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An explicit Dirichlet series is obtained, which represents an analytic function of $s$ in the half-plane $\Re s>1/2$ except for having simple poles at points $s_j$ that correspond to exceptional eigenvalues $\lambda_j$ of the non-Euclidean…

We study the discretization of a Dirichlet form on the Koch snowflake domain and its boundary with the property that both the interior and the boundary can support positive energy. We compute eigenvalues and eigenfunctions, and demonstrate…

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. For any $\epsilon\in (0,1)$ we show that for any Dirichlet eigenvalue $\lambda_k(\Omega)>\Lambda(\epsilon,\Omega)$, it holds \begin{align*} k&\le…

In this work we investigate the Sobolev space $\mathrm{H}^{1}(\partial\Omega)$ on a strong Lipschitz boundary $\partial\Omega$, i.e., $\Omega$ is a strong Lipschitz domain. In most of the literature this space is defined via charts and…

For $N\geq 4$, we let $\Omega$ to be a smooth bounded domain of $\mathbb{R}^N$, $\Gamma$ a smooth closed submanifold of $\Omega$ of dimension $k$ with $1\leq k \leq N-2$ and $h$ a continuous function defined on $\Omega$. We denote by…

This paper is dedicated to the spectral optimization problem $$ \mathrm{min}\left\{\lambda_1^s(\Omega)+\cdots+\lambda_m^s(\Omega) + \Lambda \mathcal{L}_n(\Omega)\colon \Omega\subset D \mbox{ s-quasi-open}\right\} $$ where $\Lambda>0,…

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…

An inverse problem of finding an obstacle and the boundary condition on its surface from the fixed-energy scattering data is studied. A new method is developed for a proof of the uniqueness results. The method does not use the discreteness…

We consider the zeta function $\zeta_\Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $\Omega$ bounded by a smooth closed curve of perimeter $2\pi$. We prove that $\zeta_\Omega''(0)\ge…

We consider Dirichlet problems for linear elliptic equations of second order in divergence form on a bounded or exterior smooth domain $\Omega$ in $\mathbb{R}^n$, $n \ge 3$, with drifts $\mathbf{b}$ in the critical weak $L^n$-space…

We extend \emph{Schramm's cofat uniformization theorem} to cofat domains on upper Ahlfors 2-regular metric two-spheres $X$. Specifically, we show that if $\Omega \subset X$ is a cofat domain, then there exists a…

We study a parabolic Ventsell problem for a second order differential operator in divergence form and with interior and boundary drift terms on the snowflake domain. We prove that under standard conditions a related Cauchy problem possesses…

We consider a domain $\Omega\subseteq\mathbb{R\!}^{\,2}$ with branched fractal boundary $\Gamma^{\infty}$ and parameter $\tau\in[1/2,\tau^{\ast}]$ introduced by Achdou and Tchou \cite{ACH08}, for $\tau^{\ast}\simeq 0.593465$, which acts as…

Let $n\geq2$ and $ \Omega\subset \mathbb{R}^{n+1}$ be a Lipschitz wedge- like domain . We construct positive weak solutions of the problem $$\Delta u + u^p = 0 \quad\hbox{in}\, \Omega,$$ which vanish in a suitable trace sense on…

The simplest genuinely multidimensional monopolist's problem involves minimizing a linearly perturbed Dirichlet energy among nonnegative convex functions $u$ on an open domain $X \subset [0, \infty)^2$. The geometry of the region of strict…

Let $\Gamma$ be a rectifiable Jordan curve in the complex plane, $\Omega_i$ and $\Omega_e$ respectively the interior and exterior domains of $\Gamma$, and $p\geq 2$. Let $E$ be the vector space of functions defined on $\Gamma$ consisting of…

We develop a Helmholtz-like theorem for differential forms in Euclidean space $E_{n}$ using a uniqueness theorem similar to the one for vector fields. We then apply it to Riemannian manifolds, $R_{n}$, which, by virtue of the…

An important implication of Rademacher's Differentiation Theorem is that every Lipschitz curve $\Gamma$ infinitesimally looks like a line at almost all of its points in the sense that at $\mathcal{H}^1$-almost every point of $\Gamma$, the…

Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while…

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…