Research

Published articles

1.  Sufficient Conditions for the preservation of Path-Connectedness in an arbitrary metric space (with E. Milakis) Applied General Topology, (2025) https://doi.org/10.4995/agt.2025.22779 

Abstract: It is proven that if $ (X,d) $ is an arbitrary metric space and $ U $ is a path-connected subset of $ X $ with $M:=\{x_i:\ i\in\{1,2,\dots,k\}\}\subset int(U) $, then the property of path-connectedness is also preserved in the resulting set $ U\setminus M, $ provided that the boundary of each open ball of X is a non-empty and path-connected set. Moreover, under appropriate conditions we extend the above result in the case where the set $ M $ is infinite. As a consequence, these results provide path-connectedness for domains with holes.

2.  Sufficient Conditions for the preservation of Polygonal-Connectedness in an arbitrary normed space (with E. Milakis) Ricerche di Matematica, (2025) https://doi.org/10.1007/s11587-024-00893-2 

Abstract: In this article we prove that if $ (X,\norm{\cdot}_X) $ is a normed space and $ U $ is a polygonally-connected subset of $ X $ with $M:=\{S_i:\ i\in I\}\subset \mathcal{P}\left( U\right),  $ a non-empty arbitrary family of discrete, non-empty  subsets of $ U, $ then the property of polygonal-connectedness is also preserved in the resulting set $ U\setminus\left( \bigcup_{i\in I} S_i\right),$ under appropriate conditions.

3. Systems of Fully Nonlinear Degenerate Elliptic Obstacle problems with Dirichlet boundary conditions (with E. Milakis) Annali di Matematica Pura ed Applicata, (2023) https://doi.org/10.1007/s10231-023-01343-w 

Abstract: In this paper, we prove existence and uniqueness of viscosity solutions to the following system: For $ i\in\left\lbrace 1,2,\dots,m\right\rbrace $ 

$$\min\biggl\{ F\bigl( y,x,u_{i}(y,x),D u_{i}(y,x),D^2 u_{i}(y,x)\bigl),  u_{i}(y,x)-\max_{j\neq i}\bigl( u_{j}(y,x)-c_{ij}(y,x)\bigl)\biggl\}=0, \left(y,x \right)\in\Omega_{L}\\

u_{i}(0,x)=g_{i}(x), x\in\bar{\Omega},\ u_i(y,x)=f_i(y,x),  (y,x)\in(0,L)\times\partial{\Omega} $$

where $ \Omega\subset\R^n $ is a bounded domain, $ \Omega_{L}:=(0,L)\times\Omega$ and $ F:\left[ 0,L\right] \times\R^n\times\R\times\R^n\times\mathcal{S}^n\to\R $ is a general second order partial differential operator which covers even the fully nonlinear case. Moreover, $ F $ belongs to an appropriate subclass of degenerate elliptic operators. Regarding uniqueness, we establish a comparison principle for viscosity sub and supersolutions of the Dirichlet problem. This system appears among other in the theory of the so-called optimal switching problems on bounded domains

Unpublished work

4.      A note on the Uniform limit of a family of pseudometric spaces (with E. Milakis).

Abstract: In this article, we consider a uniformly convergent family of pseudometrics $ d_s:X\times X\to\R,$ where $ s\in S:=(0,s_0) $ with $ s_0\in(0,\infty] $, such that $ d_s\xrightarrow{\text{uf.}}\tilde{d} $ on $ X\times X $ and we provide appropriate sufficient conditions, so that various properties of the family of pseudometrics $ (d_s)_{ s\in S} $ carry over to the limit pseudometric space $ (X,\tilde{d}). $ In particular, we state appropriate conditions so that various notions, such as, convergence of a sequence, topological properties of a set $ A\subset X $ (limit, accumulation, interior, boundary and isolated points of $ A $), continuity of functions, density, separability, completeness, compactness, connectedness and path-connectedness which are taken for the family of pseudometrics $ (d_s)_{s\in S} $, retain their validity for the limit pseudometric $ \tilde{d} $ as well. 

