Outline the operator $E: H_{per}^{1}left([0,L]proper) longrightarrow mathbb{R}$, given by

$$E(u)=frac{1}{2}int_{0}^{L}u_t^2+u_x^2+frac{1}{2}(1-u^2)^2,; forall ; u in H_{per}^{1}left([0,L]proper),$$

the place $L>0$ is a hard and fast fixed.

I wish to calculate the Fréchet spinoff of $E$, for this I began I began by calculating the Gateaux spinoff of $E$: we all know that $ E $ is differentiable Gateuax if there may be $ f in left(H_{per}^{1}left([0,L]proper)proper)’ $ (twin house) such that, for $u in H_{per}^{1}left([0,L]proper),$

$$v_E:=lim_{xi rightarrow 0} frac{1}{xi}left[E(u+xi h)-E(u)-f(xi h)right]=0,; forall ; h in H_{per}^{1}left([0,L]proper).$$

I did the maths I acquired to

$$v_E= int_{0}^{L} u_t h_t+hu_t+u^3h ; dx -f(h),$$

however I can’t proceed from that time, primarily as a result of I don’t know easy methods to calculate the integral

$$int_{0}^{L} u_t h_t ; dx.$$

My thought is to calculate the Gateaux spinoff (discovering such operator $f$) and use the $ E $ continuity to conclude that this spinoff coincides with Fréchet’s and consequently conclude what I need. How do I proceed?

Extra particulars of the house $H_{per}^{1}left([0,L]proper)$ will be discover on this guide