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Tanaka Mieko - One of the best experts on this subject based on the ideXlab platform.
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On subhomogeneous indefinite $p$-Laplace Equations in supercritical spectral interval
2021Co-Authors: Bobkov Vladimir, Tanaka MiekoAbstract:We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $-\Delta_p u = \lambda |u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$, where $12q$ and either $\int_\Omega a\varphi_p^q\,dx=0$ or $\int_\Omega a\varphi_p^q\,dx>0$ is sufficiently small, then such solutions do exist in a right neighborhood of $\lambda^*$. Here $\varphi_p$ is the first eigenfunction of the Dirichlet $p$-Laplacian in $\Omega$. This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case $q>p$ or in the sublinear case $q2q$ and $\int_\Omega a\varphi_p^q\,dx>0$ is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of $\lambda^*$, two of which are strictly positive in $\{x\in \Omega: a(x)>0\}$.Comment: 39 pages, 4 figure
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Generalized Picone inequalities and their applications to $(p,q)$-Laplace Equations
'Walter de Gruyter GmbH', 2021Co-Authors: Bobkov Vladimir, Tanaka MiekoAbstract:We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the $(p,q)$-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = f_\mu(x,u,\nabla u)$ in a bounded domain $\Omega \subset \mathbb{R}^N$ under certain assumptions on the nonlinearity and with a special attention to the resonance case $f_\mu(x,u,\nabla u) = \lambda_1(p) |u|^{p-2} u + \mu |u|^{q-2} u$, where $\lambda_1(p)$ is the first eigenvalue of the $p$-Laplacian.Comment: 18 pages, 1 figure. Remark 1.3 added, formulation and proof of Lemma 1.6 slightly improved, figure added, inequality (1.12) added, several minor changes according to referee's suggestions incorporate
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Generalized Picone inequalities and their applications to $(p,q)$-Laplace Equations
2020Co-Authors: Bobkov Vladimir, Tanaka MiekoAbstract:We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the $(p,q)$-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = f_\mu(x,u,\nabla u)$ in a bounded domain $\Omega \subset \mathbb{R}^N$ under certain assumptions on the nonlinearity and with a special attention to the resonance case $f_\mu(x,u,\nabla u) = \lambda_1(p) |u|^{p-2} u + \mu |u|^{q-2} u$, where $\lambda_1(p)$ is the first eigenvalue of the $p$-Laplacian.Comment: 17 page
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Generalized Picone inequalities and their applications to (p,q)-Laplace Equations
'Walter de Gruyter GmbH', 2020Co-Authors: Bobkov Vladimir, Tanaka MiekoAbstract:We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the (p,q)-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation −Δ − Δ = (, ,∇) in a bounded domain Ω ⊂ R under certain assumptions on the nonlinearity and with a special attention to the resonance case (, ,∇) = 1()||−2 + ||−2, where 1() is the first eigenvalue of the p-Laplacian
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Multiplicity of positive solutions for $(p,q)$-Laplace Equations with two parameters
2020Co-Authors: Bobkov Vladimir, Tanaka MiekoAbstract:We study the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u+\beta |u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$, with $1
Bobkov Vladimir - One of the best experts on this subject based on the ideXlab platform.
