A question on gradients and PDE

Question

Let $\Omega$ be a smooth bounded domain in Euclidean space. Let $u \in C_c^1(\Omega)$. The subscript indicates compact support. Let $1 < p < \infty$.

Can a $v \in C_c^1(\Omega)$ (or in its closure in the Sobolev space $W^{1,q}(\Omega)$, where $1/p + 1/q = 1$) be found such that $\nabla u \cdot \nabla v = |\nabla u|^p$ (a.e.) in $\Omega$?

Or merely $\int_\Omega \nabla u \cdot \nabla v = \int_\Omega |\nabla u|^p$?

Answer 1

Paraphrasing V. Barbu, Nonlinear differential equations of monotone types in Banach spaces, 2010, Springer, p.4:

Let $1 < p < \infty$ and $\Omega$ a bounded and open subset of $\mathbb{R}^n$. Write $X := W_0^{1,p}(\Omega)$. For $u \in X$ define $J(u) \in X'$ by $$J(u) = -\|u\|_X^{2-p} \sum_{i=1}^n \partial_i (|\partial_i u|^{p-2} \partial_i u).$$ Then $J : X \to X'$ is the duality mapping (it is indeed single-value by uniform convexity of $X$), for which $$\langle J(u), u \rangle = \|u\|_X^2 = \|J(u)\|_{X'}^2.$$ This (almost) answers the second part of the question.

As an aside, the duality mapping is, in general, a set-valued mapping $J : X \to 2^{X'}$ on a Banach space defined by $$J(x) = \{ f \in X' : f(x) = \|x\|_X^2 = \|f\|_{X'}^2 \},$$ and this is always non-empty by the Hahn-Banach theorem (analytic version).

End of paraphrase.

Let $u \in C_c^1(\Omega)$ be nonzero. Then $J(u)$ is a bounded linear functional on $H_0^1(\Omega)$. Let $v \in H_0^1(\Omega)$ solve $\int_\Omega \nabla v \cdot \nabla \varphi = \langle J(u), \varphi \rangle$ for all $\varphi \in C_c^1(\Omega)$. Then $\|u\|_X^2 = \langle J(u), u \rangle = \int_\Omega \nabla v \cdot \nabla u$.

Answer 2

Given a function $u\in C_c^1(\Omega)$, and a number $p\in (1,\infty)$, you want to find some partial global classical solution $v\in C_c^1(\Omega)$ of a linear inhomogeneous first order PDE with continuous variable coefficients $$\nabla\,u\cdot\nabla\,v=|\nabla\,u|^{p}\quad {\rm in}\;\,\Omega\subset\mathbb{R}^n,\;\;n\geqslant 2. $$ You want too much!  This is a classical two-and-a-half-century-old problem, generally known to be solved only locally. Whether or not its global solution exist depends completely on the given function $u$.  A simple, seemingly obvious partial global "solution" $\nabla\,v=\nabla\,u|\nabla\,u|^{p-2}\;$ is not a solution at all, excluding the case of spherical symmetry $u(x)=\varphi(|x|)$ and some other trivial cases when the vector field $\,\nabla\,u|\nabla\,u|^{p-2}\,$ proves to be potential. So all you need do is find characteristic curves for the ODE system $$\frac{dx_1}{a_1(x)}=\dots=\frac{dx_n}{a_n(x)}=\frac{dv}{b(x)}\,,\quad a_j(x)\overset{def}{=}\frac{\partial u(x)}{\partial x_j}\,,\,j=1,\dots,n,\,\;\,b(x)\overset{def}{=}|\nabla\,u(x)|^{p-2}. $$ But what you will find will be a solution corresponding to a certain given $u$.

As to your second question, follow the abvice of Tomás, i.e. take $v=\mu u$ choosing the number $$\mu\overset{def}{=}\,\frac{\int_{\Omega}|\nabla\,u(x)|^{p}dx} {\int_{\Omega}|\nabla\,u(x)|^{2}dx}\,. $$