arXiv subjects
Publications and source records attributed to Michel Willem.
This work analyzes the validity of Tonelli's theorem, overcoming the traditional assumption of $\sigma$-finiteness. We prove that the existence and equality of iterated integrals for indicator functions is a necessary and sufficient condition to define a product measure that allow to extend Tonelli's Theorem to more general measure spaces, including $s$-finite measures. Finally, we characterize the semi-finiteness of such a product measure.
This paper presents a new general formulation of the Radon-Nikodym theorem in the setting of abstract measure theory. We introduce the notion of weak localizability for a measure and show that this property is both necessary and sufficient for the validity of a Radon-Nikodym-type representation under a natural compatibility relation between measures. The proof relies solely on elementary tools, such as Markov's inequality and the monotone convergence theorem. In addition to establishing the main result, we provide a constructive approach to envelope functions for families of non-negative measurable functions supported on sets of finite measure.
We prove several classification results for $p$-Laplacian problems on bounded and unbounded domains, and deal with qualitative properties of sign-changing solutions to $p$-Laplacian equations on $\mathbb R^N$ involving critical nonlinearities. Moreover, on radial domains we characterise the compactness of possibly sign-changing Palais-Smale sequences.
We consider the forced problem $-Δ_p u - V(x)|u|^{p-2} u = f(x)$, where $Δ_p$ is the $p$-Laplacian ($1<p<\infty$) in a domain $Ω\subset \mathbb{R}^N$, $V\ge 0$ and $Q_V (u) := \int_Ω|\nabla u|^p\, dx - \int_ΩV|u|^p\,dx$ satisfies the condition (A) stated at the beginning of the paper. We show that this problem has a solution for all $f$ in a suitable space of distributions. Then we apply this result to some classes of functions $V$ which in particular include the Hardy potential and the potential $V(x)=λ_{1,p}(Ω)$, where $λ_{1,p}(Ω)$ is the Poincaré constant on an infinite strip.