \begin{...
• 1,220
Accepted

### Why are kernels often singular on the diagonal?

A partial answer is that many kernels of interest are Green's functions of some PDE, and if that PDE is translation invariant, then the Green's function will be of the form $G(x,y) = \Phi(x-y)$ and ...
• 6,119
Accepted

### Basic Solution to the Heat Equation

There is a conceptual bridge between stochastic models of motion of particles by random steps in time and the smooth Gaussian as the probability density of the final distribution the the position ...
• 3,234

### On max-min representation for the principal eigenvalue of nonsymmetric elliptic operators

This solution is due to a brilliant friend of mine. I promissed him the bounty if he wrote his solution, but I guess he is too busy to write a MSE post. If he does end up writing it I will naturally ...
• 2,234
1 vote

### Why is the term $H(x,y)u_{yx}$ omitted in every definition of a linear 2nd order PDE in two independent variables?

Clairaut's Theorem and its variants makes it so you generally are in a situation where $u_{xy}$ and $u_{yx}$ are equal, so that you only need one term to handle both of them.
• 24.3k
1 vote

$\newcommand\dd{\mathbf d}$It's safest to assume you can't take the derivative with respect to a constrained variable. You're better off using an unnormalized quaternion. Then the rotation is $g \... • 7,539 1 vote ### Saying solution of a PDE is continuous to the boundary No. Let$\Phi$be the fundamental solution to Laplace's equation (aka the Newton potential). It has a singularity at$0$. Now if$y \in \partial \Omega$then the map$u(x) = \Phi(x-y)$satisfies$\...
• 8,496

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