Is the Laplace equation quasi linear?

Question

Assume we have the Laplace equation $u_{x_1 x_1}+...+u_{x_n x_n}=0$. Then I would say that this equation is quasi-linear since the highest order terms are linear. But I cannot confirm my answer if I search on the internet. So I am probably wrong. Can someone explain to me why the equation is not quasi-linear?

Answer 1

Usually the terminology comes from operator theory. i.e. If you have an operator $L$, we say it's linear if $x,y$ and s (functions wherever they are defined and a scalar) satisfy $$ L(x+y) = L(x) + L(y) \quad \& \quad L(sx) = sL(x)$$

Quasi-linear in the context of differential equations means that your differential operator can be represented as

$$ Q(x+y) = L(x+y) + f = L(x) + L(y) + f = Q(x) + Q(y) - f$$

where $f$ is independent of $x,y$. i.e. a linear and with some extra term as a cost. Notice every linear operator is quasi-linear with $f= 0$. If $f$ depends on $x,y$, we say the operator is non-linear.

In your case you have

$$\Delta (x+y) = \Delta(x) + \Delta (y) \quad \& \quad \Delta(sx) = s\Delta(x)$$

So we'd call it linear.