Quasilinear Operators Research Articles

Inverse scattering for three-dimensional quasi-linear biharmonic operator

Abstract We consider an inverse scattering problem of recovering the unknown coefficients of a quasi-linearly perturbed biharmonic operator in the three-dimensional case. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove Saito’s formula and uniqueness theorem of recovering some essential information about the unknown coefficients from the knowledge of the high frequency scattering amplitude.

Study of weak solutions of variational inequality systems with degenerate parabolic operators and quasilinear terms arising Americian option pricing problems

<abstract><p>In this paper, we study variational inequality systems with quasilinear degenerate parabolic operators in a bounded domain. As a series of penalty problems, the existence of the solutions in the weak sense is proved by a limit process. The uniqueness of the solution is also proved.</p></abstract>

Improvements on overdetermined problems associated to the $ p $-Laplacian

<abstract><p>This work presents some improvements on related papers that investigate certain overdetermined problems associated to elliptic quasilinear operators. Our model operator is the $ p $-Laplacian. Under suitable structural conditions, and assuming that a solution exists, we show that the domain of the problem is a ball centered at the origin. Furthermore we discuss a convenient form of comparison principle for this kind of problems.</p></abstract>

Sensitivity Operator Framework for Analyzing Heterogeneous Air Quality Monitoring Systems

Air quality monitoring systems differ in composition and accuracy of observations and their temporal and spatial coverage. A monitoring system’s performance can be assessed by evaluating the accuracy of the emission sources identified by its data. In the considered inverse modeling approach, a source identification problem is transformed to a quasi-linear operator equation with the sensitivity operator. The sensitivity operator is composed of the sensitivity functions evaluated on the adjoint ensemble members. The members correspond to the measurement data element aggregates. Such ensemble construction allows working in a unified way with heterogeneous measurement data in a single-operator equation. The quasi-linear structure of the resulting operator equation allows both solving and predicting solutions of the inverse problem. Numerical experiments for the Baikal region scenario were carried out to compare different types of inverse problem solution accuracy estimates. In the considered scenario, the projection to the orthogonal complement of the sensitivity operator’s kernel allowed predicting the source identification results with the best accuracy compared to the other estimate types. Our contribution is the development and testing of a sensitivity-operator-based set of tools for analyzing heterogeneous air quality monitoring systems. We propose them for assessing and optimizing observational systems and experiments.

Global Hölder continuity of solutions to quasilinear equations with Morrey data

We deal with general quasilinear divergence-form coercive operators whose prototype is the [Formula: see text]-Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. The fairly non-regular boundary of the underlying domain is supposed to satisfy a capacity density condition which allows domains with exterior corkscrew property. We prove global boundedness and Hölder continuity up to the boundary for the weak solutions of such equations, generalizing this way the classical [Formula: see text]-result of Ladyzhenskaya and Ural’tseva to the settings of the Morrey spaces.

Second order monotone finite differences discretization of linear anisotropic differential operators

We design adaptive finite differences discretizations, which are degenerate elliptic and second order consistent, of linear and quasi-linear partial differential operators featuring both a first order term and ananisotropicsecond order term. Our approach requires the domain to be discretized on a Cartesian grid, and takes advantage of techniques from the field of low-dimensional lattice geometry. We prove that the stencil of our numerical scheme is optimally compact, in dimension two, and that our approach is quasi-optimal in terms of the compatibility condition required of the first and second order operators, in dimensions two and three. Numerical experiments illustrate the efficiency of our method in several contexts.

Lower hybrid current drive in a tokamak for correlated passes through resonance

Standard quasilinear descriptions are based on the constant magnetic field form of the quasilinear operator so improperly treat the trapped electron modifications associated with tokamak geometry. Moreover, successive poloidal transits of the Landau resonance during lower hybrid current drive in a tokamak are well correlated, and these geometrical details must be properly retained to account for the presence of trapped electrons that do not contribute to the driven current. The recently derived quasilinear operator in tokamak geometry accounts for these features and finds that the quasilinear diffusivity is proportional to a delta function with a transit or bounce averaged argument (rather than a local Landau resonance condition). The new quasilinear operator is combined with the Cordey (Nucl. Fusion, vol. 16, 1976, pp. 499–507) eigenfunctions to properly derive a rather simple and compact analytic expression for the trapped electron modifications to the driven lower hybrid current and the efficiency of the current drive.

