We explicate the difference between the function theory of one complex variable and the function theory of several complex variables. In particular, we use the inhomogeneous Cauchy-Riemann equations to explain why there is a Hartogs extension phenomenon in several complex variables but not in one complex variable.
(Wydawnictwa AGH, 2026) Beirão da Veiga, Hugo; Yang, Jiaqi
Recently, Qi S. Zhang provided examples of solutions to the Navier-Stokes equations which, under suitable hypotheses, blow-up in finite time. He considers axially symmetric solutions in a cylinder $D$ under appropriate boundary conditions and under the effect of supercritical external forces $f$. In its main result Zhang exhibits, for each $q\lt\infty$, a blow-up solution with suitable $f\in L^q(0,T;L^1(D))$. Following Zhang, we construct blow-up solutions with forcing terms in the space $L^q(0,T;L^p(D))$, for suitable pairs $(q,p)$. In particular our results contain Zhang's result and provide blow-up examples along the supercritical curve $\frac{2}{q}+\frac{3}{p}=\frac{7}{2}, \quad 1\leq p\lt 2,$ thereby approaching the classical endpoint $f\in L^1(0,T;L^2(D))$. A particularly significant case is the existence of external forces $f \in L^q(0,T;L^p(D))$, for every $p\lt 2$ and some well determined $q(p)\gt 1$, for which singularities occur in a finite time. The significant case $f \in L^1(0,T;L^2(D))$, which corresponds to the classical definition of weak solution, remains open. A particularly significant feature of our approach is that external forces, and solutions, are smooth in $[0,T)$. Blow-up occurs only as $t\rightarrow T$.
This paper investigates the oscillatory behavior of solutions to a class of second-order nonlinear neutral delay differential equations with both positive and negative terms of the form
$\left(a(\theta) z^{\prime}(\theta)\right)^{\prime} - p(\theta)x(\theta) + q(\theta)x^{\alpha}(\sigma(\theta)) = 0, \quad \theta \geq \theta_{0},$
where $z(\theta) = x(\theta) + b(\theta)x(\tau(\theta))$. To facilitate the analysis, the equation is transformed, via a positive solution of an auxiliary second-order ordinary differential equation, into a binomial form. By employing the comparison and integral averaging techniques together with the arithmetic-geometric mean inequality, we establish new sufficient conditions for the oscillation of all solutions. The results obtained extend and improve several existing criteria in the literature. Finally, illustrative examples are presented to demonstrate the effectiveness, novelty, and applicability of the proposed oscillation criteria.
The topological sensitivity analysis method has been recognized as a promising, fast, and accurate approach for solving topology optimization and inverse problems. It is based on developing an asymptotic expansion of a design functional with respect to the creation of a small hole inside the computational domain. In this work, we extend this method to the narrow escape problem. The biological process is governed by a parabolic diffusion equation. We derive a sensitivity analysis for the parabolic problem solution with respect to the creation of a small absorbing boundary subset. We develop a rigorous mathematical framework that is valid in two- and three-dimensional space. It provides an asymptotic formula that describes the behavior of the perturbed solution with respect to the location and size of an arbitrary perturbed boundary subset. The Sobolev capacity notion has been employed to measure the smallness of the boundary subset and to describe the asymptotic behavior with respect to the perturbation size. The performed mathematical analysis is general and can be adapted for a large class of partial differential equations. The obtained asymptotic formula can serve as a useful tool to perform numerical algorithms for solving optimization and control problems.
