Home > Analysis, Olympiad, Partial Differential Equations > Traian Lalescu Student Contest 2011 Problem 4

Traian Lalescu Student Contest 2011 Problem 4


Let D=(0,\infty)\times (0,\infty),\ u \in C^1(D) and \varepsilon>0 fixed.

1) Prove that x \frac{\partial u}{\partial x}(x,y)+y\frac{\partial u}{\partial y}(x,y)=u(x,y),\ \forall (x,y) \in D if and only if there exists \phi \in C^1(0,\infty) such that u(x,y)=x\phi(y/x),\ \forall (x,y) \in D.

2) Prove that \left|x \frac{\partial u}{\partial x}(x,y)+y\frac{\partial u}{\partial y}(x,y)-u(x,y)\right|\leq \varepsilon,\ \forall (x,y) \in D then there is a unique function \phi \in C^1(0,\infty) such that \left|u(x,y)-x\phi(y/x),\right| \leq \varepsilon \ \forall (x,y) \in D.

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