5.      A note on the set-theoretic properties of Jones-flat sets in an arbitrary metric space.

Abstract: In this article, we provide suitable conditions in an arbitrary metric space $ (X,d),$  under which various set-theoretic concepts fulfil the Jones-flat condition. In particular, we set conditions in a way that the union of a monotone family of Jones-flat sets, fulfils the Jones-flat condition as well. Moreover, we study in a normed linear space the dilations of Jones-flat sets and in addition, we prove that the Jones-flatness condition is preserved under surjective linear isometries. Furthermore, we provide hausdorff measure type estimates for the image of connecting curves in Jones-flat sets, under Hölder maps. Finally, we study the extension issue of the Jones-flat condition. In specific, for an open set $ \Omega\subset X, $ if $ \Omega\setminus\{x_0\} $ is a Jones-flat set, where $ x_0\in\Omega, $ then by setting appropriate conditions, we prove that $ \Omega $ is Jones-flat set as well. The last result is extended in the case where $ \Omega\setminus\mathcal{F} $ is Jones-flat set and $ \mathcal{F} $ is a nonempty arbitrary family of distinct points in $ \Omega. $

6.      Existence of an extremum point in an arbitrary metric space for a uniform limit of functions.

Abstract: In this article, we construct appropriate sufficient conditions for the existence of extrema points for a limit of functions. In specific, we prove that in an arbitrary metric space $ (X,\rho) $, if $ (x_n)_{n\in\N}\subset X $ is a sequence of global maximum (minimum) points of a sequence of functions $ f_n:X\to\R $, such that $ x_n\xrightarrow{\rho}x_0 $ and $ f_n\xrightarrow{\text{uf}}f $ in $ X, $ with $ f $ continuous on $ X, $ then the limit function $ f $ receives a global maximum (minimum) value at $ x_0. $ In addition, the last result is extended in the case where $ (X,\mathcal{B}(X),\mu) $ is a complete Borel measure space. Furthermore, we provide sufficient conditions for ensuring the existence of an extremum point for the limit function $ f $ on a suitable set $ D_0\subset X, $ in the case that the restrictions $ f_n|_{D_n} $ of the functions $ f_n $, attain a global extremum point on a suitable subset $ D_n $ of $ X. $ Last, we study analogous results for the existence of local extremum point for the limit function $ f. $ 

7.      Some properties of Convergence in Hausdorff pseudometric spaces.

Abstract: In this paper, we study within the framework of a metric space $ (X,\rho) $, several properties of the convergence of a sequence of sets $ (D_n)_{n\in\N}\subset \mathcal{K}(X) $ to a set $ D_0\in\mathcal{K}(X) $ with respect to the \emph{Hausdorff} pseudometric $ d_H:\mathcal{K}(X)\times\mathcal{K}(X)\to\R $, where $ \mathcal{K}(X) $ is the collection of non-empty, bounded subsets of $ (X,\rho). $ In particular, we prove that properties such as \emph{density, separability, total boundedness, completeness, compactness, nowhere-density} can be transferred to the limit set $ D_0 $, in the case that the sequence $ (D_n)_{n\in\N} $ satisfies suitable sufficient conditions. 

8.      A note on the stability of the first Dirichlet eigenvalue in rough domains (with E. Milakis).

Abstract: In this paper we prove that if the domains $ \Omega $ and $ \Omega' $ in $ \R^d, $ are close enough with respect to the Hausdorff distance $ d_H\left(\Omega'^c,\Omega^c \right) $ and their boundaries satisfy some geometrical and topological conditions, then

$$\max\left\lbrace\lambda_1-T_1\left(  d_H\left(\Omega'^c,\Omega^c \right)\right)  \lambda_1',\lambda_1'-T_1\left(  d_H\left(\Omega'^c,\Omega^c \right)\right)  \lambda_1\right\rbrace \leq T_2( d_H\left(\Omega'^c,\Omega^c \right))\nonumber$$

