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I just realised that i have never seen the space $W^{k,p}$, $p\neq 2$, used in showing existence/uniqueness to some PDE. Usually books/lectures build up theory about $W^{k,p}$ (like certain compact embeddings) spaces and then immediately show well-posedness to some PDE in the space $H^k$. Typically with Poisson's equation.

I am very curious about how to show existence to PDEs where their solutions live in $W^{k,p}$, $p \neq 2$. Obviously these spaces are not Hilbert so Lax-Milgram goes out of the window. Can someone give me a cool example or cite some book that goes through an existence proof?

Thanks

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Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, $p\in (1,\infty)$, $q\in (1,\infty)$, $\frac{1}{p}+\frac{1}{q}=1$, $f\in L^q(\Omega)$. Consider the problem $$\tag{1} \left\{ \begin{array}{rl} -\operatorname{div}(|\nabla u|^{p-2}\nabla u)=f &\mbox{ in $\Omega$} \\ u\in W_0^{1,p}(\Omega) &\mbox{} \end{array} \right. $$

The operator $ \Delta_pu=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is called $p$-Laplacian. We say that $u\in W_0^{1,p}(\Omega)$ is a solution (weak solution) of (1) if forall $\phi\in C_0^\infty(\Omega)$ $$\tag{2}\int_\Omega |\nabla u|^{p-2}\nabla u\nabla \phi=\int_\Omega f\phi$$

One way to solve the equation (2) is to consider the energy functional $F:W_0^{1,p}(\Omega)\rightarrow\mathbb{R}$ associated to it: $$F_p(u)=\frac{1}{p}\int_\Omega|\nabla u|^p -\int_\Omega fu$$

It is a pleasurable exercise to show that $F_p$ is a strictly convex functional and hence it must have an unique minimizer. Moreover you can show that if $u\in W_0^{1,p}$ is the minimizer, hence $$\langle F_p'(u),\phi\rangle=\int_\Omega |\nabla u|^{p-2}\nabla u\nabla \phi-\int_\Omega f\phi=0,\ \forall\ \phi\in W_0^{1,p}$$

To get a problem where the solution lies in $W^{k,p}$ with $k>1$ you need to ask more differentiability, for example, in the $p$-laplacian problem we need only that $k=1$.

Note 1: Observe that standard Dirichlet problem is included here and the same technique applies to solve this problem.

Note 2: If you ask more regularity of $f$, for example $f\in L^{r}$ with $r>\frac{N}{p}$, you can show that in fact $u\in C_{loc}^{1,\alpha}(\Omega)$ for some $\alpha\in (0,1)$.

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Thanks, very informative answer. Is there a way to do it without using calculus of variations (a topic which I am not very fond of)? –  pde_lover Jan 10 '13 at 22:18
    
I know that there is a banach space Lax-Milgram version, but i dont know if it is possible to solve this problem by using it. There are the notions of $S^+$ and pseudo-monotone operators and with it the Brezis theorem, but i think that in some way they are related with Calculus of Variations. –  Tomás Jan 10 '13 at 22:28
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