Related papers: Subelliptic $p$-Laplacian spectral problem for H\"…
In this paper, we study qualitative properties of the fractional $p$-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional $p-$Laplacian equation with Dirichlet condition. Moreover, an…
We discuss the behavior of $(\lambda_{1. p}(M))^{1/p}$ with respect to the Gromov-Hausdorff topology and the variable $p$, where $\lambda_{1, p}(M)$ is the first positive eigenvalue of the $p$-Laplacian on a compact Riemannian manifold $M$.…
We present some new problems in spectral optimization. The first one consists in determining the best domain for the Dirichlet energy (or for the first eigenvalue) of the {\it metric Laplacian}, and we consider in particular Riemannian or…
We study the optimal sets $\Omega^\ast\subset\mathbb{R}^d$ for spectral functionals $F\big(\lambda_1(\Omega),\dots,\lambda_p(\Omega)\big)$, which are bi-Lipschitz with respect to each of the eigenvalues…
We prove a Payne-Weinberger type inequality for the $p$-Laplacian Neumann eigenvalues ($p\ge 2$). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constants in Poincar\'e…
We consider the domain dependence of the best constant in the subcritical fractional Sobolev constant, $$ \lambda_{s,p}(\Omega):=\inf \left\{ [u]_{H^s(\mathbb{R}^N)}^2,\,\, u\in C^\infty_c(\Omega),\,\, \|u\|_{L^p(\Omega)}=1 \right\}, $$…
We study the eigenvalue problem for the $g-$Laplacian operator in fractional order Orlicz-Sobolev spaces, where $g=G'$ and neither $G$ nor its conjugated function satisfy the $\Delta_2$ condition. Our main result is the existence of a…
In this paper, we introduce isocapacitary constants for the $p$-Laplacian on graphs and apply them to derive estimates for the first eigenvalues of the Dirichlet $p$-Laplacian, the Neumann $p$-Laplacian, and the $p$-Steklov problem.
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 prove uniform resolvent estimates in weighted $L^2$-spaces for the sublaplacian $\mathcal{L}$ on the Heisenberg group $\mathbb{H}^d$. The proof are based on multiplier methods, and strongly rely on the use of horizontal multipliers and…
In this paper we study the obstacle problems for the Navier (spectral) fractional Laplacian $\left(-\Delta_\Omega\right)^{\!s}$ of order $s\in(0,1)$, in a bounded domain $\Omega\subset\mathbb R^n$.
An integral inequality is derived for compact submanifolds (with or without boundary) in the unit sphere. This result leads to a characterization of spheres.
New isoperimetric inequalities for lower order eigenvalues of the Laplacian on closed hypersurfaces, of the biharmonic Steklov problems and of the Wentzell-Laplace on bounded domains in a Euclidean space are proven. Some open questions for…
In this paper we discuss spectral properties of operators associated with the least-squares finite element approximation of elliptic partial differential equations. The convergence of the discrete eigenvalues and eigenfunctions towards the…
This article considers the eigenvalue problem for the Sturm-Liouville problem including $p$-Laplacian \begin{align*} \begin{cases} \left(\vert u'\vert^{p-2}u'\right)'+\left(\lambda+r(x)\right)\vert u\vert ^{p-2}u=0,\,\, x\in (0,\pi_{p}),\\…
We revisit the classical theory of linear second-order uniformly elliptic equations in divergence form whose solutions have H\"older continuous gradients, and prove versions of the generalized maximum principle, the $C^{1,\alpha}$-estimate,…
In this paper we study the Dirichlet eigenvalue problem $$ -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \quad \text{ in } \Omega,\quad u=0 \quad\text{ in } \Omega^c=\mathbb{R}^N\setminus\Omega. $$ Here $\Delta_p u$ is the standard local…
We investigate the asymptotic spectral distribution of the twisted Laplacian associated with a real harmonic 1-form on a compact hyperbolic surface. In particular, we establish a sublinear lower bound on the number of eigenvalues in a…
Elliptic partial differential equations arise in many fields of science and engineering such as steady state distribution of heat, fluid dynamics, structural/mechanical engineering, aerospace engineering and seismology etc. In three…
In this work, given $p\in (1,\infty)$, we prove the existence and simplicity of the first eigenvalue $\lambda_p$ and its corresponding eigenvector $(u_p,v_p)$, for the following local/nonlocal PDE system \begin{equation}\label{Eq0} \left\{…