NUMERICAL ANALYSIS OF THE OBSERVABILITY OF THE CITY ROAD NETWORK EMISSION SOURCES USING THE SENSITIVITY OPERATOR OF THE ATMOSPHERIC CHEMISTRY TRANSPORT AND TRANSFORMATION MODEL

Within the scenario approach, the observability of traffic emission sources is estimated from indirect monitoring data. What is new is that an approach based on sensitivity operators is used to estimate observability, which allows us to obtain a family of quasilinear operator equations for the source identification problem. This allows both solving and analyzing its properties. For the city of Novosibirsk, realistic weather scenarios, a scenario for the distribution of traffic emission sources and the location of monitoring sites are considered. In numerical experiments, the road network is identified smoothed over space. Concentration fields are restored with greater accuracy. Evaluation of the information about the sources contained in the data based on the analysis of the sensitivity operator allows one to get an express estimate of the inverse problem solution.

Blow-Up Solutions for a Class of Schrödinger Quasilinear Operators with a Local Sublinear Term

In this paper, we are concerned in establishing properties about the function $$\vartheta $$ and versions of the classical Keller–Osserman condition to prove existence of solutions to the Schrodinger quasilinear elliptic problem $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \mathrm{div}\left( \vartheta (u)\nabla u\right) -\frac{1}{2}\vartheta '(u)|\nabla u|^2=a(x)g(u)~ \text{ in }~ \Omega ,\\ u\ge 0\ \text{ in }~\Omega ,\ u(x){\mathop {\longrightarrow }\limits ^{d(x)\rightarrow 0}} \infty , \end{array} \right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N$$ , with $$N\ge 3$$ , is a bounded domain, $$a:{\bar{\Omega }} \rightarrow [0,\infty )$$ and $$g:[0,\infty ) \rightarrow [0,\infty )$$ are suitable nonnegative continuous functions, $$\vartheta :{\mathbb {R}}\rightarrow (0,\infty )$$ is a $$C^1$$ -function satisfying appropriated hypotheses, and $$d(x)=\mathrm{dist}(x,\partial \Omega )$$ stands for the distance function to the boundary of $$\Omega $$ . By exploring a dual approach and the relationship among the properties of $$\vartheta $$ with its corresponding Keller–Osserman condition, we were able to show existence of solutions for this problem.

Quasilinear treatment of wave–particle interactions in the electron cyclotron range and its implementation in a gyrokinetic code

A quasilinear operator for wave–particle interactions in the electron cyclotron range is derived using the cold plasma dispersion relation. The finite width of the beam implies a broadening of the resonance layer in the velocity space which allows the use of a numerically efficient treatment of the operator. The specific case of a pure heating injected at the outer mid-plane is then treated. It allows to treat the beam propagation and absorption easily, simplifying the implementation of the source term in the gyrokinetic code. This specific case is implemented in the gyrokinetic particle-in-cell code ORB5 using a Langevin approach. The derivation and verification of the source operator are presented.

Noncoercive quasilinear elliptic operators with singular lower order terms

We consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. The associated obstacle problem is also solved. Finally, we show higher integrability of a solution to the Dirichlet problem when the datum is more regular.

Reimagining full wave rf quasilinear theory in a tokamak

The velocity dependent resonant interaction of particles with applied radiofrequency (rf) waves during heating and current drive in the presence of pitch angle scattering collisions gives rise to narrow collisional velocity space boundary layers that dramatically enhance the role of collisions as recently shown by Catto (J. Plasma Phys., vol. 86, 2020, 815860302). The behaviour is a generalization of the narrow collisional boundary layer that forms during Landau damping as found by Johnston (Phys. Fluids, vol. 14, 1971, pp. 2719–2726) and Auerbach (Phys. Fluids, vol. 20, 1977, pp. 1836–1844). For a wave of parallel wave number ${k_{||}}$ interacting with weakly collisional plasma species of collision frequency $\nu$ and thermal speed ${v_{\textrm{th}}}$ , the effective collision frequency becomes of order $\nu {({k_{||}}{v_{th}}/\nu )^{2/3}} \gg \nu $ . The narrow boundary layers that arise because of the diffusive nature of the collisions allow a physically meaningful wave–particle interaction time to be defined that is the inverse of this effective collision frequency. The collisionality implied by the narrow boundary layer results in changes in the standard quasilinear treatment of applied rf fields in tokamaks while remaining consistent with causality. These changes occur because successive poloidal interactions with the rf are correlated in tokamak geometry and because the resonant velocity space dependent interactions are controlled by the spatial and temporal behaviour of the perturbed full wave fields rather than just the spatially local Landau and Doppler shifted cyclotron wave–particle resonance condition associated with unperturbed motion of the particles. The correlation of successive poloidal circuits of the tokamak leads to the appearance in the quasilinear operator of transit averaged resonance conditions localized in velocity space boundary layers that maintain negative definite entropy production.

A non-isotropic free transmission problem governed by quasi-linear operators

We study a free transmission problem in which solution minimizes a functional with different definitions in positive and negative phases. We prove some asymptotic regularity results when the jumps of the diffusion coefficients get smaller along the free boundary. At last, we prove a measure-theoretic result related to the free boundary.

