# Optimization Methods in Banach Spaces

does anyone know if there's a theory for the following problem: Optimize the task

\begin{align*} T_\phi(\tilde{u})&=\inf\limits_u T_\phi(u)\\ Au&=b\\ u&\in L^p(\Omega),\,\Omega\subset \mathbb{R}^n \end{align*}

for (nonlinear map) $$A:L^p(\Omega)\to Z$$, with $$Z$$ arbitrary local convex topological vector space and $$T_\phi:L^p(\Omega)\to \mathbb{R}\cup\{\pm\infty\}$$ defined by

$$\int\limits_{\Omega}\phi(u(x))\,dx$$

a weakly lower semiconinous convex function, with weakly compact level set. The map $$\phi$$ is also lower semiconinous convex.

Or exist there a theorem, that such a problem possess a optimal solution?

• My idee was to use the theory of monoton operators, but the problem is that the image of $A$ is not $(L^p)^*$. – FuncAna09 Feb 11 at 11:45
• Are there any more conditions on $A$? Without any additional conditions, you don't necessarily have minimizers. – MaoWao Feb 11 at 23:25
• The question is what conditions must be placed on a nonlinear A for the task to have a non-trivial solution. – FuncAna09 Feb 12 at 7:44
• If I replaced $Z$ with $(L^p)^*$ and suppose $A$ is monotonous, hemistic and coercive, then there should be a solution to the problem. This would follow from the theorem of Browder and Minty. – FuncAna09 Feb 12 at 7:50