Related papers: Perturbing the principal Dirichlet eigenfunction

Motivated by Euclidean boxes, we consider "thin" annular domains of the form $U=(a,b)\times U_0\subseteq \mathbb{R}^n$ in polar coordinates, where the spherical base $U_0\subseteq \mathbb{S}^{n-1}$ is an inner uniform domain. We show that,…

We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set $\Omega\subset \mathbb{R}^d$, in dimension $d\geq 2$. More precisely, consider the simple random walk on $\mathbb{Z}^d$ killed…

We study a class of Riemannian manifolds which are equipped with a singular metric. In particular we study a domain perturbation problem for the Dirichlet eigenvalues which depends on the best constant in the Hardy Inequality. However, we…

This paper is concerned with the P1 finite element approximation of the eigenvalue problem of second-order elliptic operators subject to the Dirichlet boundary condition. The focus is on the preservation of basic properties of the principal…

The spectral theory of the p-Laplacian is well developed for classical Dirichlet and Neumann boundary conditions, but the transitional Robin regime on exterior domains remains largely unexplored. This paper studies a weighted p-Laplacian…

This paper is concerned with eigenvalue problems for non-symmetric elliptic operators with large drifts in bounded domains under Dirichlet boundary conditions. We consider the minimal principal eigenvalue and the related principal…

We study the efficiency of the first Dirichlet eigenfunction $u$ on bounded convex domains $\Omega \subset \mathbb{R}^N$, defined as the ratio between the mean value of $u$ on $\Omega$ and its maximum value. By exploiting improved…

In this paper, we study the scale-invariant quantity \[\mathcal{G}(\Omega)=\frac{\|\partial_n u_1\|_{L^\infty(\partial\Omega)}}{\lambda_1},\]where $u_1$ is the first $L^2$-normalized Dirichlet Laplace eigenfunction of a Euclidean domain…

We study the Dirichlet problem for the non-local diffusion equation $u_t=\int\{u(x+z,t)-u(x,t)\}\dmu(z)$, where $\mu$ is a $L^1$ function and $``u=\phi$ on $\partial\Omega\times(0,\infty)$'' has to be understood in a non-classical sense. We…

We study a Dirichlet spectral problem for a second-order elliptic operator with locally periodic coefficients in a thin domain. The boundary of the domain is assumed to be locally periodic. When the thickness of the domain $\varepsilon$…

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain $\Omega $ in ${\mathbb{R}}^N$. We consider deformations $\phi (\Omega)$ of $\Omega $ obtained by means of a locally Lipschitz…

Let $D$ be a bounded $C^2$-domain. Consider the following Dirichlet initial-boundary problem of nonlocal operators with a drift: $$ \partial_t u={\mathscr L}^{(\alpha)}_\kappa u+b\cdot \nabla u+f\ \mathrm{in}\ \mathbb R_+\times D,\ \…

We study the behaviour of the first eigenfunction of the Dirichlet Laplacian on a planar convex domain near its maximum. We show that the eccentricity and orientation of the superlevel sets of the eigenfunction stabilise as they approach…

In this work we consider the homogeneous Neumann eigenvalue problem for the Laplacian on a bounded Lipschitz domain and a singular perturbation of it, which consists in prescribing zero Dirichlet boundary conditions on a small subset of the…

We consider the Dirichlet problem u_t &= \Delta u + f(x, u, \nabla u)+ h(x, t),& \qquad &(x, t) \in \Omega \times (0, \infty), u &= 0, & \qquad &(x, t) \in \partial\Omega \times (0, \infty), on a bounded domain $\Omega \subset…

We investigate existence and uniqueness of maximal plurisubharmonic functions on bounded domains with boundary data that are not assumed to be continuous or bounded. The result is applied to approximate (possibly unbounded from above)…

It is widely known that the spectrum of the Dirichlet Laplacian is stable under small perturbations of a domain, while in the case of the Neumann or mixed boundary conditions the spectrum may abruptly change. In this work we discuss an…

The question of unique continuation of harmonic functions in a domain $\Omega$ $\subset$ R d with boundary $\partial$$\Omega$, satisfying Dirichlet boundary conditions and with normal derivatives vanishing on a subset $\omega$ of the…

This paper provides sharp Dirichlet heat kernel estimates in inner uniform domains, including bounded inner uniform domains, in the context of certain (possibly non-symmetric) bilinear forms resembling Dirichlet forms. For instance, the…

We deepen the study of Dirichlet eigenvalues in bounded domains where a thin tube is attached to the boundary. As its section shrinks to a point, the problem is spectrally stable and we quantitatively investigate the rate of convergence of…