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My question is as follows: Let $A=\partial_x-\frac y2\partial_z$, $B=\partial_y+\frac x2\partial_z$, and $\Omega\subset \mathbb R^3$ be a smooth bounded open set. Take $g\in C^\infty(\Omega)$ (if you want even compactly supported in $\Omega$, although I don't think this is important).

Using the method of characteristics it is rather easy to solve the first order PDE $A(h)=g$, and in fact I obtained the solution (checked with mathematica) $$h(x,y,z)=\int_0^xg(\theta,y,\frac 12(xy+2z-\theta y))d\theta+H(y,\frac 12(xy+2z)).$$

Now, let $s\in C^\infty(\Omega)$ and assume $s$ to be very small in the $C^1(\Omega)$ norm, we can assume that $\|s\|_{C^1(\Omega)}\leq \varepsilon$, with $\varepsilon$ small enough. If it is of any use, we can assume that the $C^k(\Omega)$ norm of $\varepsilon$ is smaller than $\epsilon$, for $k\in\mathbb N$ big enough.

Now I would like to solve the perturbed PDE $$A(h)+B(s h)=g.$$ Again the method of characteristics works well if $s$ is a constant. With mathematica I found the explicit solution even if $s=s(x)$. For the general case however I doubt there is any hope of finding the explicit solution.

My question is: is there some general theory that ensures existence (I don't care about uniqueness) AND estimates of the solution $h$, in terms of $s$ and $g$? Even references are welcomed.

Many thanks

-Guido-

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  • $\begingroup$ Why doesn't the method of characteristics work? After all you don't want explicit formulas, but control of the solution. You can first control the characteristics and show that they don't deviate from the case when $s$ is constant. $\endgroup$
    – Fan Zheng
    Commented Oct 19, 2015 at 15:02
  • $\begingroup$ @FanZheng Could you please elaborate a little your answer? Intuitively it seems what I want to do except that.. uhmmm... I can't :) $\endgroup$ Commented Oct 19, 2015 at 15:05
  • $\begingroup$ The whole estimate is quite long but standard, but in which part did you get stuck? $\endgroup$
    – Fan Zheng
    Commented Oct 19, 2015 at 15:26
  • $\begingroup$ I'm stuck because I am not an expert in the field and I don't know where to start with such estimates :( If you can suggest me where to study these things that would be great. $\endgroup$ Commented Oct 19, 2015 at 15:27
  • $\begingroup$ I simply double checked... trackpad :) $\endgroup$ Commented Oct 19, 2015 at 15:43

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The PDE is $$ (\partial_x+s\partial_y+\frac{sx-y}2\partial_z)h=g+B(s)h. $$ The characteristic equation is $$ \frac{dy}{dx}=s,\quad \frac{dz}{dx}=\frac{sx-y}2,\quad \frac{dh}{dx}=g+B(s)h=g+O(\epsilon)h. $$ Integrating gives $$ y(x)=y(0)+O(\epsilon)x,\quad z(x)=z(0)+O(\epsilon)x^2-\frac{xy(0)}2 $$ and $$ h(x,y(x),z(x))=\exp(O(\epsilon)x)[h(0,y(0),z(0))+\int_0^x \exp(O(\epsilon)t)g(t,y(t),z(t))]dt. $$

Now substitute the expressions for $y(x)$ and $z(x)$. You may want to relate $$ g(t,y(0)+O(\epsilon)x,z(0)+O(\epsilon)x^2-\frac{xy(0)}2) $$ to $$ g(t,y(0),z(0)-\frac{xy(0)}2) $$ using the fundamental theorem of calculus, (so I guess you need the $C^1$ bound on $g$ as well.) I'll stop here as I don't want to litter the answer with too many $O(\epsilon)$'s.

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