Two-dimensional inverse scattering for quasi-linear biharmonic operator

<p style='text-indent:20px;'>The subject of this work concerns the classical direct and inverse scattering problems for quasi-linear perturbations of the two-dimensional biharmonic operator. The quasi-linear perturbations of the first and zero order might be complex-valued and singular. We show the existence of the scattering solutions to the direct scattering problem in the Sobolev space <inline-formula><tex-math id="M1">\begin{document}$ W^1_{\infty}( \mathbb{{R}}^2) $\end{document}</tex-math></inline-formula>. Then the inverse scattering problem can be formulated as follows: does the knowledge of the far field pattern uniquely determine the unknown coefficients for given differential operator? It turns out that the answer to this classical question is affirmative for quasi-linear perturbations of the biharmonic operator. Moreover, we present a numerical method for the reconstruction of unknown coefficients, which from the practical point of view can be thought of as recovery of the coefficients from fixed energy measurements.</p>

Evolution equations governed by quasilinear operators satisfying Carathéodory’s conditions

Evolution equations governed by quasilinear operators satisfying Carathéodory’s conditions

Parallel speedup analysis of an adjoint ensemble-based source identification algorithm

In the approach to the inverse problems, based on the sensitivity operators, the inverse problem is transformed into the family of quasi-linear operator equations. The sensitivity operator is constructed of the ensemble of the adjoint problem solutions corresponding to different adjoint problems source functions. Hence, from the theoretical point of view, sensitivity operator-based algorithms can be efficiently parallelized. The paper’s objective is to carry out the experimental study of the algorithm computation time for the different number of computation cores.

Mixed boundary value problem for a class of quasi-linear elliptic operators containing p-Laplacian

Mixed boundary value problem for a class of quasi-linear elliptic operators containing p-Laplacian

Soft Quasilinear Spaces and Soft Normed Quasilinear Spaces

In this study, a recent concepts of soft quasilinear spaces and soft proper quasilinear spaces are presented. Further, soft quasi vectors in soft quasilinear spaces are investigated, and several related properties are examined such as quasilinear dependent and quasilinear independent. Also, the concept of soft quasi norm of soft quasilinear spaces is given. Lastly, soft quasilinear operators on soft normed quasilinear spaces are defined, and some results about the bounded soft quasilinear operators and continuous soft quasilinear operators are obtained.

A limiting problem for local/non-local p-Laplacians with concave–convex nonlinearities

In this manuscript, we deal with an equation involving a combination of quasi-linear elliptic operators of local and non-local nature with p-structure, and concave–convex nonlinearities. The prototypical model is given by $$\begin{aligned} \left\{ \begin{array}{rclcl} -\Delta _p u + (-\Delta )^s_p u &{} = &{} \lambda _p u^q(x) + u^r(x) &{} \text{ in } &{} \Omega , \\ u(x)&{}>&{}0&{}\text{ in }&{} \Omega ,\\ u(x)&{} =&{} 0&{}\text { on } &{} \mathbb {R}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega \subset \mathbb {R}^n$$ is a bounded and smooth domain, $$s\in (0,1)$$ , $$2 \le p < \infty $$ , $$0<q(p)<p-1<r(p)<\infty $$ and $$0<\lambda _p< \infty $$ , being $$\Delta _p$$ and $$(-\Delta )_p^s$$ the p-Laplace and fractional p-Laplace operators, respectively. We study existence and global uniform and explicit boundedness results to weak solutions. Then, we perform an asymptotic analysis for the limit of a family of weak solutions $$\{u_p\}_{p\ge 2}$$ as $$p \rightarrow \infty $$ , which converges, up to a subsequence (under suitable assumptions on the problem data), to a non-trivial profile with uniform and explicit bounds, enjoying of a universal Lipschitz modulus of continuity, and verifying a nonlinear limiting PDE in the viscosity sense, which exhibits both local/non-local character.

Full and partial regularity for a class of nonlinear free boundary problems

In this paper we classify the nonnegative global minimizers of the functionalJF(u)=∫ΩF(|∇u|2)+λ2χ{u>0}, where F satisfies some structural conditions and χD is the characteristic function of a set D⊂Rn. We compute the second variation of the energy and study the properties of the stability operator. The free boundary ∂{u>0} can be seen as a rectifiable n−1 varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded. Hence one can use Allard's monotonicity formula to prove the existence of tangent cones modulo a set of small Hausdorff dimension. In particular, we prove that if n=3 and the ellipticity constants of the quasilinear elliptic operator generated by F are close to 1 then the conical free boundary must be flat.