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Edit: This first paragraph is wrong: I think (a) is fine, because $1 \in L^p$ for $p \in [0,\infty]$ and therefore $f \in L^\infty$ implies $f \in L^p$. But I think, there is an issue with (b) as it …

My answer is for $p \in (2,4)$, the other case should follow similar. Let $t \in (0,1)$ be given, such that $$\frac1p = \frac{1-t}{2}+\frac{t}{4}.$$ I will go to use the following consequence of Höld …

The minimizers cannot lie on the boundary. In fact, denote by $E_i \subset E$ the set of all points which are transported to $x_i^*$. Then, $x_i^*$ has to be the center of mass of $E_i$ and, thus, can …

You can manipulate the left-hand side to \begin{equation*} \lVert (Dv)^2 \rVert_{H^{3k-2}} = \lVert D v \rVert_{W^{3k-2,4}}^2 \le \lVert v \rVert_{W^{3k-1,4}}^2. \end{equation*} (Note that what w …

You can find both results in Theorems 2.1 and 2.4 in Lucio Boccardo, Thierry Gallouët, Luigi Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, …

This works (only?) for $p = 2$. Let us denote the solution of the PDE on $\Omega$ by $v$. Then, the variational formulations of the PDEs are $$\int_D \nabla u \cdot \nabla z - fz \,\mathrm{d}x = 0 \q …