
PDE
Can we say that the boundary integral conditions= the nonlocal boundary condition?!!
When the generalized solution = the weak solution?
What's the norm of the sobolev space H^{r} and how we can say u_{m}^{p2}u_{m} \in L^{q}(0,T;H^{r})(\omega))?
In all the lemma 4.4....4.7 they try to use the norm in H^{r})(\omega) why?

Re: PDE
Differentiation under the integral sign is a useful operation in calculus. Suppose that it is required to differentiate with respect to x the function
F(x)= int_{a(x)}^{b(x)}f(x,t) ,dt,
where the functions f(x,t) , and frac{ partial}{ partial x} ,f(x,t) , are both continuous in both t , and x , in some region of the (t,x) , plane, including a(x) leq t leq b(x) ,, x_0 leq x leq x_1 ,, and the functions a(x) , and b(x) , are both continuous and both have continuous derivatives for x_0 leq x leq x_1 ,. Then for ,x_0 leq x leq x_1 , ,:
frac{d}{dx} ,F(x) = f(x,b(x)) ,b'(x)  f(x,a(x)) ,a'(x) + int_{a(x)}^{b(x)} frac{ partial}{ partial x} , f(x,t) ; dt ,.
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