General solution of Transport equation (homogeneous): Method of Characteristics $$u_t+cu_x=0 (\star)$$ We know that if $f: \mathbb{R} \to \mathbb{R}$ is differentiable then $u(x,t)=f(x-ct)$ is a solution of $(\star)$. We will show that each solution of $(\star)$ is of the form $u(x,t)=f(x-ct)...

Let $V=C^1([a,b])$. If $J$ is a continuous functional for the norm $\|\cdot\|_\infty$ then it is continuous for the norm $\|\cdot\|_1:= ||y||_{\infty}+||y'||_{\infty}, y \in V$. But the converse is false. In my book there is the hint that we can use the functional of arc length, in order to show...

Show that the functional $J(y)=\int_a^b (\sin^3 x+y^2) dx$ is continuous in respect to the $||\cdot||_{\infty}$ norm, at any $y_0 \in C([a,b])$. Let $y_0 \in C([a,b])$. Then for $y \in C([a,b])$ we have: $$|J(y)-J(y_0)|=\left| \int_a^b (y^2-y_0^2) dx\right| \leq \int_a^b ||y-y_0||_{\infty} |y+y...