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We study the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibred boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibred boundary (aka $\phi$-) pseudodifferential operator when…

Given a $C^k$-smooth closed embedded manifold $\mathcal N\subset{\mathbb R}^m$, with $k\ge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $\partial S\neq\emptyset$, we consider $\frac 12$-harmonic maps $u\in…

We study a class of partial differential equations (PDEs) in the family of the so-called Euler-Poincar\'e differential systems, with the aim of developing a foundation for numerical algorithms of their solutions. This requires particular…

We consider Schr\"{o}dinger equations with linearly energy-depending potentials which are compactly supported on the half-line. We first provide estimates of the number of eigenvalues and resonances for such complex-valued potentials under…

The biharmonic equation with Dirichlet and Neumann boundary conditions discretized using the mixed finite element method and piecewise linear (with the possible exception of boundary triangles) finite elements on triangular elements has…

We consider the Calder\'on-Zygmund kernels $K_ {\alpha,n}(x)=(x_i^{2n-1}/|x|^{2n-1+\alpha})_{i=1}^d$ in $\mathbb{R}^n$ for $0<\alpha\leq 1$ and $n\in\mathbb{N}$. We show that, on the plane, for $0<\alpha<1$, the capacity associated to the…

In this paper we establish optimal solvability results, that is, maximal regularity theorems, for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless…

We propose, analyze, and test new robust iterative solvers for systems of linear algebraic equations arising from the space-time finite element discretization of reduced optimality systems defining the approximate solution of hyperbolic…

This article presents a mathematical framework for solving Maxwell's equations in cylindrical and spherical geometries with continuous angular indices. We extend beyond standard discrete harmonic decomposition to a continuous spectral…

For a closed subset $K$ of a compact metric space $A$ possessing an $\alpha$-regular measure $\mu$ with $\mu(K)>0$, we prove that whenever $s>\alpha$, any sequence of weighted minimal Riesz $s$-energy configurations…

This paper provides a rigorous analysis of boundary element methods for the magnetic field integral equation on Lipschitz polyhedra. The magnetic field integral equation is widely used in practical applications to model electromagnetic…

This second part of the series treats spin $\pm2$ components (or extreme components), that satisfy the Teukolsky master equation, of the linearized gravity in the exterior of a slowly rotating Kerr black hole. For each of these two…

We derive Adams inequalities for potentials on general measure spaces, extending and improving previous results obtained by the authors. The integral operators involved, which we call "Riesz subcritical", have kernels whose decreasing…

This paper is concerned with existence and multiplicity results for the semilinear subelliptic equation with free perturbation term. By using the degenerate Rellich-Kondrachov compact embedding theorem, precise lower bound estimates of…

In this paper we initiate the study of maps minimising the energy $$ \int_{D} (|\nabla \u|^2+2|\u|)\ dx. $$ which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations $$ \Delta…

The theory of string-like continuous curves and discrete chains have numerous important physical applications. Here we develop a general geometrical approach, to systematically derive Hamiltonian energy functions for these objects. In the…

We use discrete analogs of Riemann-Hilbert problem's methods to derive the discrete Bessel kernel which describes the poissonized Plancherel measures for symmetric groups. To do this we define discrete analogs of a Riemann-Hilbert problem…

We obtain two-sided heat kernel estimates for Riemannian manifolds with ends with mixed boundary condition, provided that the heat kernels for the ends are well understood. These results extend previous results of Grigor'yan and…

In this note we examine the a priori and a posteriori analysis of discontinuous Galerkin finite element discretisations of semilinear elliptic PDEs with polynomial nonlinearity. We show that optimal a priori error bounds in the energy norm…

We consider a distributed optimal control problem subject to a parabolic evolution equation as constraint. The control will be considered in the energy norm of the anisotropic Sobolev space $[H_{0;,0}^{1,1/2}(Q)]^\ast$, such that the state…