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Recently, much of the existing work in manifold learning has been done under the assumption that the data is sampled from a manifold without boundaries and singularities or that the functions of interest are evaluated away from such points.…
Asymptotic formulae for Green's kernels $G_\epsilon({\bf x}, {\bf y})$ of various boundary value problems for the Laplace operator are obtained in regularly perturbed domains and certain domains with small singular perturbations of the…
We investigate the class of continuous-state branching processes with interaction driven by a L\'evy-Khintchine type drift (CBDI). These $[0,\infty]$-valued processes capture both dynamics of branching and density-dependence, allowing for…
In this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the…
Recent applications of large network models to machine learning, and to neural network suggest a need for a systematic study of the general correspondence, (i) discrete vs (ii) continuous. Even if the starting point is (i), limit…
We provide a general framework to study invariant properties of various gradient-like and Laplace-like differential operators naturally associated to geometric structures on $\mathbb{R}^n$, which encompass Euclidean, Minkowski,…
We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvilinear polygons. It was known from previous works that the asymptotics of several first eigenvalues is essentially determined by the corner…
We study semilinear problems in general bounded open sets for non-local operators with exterior and boundary conditions. The operators are more general than the fractional Laplacian. We also give results in case of bounded $C^{1,1}$ open…
We analyse $C_0$-semigroups of contractive operators on real-valued $L^p$-spaces for $p \not= 2$ and on other classes of non-Hilbert spaces. We show that, under some regularity assumptions on the semigroup, the geometry of the unit ball of…
We consider five different abstract Cauchy problems where the operator is of fourth order. For each problem, using semigroups techniques and functional calculus, we study the invertibility and the spectral properties of the fourth order…
We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann or Robin boundary condition. We keep the presentation at a level accessible to scientists from…
Related to a semigroup of operators on a metric measure space, we define and study pseudodifferential operators (including the setting of Riemannian manifold, fractals, graphs ...). Boundedness on $L^p$ for pseudodifferential operators of…
We establish sharp pointwise estimates for the ground states of some singular fractional Schr\"odinger operators on relatively compact Euclidean subsets. The considered operators are of the type $(-\Delta)^{\alpha/2}|_\Om-c|x|^{-\alpha}$,…
Spectral boundary conditions for Laplace-type operators, of interest in string and brane theory, are partly Dirichlet, partly Neumann-type conditions, partitioned by a pseudodifferential projection. We give sufficient conditions for…
We prove an approximation result showing how operators of the type $-\Delta -\gamma \delta (x-\Gamma)$ in $L^2(\mathbb{R}^2)$, where $\Gamma$ is a graph, can be modeled in the strong resolvent sense by point-interaction Hamiltonians with an…
Given a bounded domain in the Euclidean space satisfying the uniform outer cone condition, we show that a uniformly elliptic operator of second order with continuous second order coefficients generates a holomorphic semigroup on the space…
For a finite not necessarily compact metric graph, one considers the differential expression $-\frac{d^2}{d x^2}$ on each edge. The boundary conditions at the vertices of the graph yielding quasi-m-accretive as well as m-accretive operators…
In this note we investigate the nonelliptic differential expression A=-div sgn grad on a rectangular domain in the plane. The seemingly simple problem to associate a selfadjoint operator with the differential expression A in an L^2 setting…
We discuss the properties of interacting electrons on a finite chain with open boundary conditions. We extend the Haldane Luttinger liquid description to these systems and study how the presence of the boundaries modifies various…
We apply the generalized method of separation of variables (GMSV) to solve boundary value problems for the Laplace operator in three-dimensional domains with disconnected spherical boundaries (i.e., an arbitrary configuration of…