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On subhomogeneous indefinite $p$-Laplace Equations in supercritical spectral interval
2021Co-Authors: Bobkov Vladimir, Tanaka MiekoAbstract:We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $-\Delta_p u = \lambda |u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$, where $12q$ and either $\int_\Omega a\varphi_p^q\,dx=0$ or $\int_\Omega a\varphi_p^q\,dx>0$ is sufficiently small, then such solutions do exist in a right neighborhood of $\lambda^*$. Here $\varphi_p$ is the first eigenfunction of the Dirichlet $p$-Laplacian in $\Omega$. This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case $q>p$ or in the sublinear case $q2q$ and $\int_\Omega a\varphi_p^q\,dx>0$ is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of $\lambda^*$, two of which are strictly positive in $\{x\in \Omega: a(x)>0\}$.Comment: 39 pages, 4 figure
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Generalized Picone inequalities and their applications to $(p,q)$-Laplace Equations
'Walter de Gruyter GmbH', 2021Co-Authors: Bobkov Vladimir, Tanaka MiekoAbstract:We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the $(p,q)$-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = f_\mu(x,u,\nabla u)$ in a bounded domain $\Omega \subset \mathbb{R}^N$ under certain assumptions on the nonlinearity and with a special attention to the resonance case $f_\mu(x,u,\nabla u) = \lambda_1(p) |u|^{p-2} u + \mu |u|^{q-2} u$, where $\lambda_1(p)$ is the first eigenvalue of the $p$-Laplacian.Comment: 18 pages, 1 figure. Remark 1.3 added, formulation and proof of Lemma 1.6 slightly improved, figure added, inequality (1.12) added, several minor changes according to referee's suggestions incorporate
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Generalized Picone inequalities and their applications to $(p,q)$-Laplace Equations
2020Co-Authors: Bobkov Vladimir, Tanaka MiekoAbstract:We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the $(p,q)$-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = f_\mu(x,u,\nabla u)$ in a bounded domain $\Omega \subset \mathbb{R}^N$ under certain assumptions on the nonlinearity and with a special attention to the resonance case $f_\mu(x,u,\nabla u) = \lambda_1(p) |u|^{p-2} u + \mu |u|^{q-2} u$, where $\lambda_1(p)$ is the first eigenvalue of the $p$-Laplacian.Comment: 17 page
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Generalized Picone inequalities and their applications to (p,q)-Laplace Equations
'Walter de Gruyter GmbH', 2020Co-Authors: Bobkov Vladimir, Tanaka MiekoAbstract:We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the (p,q)-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation −Δ − Δ = (, ,∇) in a bounded domain Ω ⊂ R under certain assumptions on the nonlinearity and with a special attention to the resonance case (, ,∇) = 1()||−2 + ||−2, where 1() is the first eigenvalue of the p-Laplacian
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Multiplicity of positive solutions for $(p,q)$-Laplace Equations with two parameters
2020Co-Authors: Bobkov Vladimir, Tanaka MiekoAbstract:We study the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u+\beta |u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$, with $1
S K Zaripov - One of the best experts on this subject based on the ideXlab platform.
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modeling of fluid flow in periodic cell with porous cylinder using a boundary element method
Engineering Analysis With Boundary Elements, 2016Co-Authors: R F Mardanov, Sarah J Dunnett, S K ZaripovAbstract:Abstract The problem of viscous incompressible flow past a periodic array of porous cylinders (a model of flow in an aerosol filter) is solved. The approximate periodic cell model of Kuwabara is used to formulate the fluid flow problem. The Stokes flow model is then adopted to model the flow outside the cylinder and the Darcy law of drag is applied to find the filtration velocity field inside the porous cylinder. The boundary value problems for biharmonic and Laplace Equations for stream functions outside and inside the porous cylinder are solved using a boundary elements method. A good agreement of numerical and analytical models is shown. The analytical formulas for the integrals in the expressions for the stream function, vorticity and Cartesian velocity components are obtained. It is shown that the use of analytical integration gives considerable advantage in computing time.
Shravan Veerapaneni - One of the best experts on this subject based on the ideXlab platform.
