Divergence Form Research Articles

Linear stabilization for a degenerate wave equation in non divergence form with drift

Linear stabilization for a degenerate wave equation in non divergence form with drift

Propagation of smallness for solutions of elliptic equations in the plane

<p>We explore quantitative propagation of smallness for solutions of two-dimensional elliptic equations from sets of positive $ \delta $-dimensional Hausdorff content for any $ \delta > 0 $. In particular, the gradients of solutions to divergence form equations with Hölder continuous coefficients, as well as those of nondivergence form equations with measurable coefficients, can be quantitatively estimated from the small sets.</p>

Global well-posedness, ill-posedness and long time behavior of solutions of the surface electromigration equation

Abstract This paper is concerned with the Cauchy problem and the long time behavior of the two-dimensional surface electromigration (SEM) equation. Firstly, we prove that the SEM equation is locally well-posed in H s ( R 2 ) with s > − 1 4 , and globally well-posed in L 2 ( R 2 ) without smallness assumption. Secondly, we prove that the SEM equation is ill-posed in H s ( R 2 ) with s < − 1 4 in the sense that the solution map is not C 2. Finally, we get a local energy decay result of the global solution in a growing unbounded in time region but not containing the soliton region. Our well-posedness results extend the latest findings in Linares et al (2021 Nonlinearity 34 5213–33), where the authors proved that the SEM equation is locally well-posed in H s ( R 2 ) with s > 1 2 . The key ingredient in this paper is to rewrite the linear and nonlinear part in the SEM equation into symmetric separate dispersive terms on the two space variables and some divergence form, respectively. After this invertible process, we use two bilinear estimates in the modified Fourier restriction space introduced in Kinoshita (2021 Ann. Inst. Henri Poincare C 38 451–505) to relax the local well-posedness index from s > 1 2 to s > − 1 4 .

Regularity of Weak Solutions to a Class of Nonlinear Parabolic Equations in Fractional Sobolev Spaces

In this article, we study regularity of weak solutions to a class of nonlinear parabolic equations in divergence form. The main purpose is to present a regularity estimate with more general conditions on coefficients, N-functions and non-homogeneous terms in the fractional Sobolev spaces. By deriving a higher integrability estimate of weak solutions, we obtain the desired regularity estimate. In addition, the results of this article expand the regularity theory of parabolic equations in fractional Sobolev spaces and Besov spaces.

Decolonization of English language assessment for equitable learning: Teachers’ agency perspective

This paper examines the agency of English language teachers in decolonizing assessment practices within their respective school contexts. Employing a phenomenological approach, this study draws on data collected through in-depth interviews with six secondary-level English language teachers from six purposively selected public schools in the Kathmandu Valley. The findings reveal that divergent forms of classroom instruction, normative assessment practices perpetuate epistemic injustice. English language teachers find themselves caught between top-down policies and bottom-up practices, despite their awareness and willingness to transition language assessment from conventional monolingual to plurilingual processes. This constraint hinders their ability to contextualize assessment according to learners' diversities and needs. The situation underscores the persistence of colonial-sounding practices, such as normative standards in assessment, necessitating teachers' full autonomy in assessment design and implementation to advance decolonization efforts in assessment and pedagogy. These findings significantly affect future policy initiatives in English language teaching in Nepal, particularly concerning the assessment integrated with pedagogical practices.

A novel selenoglycoside compound GlcSeCys alleviates diets-induced obesity and metabolic dysfunctions with the modulation of Galectin-1 and selenoproteins

Selenium, an essential trace element in human, has been shown to play protective roles in obesity and metabolic disorders despite insufficient understanding of mechanisms. Moreover, it's well known that biological actions of selenium compounds differed greatly due to divergent chemical forms. Selenoglycoside is a type of organoselenium compounds with excellent hydrophilicity, but biological activity of which in vivo are almost unknown. We have designed and synthesized Se-β-D-glucopyranosyl-D-selenocysteine, a novel selenoglycoside compound named GlcSeCys. Herein, GlcSeCys was given to high fat high cholesterol (HFHC) fed mice to determine its actions as well as relevant molecular mechanisms using transcriptome and multiple molecular biological methods. It was revealed that GlcSeCys displayed pronounced anti-obesity effect and significantly alleviated hyperglycemia, hyperinsulinemia along with hepatic steatosis in HFHC diets-induced mice. Mechanistically, GlcSeCys was found to inhibit lipogenesis, lipid uptake and inflammation in liver, along with attenuation of Galectin-1 and induction of selenoprotein S (SELENOS). With regard to adipose tissues, GlcSeCys ameliorated hypertrophy of adipocytes, suppressed lipids biosynthesis and stimulated WAT browning along with abrogated WAT inflammation activation, which were in line with repression of Galectin-1 and increase of GPx3. Collectively, our results uncovered, for the first time, that selenoglycoside compound GlcSeCys possessed excellent protective effects against obesity and metabolic disorders, and the mechanisms were correlated with modulation of Galectin-1 and selenoproteins, shedding lights upon molecular biology of selenium and novel therapeutic for obesity and relevant metabolic disorders.

