Let $\Sigma$ be a hypersurface in an $n$-dimensional Riemannian manifold $M$, $n\geq 2$. We study the isometric extension problem for isometric immersions $f\colon\Sigma\to\mathbb{R}^n$, where $\mathbb{R}^n$ is equipped with the Euclidean standard metric. We prove a general curvature obstruction to the existence of merely differentiable extensions and an obstruction to the existence of Lipschitz extensions

In this manuscript, we consider the compressible Hall-MHD system, which is a mathematical model to describe magnetic reconnection in space plasmas, star formation, neutron stars, and geo-dynamics. In the system, there are two viscosity constants and a resistivity coefficient. Uniform regularity for the compressible isentropic Hall-MHD system is established in terms of these coefficients.

Let $\Omega\subset\mathbb{R}^d$ ($d\geq 2$) be a bounded Lipschitz domain. In this article, the author mainly studies complex interpolation of Besov-type spaces on the domain $\Omega$, namely, we investigate the interpolation $$ [B_{p_0,q_0}^{s_0,\tau_0}(\Omega),B_{p_1,q_1}^{s_1,\tau_1}(\Omega)]_\Theta = B_{p,q}^{\diamond s,\tau}(\Omega) $$ under certain conditions on the parameters, where $B_{p,q}^{\diamond

We study the existence of solutions for the $A$-Laplace system of the form $$ \left\{\begin{aligned} -\Delta_Au&=f &&\text{in}\ \Omega,\\ u&=0 && \text{on}\ \partial\Omega, \end{aligned}\right. $$ where $-\Delta_A u:=-\operatorname{div}(a(\lvert Du\rvert)Du)$. The existence result is proved by using the technique of Young measures.

We provide a new and simple proof based on Harnack’s inequality to the Lipschitz continuity of the solutions of a class of free boundary problems.

In this paper, new estimates of the sequential Caputo fractional derivatives of a function at its extremum points are obtained.We derive comparison principles for the linear fractional differential equations, then apply these principles to obtain lower and upper bounds of solutions of linear and nonlinear fractional differential equations. The extremum principle is then applied to show that the in

In this paper, we study the numerical stabilization of a 1D system of two wave equations coupled by velocities with an internal, local control acting on only one equation. In the theoretical part of this study [Z. Anal. Anwend. 40 (2021)(1), 67–96] we distinguished two cases. In the first one, the two waves assumed propagate at the same speed. Under appropriate geometric conditions, we had proved that

We study an initial boundary value problem of two-dimensional nonhomogeneous Bénard system with nonnegative density. We derive the global existence of a unique strong solution. In particular, the initial data can be arbitrarily large.

The aim of this work is to study the existence of weak solutions for a nonhomogeneous singular $p(x)$-Kirchhoff problem of the following form \begin{equation*} (\textbf{P}_{\pm \lambda}) \quad \left\{ \begin{aligned} M(t)\Delta(|\Delta u|^{p(x)-2}\Delta u) &=a(x) u^{-\gamma (x)}\pm \lambda u^{q(x)-2}u, &\ &\mbox{in }\Omega, \\ \Delta u&=u=0, & \ &\mbox{on }\partial\Omega, \end{aligned} \right. \end{equation*}

Morrey spaces are the powerful tool for the study of partial differential equations. Recently, weak Morrey spaces and weak Herz spaces are known to be useful for the study of Navier–Stokes equations. In this paper we introduce weak central Herz–Morrey spaces $W\mathcal H^{p(\cdot),q,\omega}(\mathbf B)$ and establish the weak estimate for the maximal and Riesz potential operators in the central Herz–Morrey

In this article we establish a Trudinger–Moser inequality of Adachi–Tanaka type in nonradial weighted Sobolev spaces for functions defined on the whole $\mathbb{R}^{N}$. The main tools are the Besecovitch covering lemma and a Trudinger–Moser inequality on the whole space established by S. Adachi and K. Tanaka.

A carefully written Nirenberg's proof of the famous Gagliardo–Nirenberg interpolation inequality for intermediate derivatives in $\mathbb R^n$ seems, surprisingly, to be missing in literature. In our paper, we shall first introduce this fundamental result and provide information about its historical background. Afterwards, we present a complete, student-friendly proof. In our proof, we use the architecture

The aim of this article is to study the behavior of the multifractal packing function under slices in Euclidean space. We discuss the relationship between the multifractal packing and pre-packing functions of a compactly supported Borel probability measure and those of slices or sections of the measure. More specifically, we prove that if $\mu$ satisfies a certain technical condition and $q$ lies in

We study the description by means of the $J$-functional of logarithmic interpolation spaces $(A_0, A_1)_{1,q,\mathbb{A}}$ in the category of the $p$-normed quasi-Banach couples ($0 < p \leq 1$). When $(A_0, A_1)$ is a Banach couple, it is known that the description changes depending on the relationship between $q$ and $\mathbb{A}$. In our more general setting, the parameter $p$ also has an important

This paper is mainly focused upon the pseudo $S$-asymptotically Bloch type periodicity and its applications. Firstly, a new notion of pseudo $S$-asymptotically Bloch type periodic functions is introduced, and some fundamental properties on pseudo $S$-asymptotically Bloch type periodic functions are established. Then, the notion and properties of weighted pseudo $S$-asymptotically Bloch type periodic

