Divergence Form Research Articles

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

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, .

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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.

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.

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.

Numerical solution of the boundary value problem of elliptic equation by Levi function scheme

AbstractFor boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the parametrix can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the parametrix framework. The well‐posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the parametrix, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2‐dimensional and 3‐dimensional cases are presented to show the validity of the proposed schemes.

The role of the environment within a constellation of interacting causal factors shaping mobility—the example of three oases in Northern Africa

Northern Africa’s oases are fragile agroecosystems that, through environmental limitations on agricultural development, such as insufficient amounts of water for irrigation, have had to face a profound transformation of the environment on which livelihoods depend. A common strategy in such a situation is mobility. Though mobility responds to similar environmental limitations, the varying economic, social, demographic, and political contexts lead to divergent forms of it. Based on the mixed-method research conducted in the three oases located on the edge of the Sahara Desert (Mhamid in Morocco, El Faouar, and El Ferch in Tunisia), this paper explores the complex interplay of environmental and non-environmental factors shaping mobility. It highlights the need for a contextualization analysis of the mobility-environment nexus, illustrating how different factors’ constellations make mobility part of adaptation, coping, or escape strategies. The results reveal the dynamics of intertwined factors influencing mobility in the oases, sometimes dominated by environmental factors, other times by economic ones, and so forth. This makes their residents subject to fluid categorisation as environmental migrants, and the mobility-environmental nexus itself is highly complex.

Lattice Boltzmann method for the linear complementarity problem arising from American option pricing

In this paper, a lattice Boltzmann method is proposed for solving the linear complementarity problem (LCP) arising in single and multi-asset American put option pricing. The LCP for American option is a variable coefficient parabolic model defined on an unbounded domain. Initially, using the far field estimate method and the penalty method respectively, the LCP could be reformulated into a nonlinear parabolic partial differential equation on a bounded domain. To construct a unified lattice Boltzmann model for the option pricing problems, the above transformation equations are rewritten into an equivalent divergence form. Then, through the incorporation of an amending function into the evolution equation, which assists in recovering the source term and eliminating the error term, the lattice Boltzmann model with spatial second-order accuracy is constructed. Finally, the present model is validated using numerical simulations, and the numerical results agree well with the option values obtained by existing methods, which indicates that the present lattice Boltzmann model is efficient for solving the American put option pricing problem.

Gradient estimates for the insulated conductivity problem: The non-umbilical case

We study the insulated conductivity problem with inclusions embedded in a bounded domain in Rn, for n≥3. The gradient of solutions may blow up as ε, the distance between the inclusions, approaches to 0. We established in a recent paper optimal gradient estimates for a class of inclusions including balls. In this paper, we prove such gradient estimates for general strictly convex inclusions. Unlike the perfect conductivity problem, the estimates depend on the principal curvatures of the inclusions, and we show that these estimates are characterized by the first non-zero eigenvalue of a divergence form elliptic operator on Sn−2.

Counterexamples to maximal regularity for operators in divergence form

In this paper, we present counterexamples to maximal Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-regularity on H-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^{-1}$$\\end{document} under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-regularity on H-1(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^{-1}(\\mathbb {R}^d)$$\\end{document} or L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-regularity on L2(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\mathbb {R}^d)$$\\end{document}.

Elliptic problems with superlinear convection terms

In this manuscript we deal with elliptic equations with superlinear first order terms in divergence form of the following type−div(M(x)∇u)=−div(h(u)E(x))+f(x), where M is a bounded elliptic matrix, the vector field E and the function f belong to suitable Lebesgue spaces, and the function s→h(s) features a superlinear growth at infinity. We provide some existence and non existence results for solutions to the associated Dirichlet problem and a comparison principle.

The role of social capital in building community disaster resilience –empirical evidences from rural China

The paper investigates the role of social capital in hazard management in a case area which was overwhelmingly swept by flood in mid-2021. The interplay among social capital, hazard preparedness, response and recovery are explored by using structure equation modeling. Data were gathered from over 413 villages from August 2021 to September 2021, which is one month after the flood occurred. Research findings show that villagers in region with abundant social capital are more likely to contribute to hazard preparedness. We have also confirmed the positive role of social capital in restraining the pursuit of self-interests. Villagers are more willing to help others, and take care of vulnerable population when the hazard strikes. However, it is surprising to note that social capital fails to motivates residents to make preparedness plans for future risks. This retreat effect exists when big loss occurred and failed to meet local people's expectations. Overall, this study implies that the impact of social capital on hazard risk management is extremely complex and such complexity should not be attribute solely to divergent forms of network relationship. The above conclusion enables policy makers to pay attention to the psychological foundation of social capital while cultivating it.