Stabilization for degenerate equations with drift and small singular term

We consider a degenerate/singular wave equation in lone dimension, with drift and in presence of a leading operator that is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.

Qualitative Analysis of General Elliptic Operator in Divergence Form

Qualitative Analysis of General Elliptic Operator in Divergence Form

Testing for allocation strategies and evolutionary tradeoffs in the bivalve shell

ABSTRACT Tradeoffs are a fundamental aspect of evolution, such that organismal morphologies often reflect a combination of developmental constraints and functional requirements. The net result is a striking variety of allocation strategies—where and how resources are shunted into aspects of the body plan to meet these competing demands. The bivalve shell records aspects of the animal's allocation strategies as the relative investment in the carbonate valves and the internal soft parts, generally reflected by the volume of the internal cavity. The carbonate volume can be further partitioned among different components of the shell, including the sculptural elements thought to evolve partly under ecological controls. Assuming there are general limits to energy acquisition, bivalve individuals may show different allocations between the volumes of the inner cavity and shell, and possibly for the sculpture. Using 3D scans of 385 valves representing 339 extant marine species occurring in the Florida Keys, USA, we find a variety of allocation strategies but that tradeoffs in shell construction are uncommon. Instead, species with relatively high allocations to their shell thickness also have prominent sculpture, suggesting a structural relationship. Neither outer shell mineralogy nor microstructure appears to partition the tradeoff space, suggesting varied pathways to the fabrication of both convergent and divergent shell forms. Framing new morphological and physiological analyses of bivalves by allocation strategies recorded in their well-fossilized shells opens new opportunities for understanding an under-studied, macroevolutionary dynamic: how tradeoffs have shaped the class's 500 million-year evolutionary and ecological history.

Aerodynamic and flutter performance of the bridge deck with various countermeasures using computational fluid dynamics

Aerodynamic layouts of bridge decks considerably affect aerostatic torsional divergence and flutter for bridges. The crucial operation is choosing the optimal bridge deck shape considering the aerodynamic effect. The present study intends to provide a better solution for the flutter control scheme by changing five aspect ratios and four passive aerodynamic countermeasures. The countermeasures are an eccentric wing, inspection rail, down vertical central stabilizer and wind barrier, which have been used for three deck shapes: streamlined, semi-streamline and bluff body sections with large angles of attack. Nowadays, wind barriers are frequently installed on bridges to shield vehicles from the harmful effects of crosswinds, which can influence the flutter characteristics of the bridge. Therefore, the present study contemplates reducing the flutter effect with countermeasures. These countermeasures adversely affect the deck’s dynamic stability, mainly manifested in the different divergence forms of the structure. Finally, the eccentric wing, which runs parallel to the bridge deck, increases the critical flutter speed. However, the flow that passes through the eccentric wing extracts considerable energy from the flow field of the countermeasure. Therefore, the framework and solution of the present investigation are sustainable and safe for bridge design.

A note on limiting Calderon–Zygmund theory for transformed n$n$‐Laplace systems in divergence form

AbstractWe consider rotated ‐Laplace systems on the unit ball of the form where , , and for some . We prove that with estimates. As a corollary, we obtain that solutions to , where is the Hardy space, have a higher integrability, namely, .

Extrapolation of solvability of the regularity and the Poisson regularity problems in rough domains

Let Ω⊂Rn+1, n≥2, be an open set satisfying the corkscrew condition with n-Ahlfors regular boundary ∂Ω, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Hajłasz-Sobolev space M1,1(∂Ω) and the weak-A∞ property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in M1,1(∂Ω) is equivalent to the solvability of the regularity problem in M1,p(∂Ω) for some p>1. We also prove analogous extrapolation results for the Poisson regularity problem defined on tent spaces. Moreover, under the hypothesis that ∂Ω supports a weak (1,1)-Poincaré inequality, we show that the solvability of the regularity problem in the Hajłasz-Sobolev space M1,1(∂Ω) is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives.

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We investigate the interior regularity to nonlinear subelliptic systems in divergence form with drift term for the case of superquadratic controllable structure conditions in the Heisenberg group. On the basis of a generalization of the A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{A}$\\end{document}-harmonic approximation technique, C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$C^{1}$\\end{document}-regularity is established for horizontal gradients of vector-valued solutions to the subelliptic systems with drift term. Specially, our result is optimal in the sense that in the case of Hölder continuous coefficients we directly attain the optimal Hölder exponent for the horizontal gradients of weak solutions on the regular set.