In this work, we discuss the existence and uniqueness of positive solutions for the second order integral boundary value problem $$ \left\{ \begin{aligned} x''(t) + f(t,x(t),(Hx)(t)) &=0, \quad 0 < t < 1,\\ x(0)=0,\quad x(1)&= \int_{0}^{1}a(s)x(s)ds, \end{aligned} \right. $$ where the function $f$ has a singularity at $t_{0}=0$. Our main tool is a fixed point theorem of Wardowski (2012). Moreover,

In this paper, we study the exact controllability and stabilization of a system of two wave equations coupled by velocities with an internal, local control acting on only one equation. We distinguish two cases. In the first one, when the waves propagate at the same speed: using a frequency domain approach combined with multiplier technique, we prove that the system is exponentially stable when the

In this paper, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet $p$-sub-Laplacian for general vector fields. As a byproduct, we establish the Caccioppoli inequalities and also discuss the particular cases on the Grushin plane and on the Heisenberg group.

In this paper, we study strongly coupled elliptic systems in non-variational form with negative exponents involving fractional Laplace operators. We investigate the existence, nonexistence, and uniqueness of the positive classical solution. The results obtained here are a natural extension of the results obtained by Ghergu [J. Funct. Anal. 258 (2010), 3295–3318] for the fractional case.

Under certain conditions, an element of a tensor product space can be identified with a compact operator and the singular value decomposition (SVD) applies to the latter. These conditions are not fulfilled in Sobolev spaces. In the previous part of this work (part I) [Z. Anal. Anwend. 39 (2020), 349–369], we introduced some preliminary notions in the theory of tensor product spaces. We analyzed low-rank

We prove the global solutions to the incompressible Hall-magnetohydrodynamics (Hall-MHD) equations with small or some large initial data in $\mathbb R^n (n \geq 3)$. Especially, Fujita–Kato type initial data as the incompressible Navier–Stokes equations are allowed. We also study the large time behavior of the solutions and obtain an optimal decay rate in the general Besov spaces. Different from all

Stimulated by the category theorems of Eisner and Serény in the setting of unitary and isometric $C_0$-semigroups on separable Hilbert spaces, we prove category theorems for Schrödinger semigroups. Speci cally, we show that, to a given class of Schrödinger semigroups, Baire generically the semigroups are strongly stable but not exponentially stable. We also present a typical spectral property of the

In this paper, by using the idea of linearizing maximal operators originated by Charles Fefferman and the $TT*$ method of Stein–Wainger, we establish a weighted inequality for vector valued maximal Carleson type operators with singular kernels proposed by Andersen and John on the weighted Lorentz spaces with vector-valued functions.

In the present note, we examine the behavior of some homothecy-invariant differentiation basis of rectangles in the plane satisfying the following requirement: for a given rectangle to belong to the basis, the ratio of the largest of its side-lengths by the smallest one (which one calls its shape) has to be a fixed real number depending on the angle between its longest side and the horizontal line

We consider the nonlinear eigenvalue problem $$Lx + \varepsilon N(x) = \lambda Cx, \quad \|x\|=1,$$ where $\varepsilon,\lambda$ are real parameters, $L, C\colon G \to H$ are bounded linear operators between separable real Hilbert spaces, and $N\colon S \to H$ is a continuous map defined on the unit sphere of $G$. We prove a global persistence result regarding the set $\Sigma$ of the solutions $(x,\varepsilon

In this paper we build up a criteria for fractional Orlicz–Sobolev extension and imbedding domains on Ahlfors $n$-regular domains.

Let $X = [0, 1]^n, n \geq 1$. We show that the typical (in the sense of Baire category) compact subset of $X$ is not only a zero dimensional Cantor space but it satisfies the property of being strongly microscopic, which is stronger than dimension zero.

We deal with the following nonlinear problem involving fractional $p\&q$ Laplacians: \begin{equation*} (-\Delta)^{s}_{p}u+(-\Delta)^{s}_{q}u+|u|^{p-2}u+|u|^{q-2}u=\lambda h(x) f(u)+|u|^{q^{*}_{s}-2}u \quad \mbox{in } \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $1 < p < q < \frac{N}{s}$, $\q=\frac{Nq}{N-sq}$, $\lambda > 0$ is a parameter, $h$ is a nontrivial bounded perturbation and $f$ is a

We present a partial Hölder regularity result for the gradient of solutions to quasimonotone systems: $$\mathrm div \:\mathbf A(\cdot,D\mathbf u) = \mathbf B(\cdot, D\mathbf u) \quad \text{in } \Omega,$$ on bounded domains in the weak sense. Here certain continuity, uniformly strictly quasimonotonicity, growth conditions are imposed on the coefficients, including an asymptotic Uhlenbeck behaviour close

A well known result from functional analysis states that any compact operator between Hilbert spaces admits a singular value decomposition (SVD). This decomposition is a powerful tool that is the workhorse of many methods both in mathematics and applied fields. A prominent application in recent years is the approximation of high-dimensional functions in a low-rank format. This is based on the fact