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Solution of a Biharmonic Equation [PDF]
Nature, 1964 As is well known, a harmonic function φ(P), satisfying the Laplace's equation ▿2φ = 0 at any point P in a simply connected domain D bounded by a contour Γ, may be represented by: where q is a point on the contour Γ. On the boundary this equation attains the value φ(p) ≡ g(p) where
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Nonradiating sources of the biharmonic wave equation
Inverse Problems and Imaging, 2023 This paper offers an extensive exploration of nonradiating sources for the two- and three-dimensional biharmonic wave equations. Various equivalent characterizations are derived to reveal the nature of a nonradiating source. Additionally, we establish the connection between nonradiating sources in the biharmonic wave equation and those in the Helmholtz
Li, Peijun, Wang, Jue
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Biharmonic submanifolds of $\mathbb{C}P^n$ [PDF]
, 2009 We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature ...
Fetcu, D. +3 more
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Наука и техника, 2022 . Many important questions in the theory of elasticity lead to a variational problem associated with a biharmonic equation and to the corresponding boundary value problems for such an equation.
I. N. Meleshko, P. G. Lasy
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Radial Symmetry of Entire Solutions of a Biharmonic Equation with Supercritical Exponent
Advanced Nonlinear Studies, 2019 Necessary and sufficient conditions for a regular positive entire solution u of a biharmonic ...
Guo Zongming, Wei Long
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Solving a Class of High-Order Elliptic PDEs Using Deep Neural Networks Based on Its Coupled Scheme
Mathematics, 2022 Deep learning—in particular, deep neural networks (DNNs)—as a mesh-free and self-adapting method has demonstrated its great potential in the field of scientific computation. In this work, inspired by the Deep Ritz method proposed by Weinan E et al.
Xi’an Li +3 more
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Biharmonic nonlinear scalar field equations
, 2021 We prove a Brezis-Kato-type regularity result for weak solutions to the biharmonic nonlinear equation $$\Delta^2u=g(x,u)\qquad\text{ in }\mathbb{R}^N$$ with a Carathéodory function $g:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$, $N\ge5$. The regularity results give rise to the existence of ground state solutions provided that g has a general subcritical
Mederski, Jarosław, Siemianowski, Jakub
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Solutions to a ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -biharmonic elliptic problem on a bounded domain
Boundary Value Problems, 2023 Using variational methods and critical point results, we prove the existence and multiplicity of weak solutions of a ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -biharmonic elliptic equation along with a singular term under Navier boundary conditions.
Ali Khaleghi, Abdolrahman Razani
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Singular standing-ring solutions of nonlinear partial differential equations
, 2009 We present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d-dimensional sphere, where d>1. The asymptotic profile and blowup rate of these solutions are the same as those of solutions of
Bricmont +30 more
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Advances in Nonlinear Analysis, 2018 In the present paper, we investigate the existence of solutions for the following inhomogeneous singular equation involving the p(x){p(x)}-biharmonic operator:
Kefi Khaled, Saoudi Kamel
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