ETD

• capacity
• Hardy inequality
• p-Laplacian
• sharp constants
• Sobolev embeddings

This thesis investigates sharp functional inequalities and optimal constants within the Calculus of Variations and Partial Differential Equations theory, building directly upon the foundational framework developed by V. G. Maz'ya.

Utilizing $p$-capacity as a refining tool to analyze the fine properties of Sobolev functions, we address the classical problem of characterizing continuous Sobolev embeddings
\[
\mathcal{D}^{1,p}_0(\Omega) \hookrightarrow L^q(\Omega), \qquad 1 \leq p , q \leq \infty,
\]
for arbitrary open sets $\Omega \subset \mathbb{R}^N$, where $\mathcal{D}^{1,p}_0(\Omega)$ denotes the \emph{homogeneous Sobolev space}. While classical results typically suppose some regularity of the domain, Maz'ya established that these functional embeddings are fundamentally equivalent to geometric isocapacitary inequalities, opening a pathway via measure-theoretic tools.

The central focus of this work is the \textit{capacitary inequality}:
\[
\int_{0}^{+\infty} \mathrm{cap}_p(\{ x \in \Omega : |u(x)| > t \}; \Omega) \, d(t^p) \leq \frac{p^p}{(p-1)^{p-1}} \int_{\Omega} |\nabla u|^p \, dx
\]
for all $u \in C^{\infty}_c(\Omega)$. We complete Maz’ya’s result by proving the optimality of the constant in the above inequality for all $p \geq 1$, in particular by handling the \textit{conformal} and \textit{superconformal} cases, i.e. $p=N$ and $p > N$. This fact requires a careful analysis of some sharp Hardy-type inequalities, some of which are new.

An application of the capacitary inequality leads to the characterization of the admissibility of a continuous embedding in terms of a capacitary analogue of the Cheeger constant, which we term \emph{Maz’ya’s constant}:
\[
\mathcal{M}_{p,q}(\Omega):=\inf_{\substack{E\subset\Omega\\ E\text{ compact}}}\frac{\mathrm{cap}_p(E;\Omega)}{|E|^{\frac{p}{q}}}.
\]

With this geometric constant in hand, one can prove that for $p \leq q$ the following holds:
\[
\mathcal{D}^{1,p}_0(\Omega) \hookrightarrow L^q(\Omega) \iff \mathcal{M}_{p,q}(\Omega) >0.
\]

In particular, denoting by $\lambda_{p,q}(\Omega)$ the sharp embedding constant for the relevant embedding, we review the proof of the following two-sided estimate:

\[
\frac{(p-1)^{p-1}}{p^p}\,\mathcal{M}_{p,q}(\Omega)\leq \lambda_{p,q}(\Omega)\leq \mathcal{M}_{p,q}(\Omega).
\]

In the final part of the thesis we start investigating the optimality of these estimates, beginning from some limiting cases. In particular, in the case $p>N$ and $q= + \infty$, we are lead to analyze in detail the $p$-capacitary problem in a general open set $\Omega$, which in turn leads us to give a characterization of the homogeneous Sobolev space in the superconformal case.