On a nonlinear general eigenvalue problem in Musielak–Orlicz spaces

Abstract In this paper, we concern the existence result of the following general eigenvalue problem: { 𝒜 ⁢ ( u ) = λ ⁢ ℬ ⁢ ( u ) in ⁢ Ω , D α ⁢ ( u ) = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}\mathcal{A}(u)&\displaystyle={\lambda}% \mathcal{B}(u)&&\displaystyle\phantom{}\text{in }{\Omega},\\ \displaystyle D^{\alpha}(u)&\displaystyle=0&&\displaystyle\phantom{}\text{on }% {\partial\Omega},\end{aligned}\right. in an arbitrary Musielak–Orlicz spaces, where 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} are quasilinear operators in divergence form of order 2 ⁢ n {2n} and 2 ⁢ ( n - 1 ) {2(n-1)} , respectively. The main assumptions in this case are that 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} are potential operators with 𝒜 {\mathcal{A}} being elliptic and monotone. In this study, we intentionally avoid imposing constraints on the growth of a generalized N-function, including the Δ 2 {\Delta_{2}} -condition for both the generalized N-function and its conjugate. Consequently, this necessitates the formulation of the approximation theorem and the extensive utilization of modular convergence concepts.

Strong barriers for weighted quasilinear equations

In potential theory, use of barriers is one of the most important techniques. We construct strong barriers for weighted quasilinear elliptic operators. There are two applications: (i) solvability of Poisson-type equations with boundary singular data, and (ii) a geometric version of Hardy inequality. Our construction method can be applied to a general class of divergence form elliptic operators on domains with rough boundary.

Algorithmic models through a representational lens

Although algorithms are imbued with a sense of objectivity and reliability, numerous high-profile incidents have demonstrated their fallibility. In response, many have called for algorithmic governance that mitigates their potential harms. Further, these incidents have inspired studies that consider algorithms as part of wider sociotechnical systems. In this article, we build on such work and focus on how the specific forms of algorithms may facilitate or constrain the ways in which they become embedded within these systems. More specifically, we suggest that (a) algorithms should be understood as models, with (b) divergent forms, and (c) associated representational qualities. We showcase this approach in three critical case studies of algorithmic models used in government: the SAFFIER II model that underpins the Netherlands government’s spending, the Ofqual DCP A-Level grading algorithm that was used (and later abandoned) in lieu of actual secondary school exams in the United Kingdom, and the Risk Classification Model used by the Dutch Tax and Customs Administration to identify social benefit fraud. With the three case studies, we show how the divergent forms of algorithms have implications for their responsiveness and ultimately their solidification in – or dissolution from – socio-technical systems.

Elliptic equations in divergence form with discontinuous coefficients in domains with corners

Elliptic equations in divergence form with discontinuous coefficients in domains with corners

Large deviation principle for pseudo-monotone evolutionary equation

In this article, we explore Freidlin-Wentzell’s large deviation principle for a stochastic partial differential equation with the nonlinear diffusion-convection operator in divergence form satisfying p-type growth, involving coercivity assumptions, perturbed by small multiplicative Brownian noise. We use the weak convergence method to prove the Laplace principle, which is equivalent to the large deviation principle in our framework.

Schauder and Cordes–Nirenberg estimates for nonlocal elliptic equations with singular kernels

AbstractWe study integro‐differential elliptic equations (of order ) with variable coefficients, and prove the natural and most general Schauder‐type estimates that can hold in this setting, both in divergence and non–divergence form. Furthermore, we also establish Hölder estimates for general elliptic equations with no regularity assumption on , including for the first‐time operators like , provided that the coefficients have “small oscillation.”

First Report and Molecular Characterization of Umbra-like Virus on Ficus carica Caprifig Trees in Crimea.

Fig mosaic is the most serious viral disease affecting figs. A fig germplasm collection from the Nikita Botanical Garden on the Crimean Peninsula was surveyed for viruses using high-throughput sequencing and RT-PCR with primers specific to known fig viruses. Reads related to fig umbra-like virus (FULV) were generated in samples from Ficus carica caprifig (pollinator) trees of the cultivar Belle dure. F. carica trees of other cultivars, as well as F. afghanistanica, F. palmata, and F. virgata trees, tested negative for FULV. Near-complete genomes of five Crimean fig umbra-like virus (FULV-CR) isolates shared 99.4% to 99.9% identity and were most closely related (85.2% identity) to the Hawaiian FULV isolate Oahu1 (MW480892). Based on their genome structure and a phylogenetic analysis, the FULV-CR isolates were determined to be dicot-infecting Class 2 umbra-like viruses and seem to be highly divergent forms of the same virus found recently in Hawaii, USA. This is the first report of an umbra-like virus found on figs in Crimea and outside of Hawaii, expanding information on the geographical distribution and genetic diversity of FULV. All of the Crimean FULV-positive plants were also co-infected with fig mosaic virus, fig badnavirus 1, and grapevine badna FI virus.