arXiv subjects

Lorenzo Carletti

Publications and source records attributed to Lorenzo Carletti.

5 recordsLinked to original sources

arXiv2026

A priori bounds for energy-bounded solutions of critical polyharmonic equations

We investigate critical polyharmonic equations of the type: $$ Lu = |u|^{2^\sharp-2} u \quad \text{ in } Ω$$ with Dirichlet boundary conditions, in a smooth bounded domain $Ω$ of $\mathbb{R}^n$. Here $L$ is an elliptic differential operator of integer order $2 \le 2k < n$ whose leading order term is $(-Δ)^k$ and $2^\sharp = \frac{2n}{n-2k}$ is the critical Sobolev exponent. Our main result establishes, in large dimensions, uniform \emph{a priori} bounds in $C^{2k}(\overlineΩ)$ for bounded-energy solutions of this problem, that only depend on an upper bound on the energy. We prove this under a coercivity assumption of sorts on the lower-order terms of $L$. Our results are sharp, at least when $k=1$. Our approach uses asymptotic analysis techniques and in the course of the proof we obtain in particular a new global pointwise description of bounded-energy blowing-up solutions for this problem, which is of independent interest.

math.AP↗

arXiv2025

Optimal Sobolev inequalities of high order with $L^2$-remainder

We investigate the validity of the optimal higher-order Sobolev inequality $H_k^2(M^n)\hookrightarrow L^{\frac{2n}{n-2k}}(M^n)$ on a closed Riemannian manifold when the remainder term is the $L^2-$norm. Unlike the case $k=1$, the optimal inequality does not hold in general for $k>1$. We prove conditions for the validity and non-validity that depend on the geometry of the manifold. Our conditions are sharp when $k=2$ and in small dimensions.

math.AP↗

arXiv2025

Attaining the optimal constant for higher-order Sobolev inequalities on manifolds via asymptotic analysis

Let $(M,g)$ be a closed Riemannian manifold of dimension $n$, and $k\geq 1$ an integer such that $n>2k$. We show that there exists $B_0>0$ such that for all $u \in H^{k}(M)$, \[\|u\|_{L^{2^\sharp}(M)}^2 \leq K_0^2 \int_M |Δ_g^{k/2} u|^2 \,dv_g + B_0 \|u\|_{H^{k-1}(M)}^2,\] where $2^\sharp = \frac{2n}{n-2k}$ and $Δ_g = -\operatorname{div}_g(\nabla\cdot)$. Here $K_0$ is the optimal constant for the Euclidean Sobolev inequality $\big(\int_{\mathbb{R}^n} |u|^{2^\sharp}\big)^{2/2^\sharp} \leq K_0^2 \int_{\mathbb{R}^n} |\nabla^k u|^2$ for all $u \in C_c^\infty(\mathbb{R}^n)$. This result is proved as a consequence of the pointwise blow-up analysis for a sequence of positive solutions $(u_α)_α$ to polyharmonic critical non-linear equations of the form $(Δ_g + α)^k u = u^{2^\sharp-1}$ in $M$. We obtain a pointwise description of $u_α$, with explicit dependence in $α$ as $α\to \infty$.

math.AP↗

arXiv2024

The Green's function of polyharmonic operators with diverging coefficients: Construction and sharp asymptotics

We show existence, uniqueness and positivity for the Green's function of the operator $(Δ_g + α)^k$ in a closed Riemannian manifold $(M,g)$, of dimension $n>2k$, $k\in \mathbb{N}$, $k\geq 1$, with Laplace-Beltrami operator $Δ_g = -\operatorname{div}_g(\nabla \cdot)$, and where $α>0$. We are interested in the case where $α$ is large : We prove pointwise estimates with explicit dependence on $α$ for the Green's function and its derivatives. We highlight a region of exponential decay for the Green's function away from the diagonal, for large $α$.

math.AP↗

arXiv2023

The Importance of Worst-Case Memory Contention Analysis for Heterogeneous SoCs

Memory interference may heavily inflate task execution times in Heterogeneous Systems-on-Chips (HeSoCs). Knowing worst-case interference is consequently fundamental for supporting the correct execution of time-sensitive applications. In most of the literature, worst-case interference is assumed to be generated by, and therefore is estimated through read-intensive synthetic workloads with no caching. Yet these workloads do not always generate worst-case interference. This is the consequence of the general results reported in this work. By testing on multiple architectures, we determined that the highest interference generation traffic pattern is actually hardware dependant, and that making assumptions could lead to a severe underestimation of the worst-case (in our case, of more than 9x).

cs.PF↗