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In this paper, we investigate the existence of a "weak solutions" for a Neumann problems of $p(x)$-Laplacian-like operators, originated from a capillary phenomena, of the following form \begin{equation*}…
We consider the evolutionary $p$-Laplace equation in $\mathbb{R}^n$. For $p>n$, we construct a solution $u$ with a moving gradient singularity in the sense that $|\nabla u(x,t)|\to \infty$ for each $t$ as $x\to\xi(t)$, where…
Let $(M,g)$ be a compact, smooth Riemannian manifold and $\{u_h\}$ be a sequence of $L^2$-normalized Laplace eigenfunctions that has a localized defect measure $\mu$ in the sense that $ M \setminus \text{supp}(\pi_* \mu) \neq \emptyset$…
\[ \left\{ \begin{array} [c]{lll} -\left( \Delta_{p}+\Delta_{q(p)}\right) u=\lambda_{p}\left\vert u(x_{u})\right\vert ^{p-2}u(x_{u})\delta_{x_{u}} & \mathrm{in} & \Omega\\ u=0 & \mathrm{on} & \partial\Omega, \end{array} \right. \] where…
Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with $L^p$-boundary data for $p$ near $2$ (more precisely, in…
The ordering problem in quantum systems with position-dependent mass (PDM) is treated by inclusion of the classically fictitious similarity transformation into the kinetic term. This provides a generation of supersymmetry with the first…
We obtain upper bounds for the eigenvalues of the Schr\"odinger operator $L=\Delta_g+q$ depending on integral quantities of the potential $q$ and a conformal invariant called the min-conformal volume. Moreover, when the Schr\"odinger…
We consider the Euler system set on a bounded convex planar domain, endowed with impermeability boundary conditions. This system is a model for the barotropic mode of the Primitive Equations on a rectangular domain. We show the existence of…
Let $\Omega$ be an open set. We consider the supremal functional \[ \tag{1} \label{1} \ \ \ \ \ \ \mathrm{E}_\infty (u,\mathcal{O})\, :=\, \| \mathrm D u \|_{L^\infty( \mathcal{O} )}, \ \ \ \mathcal{O} \subseteq \Omega \text{ open}, \]…
The Laplace--P\'olya integral, defined by $J_n(r) = \frac1\pi\int_{-\infty}^\infty \mathrm{sinc}^n t \cos(rt) \mathrm{d} \, t$, appears in several areas of mathematics. We study this quantity by combinatorial methods; accordingly, our…
This undergraduate thesis is concerned with developing the tools of differential geometry and semiclassical analysis needed to understand the the quantum ergodicity theorem of Schnirelman (1974), Zelditch (1987), and Colin de Verdi\`ere…
Measuring the similarity between data points often requires domain knowledge, which can in parts be compensated by relying on unsupervised methods such as latent-variable models, where similarity/distance is estimated in a more compact…
In this paper, the author derives an $O(h^4)$-superconvergence for the piecewise linear Ritz-Galerkin finite element approximations for the second order elliptic equation $-\nabla \cdot(A\nabla u)= f$ equipped with Dirichlet boundary…
We establish the first extension results for divergence-free (or solenoidal) elements of $\mathrm{L}^{1}$-based function spaces. Here, the key point is to preserve the solenoidality constraint while simultaneously keeping the underlying…
If $u : \Omega\subset \mathbb{R}^d \to {\rm X}$ is a harmonic map valued in a metric space ${\rm X}$ and ${\sf E} : {\rm X} \to \mathbb{R}$ is a convex function, in the sense that it generates an ${\rm EVI}_0$-gradient flow, we prove that…
We study here the sub-Riemannian geometry on a manifold $M$ induced by a finite family $F$ of vector fields satisfying the H{\"o}rmander condition, as well as the differential operators obtained as polynomials in the elements of $F$. Such…
It is shown that for each finite number of Dirac measures supported at points $s_n$ in three-dimensional Euclidean space, with given amplitudes $a_n$, there exists a unique real-valued Lipschitz function $u$, vanishing at infinity, which…
Recently, several works have been carried out in attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In…
We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimising frame, also known as the Normal, Parallel or Bishop frame. Such systems have previously been…
We define a number of natural (from geometric and combinatorial points of view) deformation spaces of valuations on finite graphs, and study functions over these deformation spaces. These functions include both direct metric invariants…