# partial differential equations with mixed boundary conditions

hi,

does anyone know some good references (books, papers) on partial differential equations with mixed boundary conditions ?

actually I am intrested in the following: Let $f(x)=(f_{1}(x),...,f_{n}(x))$ such that $x \in \mathbb{R}^{n}$ be an unknown function and denote by $J(f)$ the Jacobian of $f$. There is a first order partial non-linear equation, where $f$ is the unknown function, i.e. $F(J(f)(x),f(x),x)=g(x)$ (where one can assume that $F$ is "nice" and $g$ is some given "nice" function) on a domain $D$ in $\mathbb{R}^{n}$ such that $\partial D=C_{1} \cup C_{2}$. And the boundary conditions are: $f$ restricted to $C_{1}$ is zero and $J(f)$ restricted to $C_{2}$ is zero (as a matrix). Are there some references on such kind of equations ?

Does anyone have an idea, about some references or this is to be solved ???

pascal

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Yes, I do...... –  András Bátkai Apr 24 '12 at 21:52
What do you mean by mixed boundary conditions? –  timur Apr 24 '12 at 22:07
Your question is far to vague. Can you specify what you are interested in? –  András Bátkai Apr 25 '12 at 6:06
actually I am intrested in the following: Let $f(x) = (f_{1}(x), ..., f_{n}(x))$ such that $x \in \mathbb{R}^{n}$ be an unknown function and denote by $J(f)$ the Jacobian of $f$. There is a first order partial non-linear equation, where $f$ is the unknown function, i.e. $F(J(f)(x), f(x), x) = g(x)$ (where one can assume that $F$ is "nice" and $g$ is some given "nice" function) on a domain $D$ such that $\partial D = C_{1} \cup C_{2}$. And the boundary conditions are: $f|_{C_{1}} = 0$ and $J(f)|_{C_{2}} = 0$. Are there some references on such kind of equations ? –  pascal Apr 25 '12 at 14:13
I think it should be clear now, what I mean ? Or ? –  pascal Apr 26 '12 at 6:11