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Spectrally-accurate quadratures for evaluation of layer potentials close to the boundary for the 2D Stokes and Laplace Equations
2016Co-Authors: Alex Barnett, Shravan VeerapaneniAbstract:Abstract. Dense particulate flow simulations using integral equation methods demand accurate evaluation of Stokes layer potentials on arbitrarily close interfaces. In this paper, we generalize techniques for close evaluation of Laplace double-layer potentials in J. Helsing and R. Ojala, J. Comput. Phys. 227 (2008) 2899–2921. We create a “globally compensated ” trapezoid rule quadrature for the Laplace single-layer potential on the interior and exterior of smooth curves. This exploits a complex representation, a product quadrature (in the style of Kress) for the sawtooth function, careful attention to branch cuts, and second-kind barycentric-type formulae for Cauchy integrals and their derivatives. Upon this we build accurate single- and double-layer Stokes potential evaluators by expressing them in terms of Laplace potentials. We test their convergence for vesicle-vesicle interactions, for an extensive set of Laplace and Stokes problems, and when applying the system matrix in a boundary value problem solver in the exterior of multiple close-to-touching ellipses. We achieve typically 12 digits of accuracy using very small numbers of discretization nodes per curve. We provide documented codes for other researchers to use. Key words. Stokes Equations, quadrature, nearly singular integrals, spectral methods, boundary integral Equations, barycentric. 1. Introduction. Dens
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spectrally accurate quadratures for evaluation of layer potentials close to the boundary for the 2d stokes and Laplace Equations
SIAM Journal on Scientific Computing, 2015Co-Authors: Alex H Barnett, Shravan VeerapaneniAbstract:Dense particulate flow simulations using integral equation methods demand accurate evaluation of Stokes layer potentials on arbitrarily close interfaces. In this paper, we generalize techniques for close evaluation of Laplace double-layer potentials in [J. Helsing and R. Ojala, J. Comput. Phys., 227 (2008), pp. 2899--2921]. We create a “globally compensated” trapezoid rule quadrature for the Laplace single-layer potential on the interior and exterior of smooth curves. This exploits a complex representation, a product quadrature (in the style of Kress) for the sawtooth function, careful attention to branch cuts, and second-kind barycentric-type formulae for Cauchy integrals and their derivatives. Upon this we build accurate single- and double-layer Stokes potential evaluators by expressing them in terms of Laplace potentials. We test their convergence for vesicle-vesicle interactions, for an extensive set of Laplace and Stokes problems, and when applying the system matrix in a boundary value problem solver...
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spectrally accurate quadratures for evaluation of layer potentials close to the boundary for the 2d stokes and Laplace Equations
arXiv: Numerical Analysis, 2014Co-Authors: Alex H Barnett, Shravan VeerapaneniAbstract:Dense particulate flow simulations using integral equation methods demand accurate evaluation of Stokes layer potentials on arbitrarily close interfaces. In this paper, we generalize techniques for close evaluation of Laplace double-layer potentials in J. Helsing and R. Ojala, J. Comput. Phys. 227 (2008) 2899-2921. We create a "globally compensated" trapezoid rule quadrature for the Laplace single-layer potential on the interior and exterior of smooth curves. This exploits a complex representation, a product quadrature (in the style of Kress) for the sawtooth function, careful attention to branch cuts, and second-kind barycentric-type formulae for Cauchy integrals and their derivatives. Upon this we build accurate single- and double-layer Stokes potential evaluators by expressing them in terms of Laplace potentials. We test their convergence for vesicle-vesicle interactions, for an extensive set of Laplace and Stokes problems, and when applying the system matrix in a boundary value problem solver in the exterior of multiple close-to-touching ellipses. We achieve typically 12 digits of accuracy using very small numbers of discretization nodes per curve. We provide documented codes for other researchers to use.
Shengping Shen - One of the best experts on this subject based on the ideXlab platform.
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a theory of flexoelectricity with surface effect for elastic dielectrics
Journal of The Mechanics and Physics of Solids, 2010Co-Authors: Shengping ShenAbstract:Abstract The flexoelectric effect is very strong for nanosized dielectrics. Moreover, on the nanoscale, surface effects and the electrostatic force cannot be ignored. In this paper, an electric enthalpy variational principle for nanosized dielectrics is proposed concerning with the flexoelectric effect, the surface effects and the electrostatic force. Here, the surface effects contain the effects of both surface stress and surface polarization. From this variational principle, the governing Equations and the generalized electromechanical Young–Laplace Equations are derived and can account for the effects of flexoelectricity, surface and the electrostatic force. Moreover, based on this variational principle, both the generalized bulk and surface electrostatic stresses can be obtained and are composed of two parts: the Maxwell stress corresponding to the polarization and strain and the remainder relating to the polarization gradient and the strain gradient. The theory developed in this paper provides the underlying framework for the analyses and computational solutions of electromechanical problems in nanodielectrics.
