# Questions tagged [partial-differential-equations]

Questions on partial (as opposed to ordinary) differential equations - equations involving partial derivatives of one or more dependent variables with respect to more than one independent variables.

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### PDE on the probability that the brownian motion stays in [a,b]

This was part of an exam question. Let $a<b$, $(X_t)$ a brownian motion and $$\forall x \in \mathbb R, t\ge 0, \quad \pi(x,t):=P(\forall s \in [0,t], x+X_t \in [a,b]).$$ Given that $\pi$ is $C^2$ (...
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### General solution to $\frac{\partial^n}{\partial x^n}u(x,y)=\frac{\partial^m}{\partial y^m}u(x,y)$

I have never taken a Partial Differential Equations course, so bare with me. I want to find the general solution to $$\frac{\partial^n}{\partial x^n}u(x,y)=\frac{\partial^m}{\partial y^m}u(x,y)$$out ...
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### An isoperimetric-type inequality

I am reading some notes on de Giorgi's methods in the regularity of elliptic equations, and have come across a step which I can't make sense of. The claim is as follows (see Lemma 10 in the linked ...
How to prove that if the initial density $\rho(x,0)$ of the traffic flow is decreasing, then there is $lim_{t->\infty}\rho(x,t)$