where $ T_1,T_2 $ are appropriate functions and $ \lambda_1 $ (resp. $ \lambda_1' $) is the first Dirichlet eigenvalue of the following general elliptic operator in $ \Omega $ (resp. in $ \Omega' $)

$$L[u](x):=-\sum_{i,j=1}^d\frac{\partial}{\partial x_j}\left( k(x)\delta_{ij}\frac{\partial}{\partial x_i} u(x)\right) +c(x)u(x),\ \text{for all}\ x\in\Omega\ (\text{resp.}\ x\in\Omega')$$.

9.      Extendibility of uniform domains and Hausdorff estimates for uniform curves in metric spaces.

Abstract: In this article, we provide suitable conditions in an arbitrary metric space $ (X,d),$  under which various set$\backslash$function-theoretic concepts fulfil the uniform condition. In particular, we set conditions in a way that the union of a monotone family of uniform domains, fulfils the uniform condition as well. Moreover, we study in a normed linear space the dilations of uniform domains and in addition, we prove in the setting of metric spaces, that the uniform condition of a domain, is preserved under bi-Lipschitz maps. Next, we provide hausdorff measure type estimates for uniform curves, under Hölder maps. As a consequence of these estimates, we extract local hausdorff estimates for uniform curves, under the act of functions that belong to a Sobolev space $M\left(X,d,\mu \right) $ defined on a metric measure space $\left(X,d,\mu \right) $. Finally, we study the extendibility of the uniform condition. In specific, for a $d-$open set $ \Omega\subset X, $ if $ \Omega\setminus\{x_0\} $ is a uniform domain in $X$, where $ x_0\in\Omega, $ then by setting appropriate sequential type conditions, we prove that $ \Omega $ is uniform domain as well. The last result is extended in the case where $ \Omega\setminus\mathcal{F} $ is uniform domain in $X$, where $ \mathcal{F}\subset\Omega $ is a $d-$ closed set with $\mathcal{F}=iso_d\left(\mathcal{F} \right) $.

10.  A note on the length space property in an arbitrary metric space. 

Abstract: In this article, we focus on the preservation of length space property. In specific, if $ (X,d)$ is a metric vector space and $\mathcal{M}\subset U\subset X$, where $U$ is a length metric subspace of $X$, then under appropriate mild conditions, we prove that $U\setminus\mathcal{M}$ carries the length space property. Next, we prove various set$\backslash$function-theoretic concepts that preserve the length space property. In particular, in a normed linear space $\left( X,\norm{\cdot}_X\right) $, it is proven that the length space property on subsets of $X$, is preserved under the act of dilation. Also, in the setting of an abstract metric space, it is proven that the length space property is preserved under isometries. Furthermore, we derive some results related with the extendibility of the length space property. Finally, we construct various hausdorff measure type estimates for shortest paths in length metric spaces, under Hölder maps.  As a consequence of these estimates, we extract local hausdorff estimates for shortest paths in length metric spaces, under the act of functions that belong to a Sobolev space $M\left(X,d,\mu \right) $ defined on a metric measure space $\left(X,d,\mu \right) $.

11. A note on the preservation of C-quasiconvexity in an arbitrary metric vector space.

Abstract: In this article, we focus on the preservation of $C-$quasiconvexity property in the metric setting. In specific, if $ (X,d)$ is a metric vector space and $\mathcal{M}\subset U\subset X$, where $\mathcal{M}$ is a $d-$closed set and $U$ is a $C-$quasiconvex subset of $X$ with $C\geq 1$, then under appropriate mild sufficient conditions, we prove that $U\setminus\mathcal{M}$ is still a $\tilde{C}-$quasiconvex subset of $X$ for any $\tilde{C}>C$.

   Work in Progress

12. Spectral stability estimates for the Dirichlet elliptic operator in Reifenberg-flat domains (with E. Milakis).