Related papers: Optimal boundary holes for the Sobolev trace const…
In this paper we study two optimal design problems associated to fractional Sobolev spaces $W^{s,p}(\Omega)$. Then we find a relationship between these two problems and finally we investigate the convergence when $s\uparrow 1$.
Sobolev trace inequalities on nonhomogeneous fractional Sobolev spaces are established.
Given a corkscrew domain with uniformly rectifiable boundary, we construct a surjective trace map onto the $L^p$ Hajlasz-Sobolev space on the boundary from the space of functions on the domain with $L^p$ norm involving the non-tangential…
In this paper we prove the existence of an optimal domain $\Omega_{opt}$ for the shape optimization problem $$\max\Big\{\lambda_q(\Omega)\ :\ \Omega\subset D,\ \lambda_p(\Omega)=1\Big\},$$ where $q<p$ and $D$ is a prescribed bounded subset…
In this paper we consider the scale invariant shape functional $${\mathcal{F}}_{p,q}(\Omega)=\frac{\lambda_p^{1/p}(\Omega)}{\lambda_q^{1/q}(\Omega)},$$ where $1\le q<p\le+\infty$ and $\lambda_p(\Omega)$ (respectively $\lambda_q(\Omega)$) is…
Let $n \geq 2$, let $\Omega \subset \mathbf{R}^n$ be a bounded domain with smooth boundary, and let $1 \leq p \leq 2$. We prove a reverse-Holder inequality for functions $u$ realizing the best constant in the Sobolev inequality, that is…
Our main objective in this work is to show how Sobolev orthogonal polynomials emerge as a useful tool within the framework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary…
We develop a sharp maximal regularity theory for the resolvent and evolution Stokes equations with no-slip boundary conditions, focusing on bounded domains of low regularity. Our framework covers the full scales of Besov and Sobolev spaces,…
We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\Omega \subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues…
We study a nonlinear generalization of a free boundary problem that arises in the context of thermal insulation. We consider two open sets $\Omega\subseteq A$, and we search for an optimal $A$ in order to minimize a non-linear energy…
Solutions to $p$-Laplace equations are not, in general, of class $C^2$. The study of Sobolev regularity of the second derivatives is, therefore, a crucial issue. An important contribution by Cianchi and Maz'ya shows that, if the source term…
For functions from the Sobolev space $H^s(\Omega)$, 1/2<s<3/2, definitions of non-unique generalised and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and…
We show that, when $sp>N$, the sharp Hardy constant $\mathfrak{h}_{s,p}$ of the punctured space $\mathbb R^N\setminus\{0\}$ in the Sobolev-Slobodecki\u{\i} space provides an optimal lower bound for the Hardy constant…
Several different problems make the study of the so called Lyapunov type inequalities of great interest, both in pure and applied mathematics. Although the original historical motivation was the study of the stability properties of the Hill…
We use form methods to define suitable realisations of the Laplacian on a domain $\Omega$ with Wentzell boundary conditions, i.e. such that $\partial_{\mathrm{n}}u + \beta u + \Delta u = 0$ holds in a suitable sense on the boundary of…
Let $\Omega$ be an open, possibly unbounded, set in Euclidean space $\R^m$ with boundary $\partial\Omega,$ let $A$ be a measurable subset of $\Omega$ with measure $|A|$, and let $\gamma \in (0,1)$. We investigate whether the solution…
We derive and implement a new way to find lower bounds on the smallest limiting trace-to-degree ratio of totally positive algebraic integers and improve the previously best known bound to 1.80203. Our method adds new constraints to Smyth's…
The paper is devoted to proving Allard-Michael-Simon-type $L^p$-Sobolev inequalities $(p>1)$ with explicit constants in the setting of Euclidean minimal submanifolds of arbitrary codimension. Our results require separate discussions for the…
We consider a bounded Lipschitz domain $\Omega\subseteq\mathbb{R}^3$ with sufficiently smooth boundary and prove piecewise Sobolev regularity of vector fields that have piecewise regular curl and divergence, but may be discontinuous across…
We describe an approach for finding upper bounds on an ODE dynamical system's maximal Lyapunov exponent among all trajectories in a specified set. A minimization problem is formulated whose infimum is equal to the maximal Lyapunov exponent,…