Related papers: Optimizing the first Dirichlet eigenvalue of the L…

For $\Omega \subset \mathbb{R}^n$, a convex and bounded domain, we study the spectrum of $-\Delta_\Omega$ the Dirichlet Laplacian on $\Omega$. For $\Lambda\geq0$ and $\gamma \geq 0$ let $\Omega_{\Lambda, \gamma}(\mathcal{A})$ denote any…

We prove that there exists a bounded convex domain $\Omega \subset \mathbf{R}^3$ of fixed volume that minimizes the first positive curl eigenvalue among all other bounded convex domains of the same volume. We show that this optimal domain…

Let $\Omega\subset\mathbb{R}^N$, $N\geq 2$, be an open bounded connected set. We consider the fractional weighted eigenvalue problem $(-\Delta)^s u =\lambda \rho u$ in $\Omega$ with homogeneous Dirichlet boundary condition, where…

In this paper, we study the first eigenvalue of the Laplacian on doubly connected domains when Robin and Dirichlet conditions are imposed on the outer and the inner part of the boundary, respectively. We provide that the spherical shell…

This paper aims to show that, in the limit of strong magnetic fields, the optimal domains for eigenvalues of magnetic Laplacians tend to exhibit symmetry. We establish several asymptotic bounds on magnetic eigenvalues to support this…

We consider a shape optimization problem related to the persistence threshold for a biological species, the unknown shape corresponding to the zone of the habitat which is favorable to the population. Analytically, this translates in the…

We prove existence results for optimization problems for the $k$th Laplace eigenvalue on closed Riemannian manifolds of dimension $m \geq 3$, depending on the choice of normalization. One such normalization leads to eigenvalue optimization…

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 consider the problem of minimising or maximising the quantity $\lambda(\O)T^q(\O)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $\lambda(\O)$ denotes the first eigenvalue of the Dirichlet Laplacian on…

We settle the issue of well-posedness for the Dirichlet problem for a higher order elliptic system ${\mathcal L}(x,D_x)$ with complex-valued, bounded, measurable coefficients in a Lipschitz domain $\Omega$, with boundary data in Besov…

A generic model for the shape optimization problems we consider in this paper is the optimization of the Dirichlet eigenvalues of the Laplace operator with a volume constraint. We deal with an obstacle placement problem which can be…

In this paper, by extending the notions of harmonic transplantation and harmonic radius in the Heisenberg group, we give an upper bound for the first eigenvalue for the following Dirichlet problem: $$(P_{\Omega}) \left\{…

This is a continuation of the paper 'Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes' by S. Chanillo, D. Grieser, M. Imai, K. Kurata, and I. Ohnishi. Again, we consider the following…

In this paper we study the maximum principle, the existence of eigenvalue and the existence of solution for the Dirichlet problem for operators which are fully-nonlinear, elliptic but presenting some singularity or degeneracy which are…

Let $\Omega\subset\mathbb{R}^N$, $N\geq 1$, be an open bounded connected set. We consider the indefinite weighted eigenvalue problem $-\Delta u =\lambda m u$ in $\Omega$ with $\lambda \in \mathbb{R}$, $m\in L^\infty(\Omega)$ and with…

We study the minimizers of \begin{equation} \lambda_k^s(A) + |A| \end{equation} where $\lambda^s_k(A)$ is the $k$-th Dirichlet eigenvalue of the fractional Laplacian on $A$. Unlike in the case of the Laplacian, the free boundary of…

Comparing Neumann and Dirichlet eigenvalues of the Laplacian on a bounded domain $\Omega\subseteq\Rbb^n$ is a topic that goes back at least to the work of P\'olya \cite{polya}. We study the effect of the isoperimetric ratio of $\Omega$ on…

This paper addresses the geometric optimization problem of the first Robin eigenvalue in exterior domains, specifically the lowest point of the spectrum of the Laplace operator under Robin boundary conditions in the complement of a bounded…

Let $\Omega$ be a bounded Lipshcitz domain in $\mathbb{R}^n$ and we study boundary behaviors of solutions to the Laplacian eigenvalue equation with constant Neumann data. \begin{align} \label{cequation0} \begin{cases} -\Delta u=cu\quad…

A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\geq2$, there exists a constant $c>0$, depending only on $n$,…