Does $[F'(u^*)]^{-1}$ exist if $u^*$ is a solution of $partial_t u + [u+1-t] partial _x u -1 = 0$?
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Given
$$F(u):=partial_t u(x,t) + [u(x,t)+1-t] partial_x u(x,t) -1 $$
we can calculate the derivative of $F$ and get that the derivative of $F'(u)$ applied to $v$ gives us
$$F'(u)(v)= partial_t v(x,t) + (u(x,t) +1 - t) partial_x v(x,t) + v(x,t) partial_x u (x,t).$$
So if we now know that $u^*$ is a solution of
$$partial_t u + [u+1-t] partial _x u -1 = 0$$
how could we show, that $[F'(u^*)]^{-1}$ exists?
functional-analysis frechet-derivative
asked Nov 21 at 22:14
cptflint
114
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up vote
0
down vote
favorite
Given
$$F(u):=partial_t u(x,t) + [u(x,t)+1-t] partial_x u(x,t) -1 $$
we can calculate the derivative of $F$ and get that the derivative of $F'(u)$ applied to $v$ gives us
$$F'(u)(v)= partial_t v(x,t) + (u(x,t) +1 - t) partial_x v(x,t) + v(x,t) partial_x u (x,t).$$
So if we now know that $u^*$ is a solution of
$$partial_t u + [u+1-t] partial _x u -1 = 0$$
how could we show, that $[F'(u^*)]^{-1}$ exists?
functional-analysis frechet-derivative
asked Nov 21 at 22:14
cptflint
114
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given
$$F(u):=partial_t u(x,t) + [u(x,t)+1-t] partial_x u(x,t) -1 $$
we can calculate the derivative of $F$ and get that the derivative of $F'(u)$ applied to $v$ gives us
$$F'(u)(v)= partial_t v(x,t) + (u(x,t) +1 - t) partial_x v(x,t) + v(x,t) partial_x u (x,t).$$
So if we now know that $u^*$ is a solution of
$$partial_t u + [u+1-t] partial _x u -1 = 0$$
how could we show, that $[F'(u^*)]^{-1}$ exists?
functional-analysis frechet-derivative
asked Nov 21 at 22:14
cptflint
114
Given
$$F(u):=partial_t u(x,t) + [u(x,t)+1-t] partial_x u(x,t) -1 $$
we can calculate the derivative of $F$ and get that the derivative of $F'(u)$ applied to $v$ gives us
$$F'(u)(v)= partial_t v(x,t) + (u(x,t) +1 - t) partial_x v(x,t) + v(x,t) partial_x u (x,t).$$
So if we now know that $u^*$ is a solution of
$$partial_t u + [u+1-t] partial _x u -1 = 0$$
how could we show, that $[F'(u^*)]^{-1}$ exists?
functional-analysis frechet-derivative
functional-analysis frechet-derivative
asked Nov 21 at 22:14
cptflint
114
asked Nov 21 at 22:14
cptflint
114
asked Nov 21 at 22:14
cptflint
114
asked Nov 21 at 22:14
cptflint
114
asked Nov 21 at 22:14
cptflint
114
114
add a comment |
add a comment |
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