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In this paper, we address the Robin inverse problem for the biharmonic equation in a 2D$$ 2D $$ simply connected domain, to reconstruct the geometric shape of a non‐accessible part of the boundary from a single measurement of Riquier–Neumann data on the accessible part of that boundary. Our approach extends the nonlinear boundary integral equation, to recover the shape of the boundary. We propose the Newton iterative technique based on the Fréchet derivatives to linearize the system and then establish an injectivity of the linearized system for certain Robin coefficients, as well as the iteration scheme to describe the inverse algorithm for recovering the shape with respect to the unknowns. The mathematical spirit of the proposed method will be presented, and to illustrate its feasibility, some numerical examples will be provided. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the inverse problem for the biharmonic equation posed on a rectangle, which is of great importance in many areas of industry and engineering. We show that the problem under consideration is ill‐posed; therefore, to solve it, we opted for a regularization method via modified auxiliary boundary conditions. The numerical implementation is based on the application of the semidiscrete finite difference method for a sequence of well‐posed direct problems depending on a small parameter of regularization. Numerical results are performed for a rectangle domain showing the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear biharmonic equation. The problem is ill‐posed in the sense of Hadamard. To obtain a stable numerical solution, we consider a regularization method. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in ℒ2 uniformly with respect to the space coordinate under some a priori assumptions on the solution. [ABSTRACT FROM AUTHOR]

Let Hom+(T) be the class of all sense‐preserving homeomorphic self‐mappings of T={z=x+iy∈C:|z|=1}. The aim of this paper is twofold. First, we obtain Heinz‐type inequality for (K,K′)‐quasiconformal mappings satisfying inhomogeneous biharmonic equation Δ(Δω)  =  g in unit disk D with associated boundary value conditions Δω|T=φ∈C(T) and ω|T=f∗∈Hom+(T). Second, we establish biLipschitz continuity for (K,K′)‐quasiconformal mappings satisfying aforementioned inhomogeneous biharmonic equation when K′,||φ||∞:=supz∈D|φ(z)| and ||g||∞:=supz∈D|g(z)| are small enough. [ABSTRACT FROM AUTHOR]

We consider the problem Δ2u = V(x)up + ϵ in RN with u,Δu→0 as |x|→ + ∞, where p=N+4N−4, N ≥ 5, V is a positive continuous potential. Our aim is to construct high‐energy solutions for this equation by applying the finite‐dimensional reduction method and the penalization method. [ABSTRACT FROM AUTHOR]

In this paper, we study the following biharmonic equation: BvΔ2u+Vλ(x)u=α(x)f(u)+νK(x)|u|q−2uinRN,u∈H2(RN),where N⩾5,ν ∈ (0,ν0],1  <  q  <  2,Δ2u  =  Δ(Δu) and Vλ(x)  =  λa(x) − b(x) with λ > 0. Firstly, we prove the bipolar Rellich inequality. Secondly, by using bipolar Rellich inequality, Gigliardo‐Nirenberg inequality, and Ekeland variational principle, we prove the existence of nontrivial solution for problem Bν. [ABSTRACT FROM AUTHOR]

In this paper, we study the following biharmonic equation [ABSTRACT FROM AUTHOR]

We consider a commutative algebra [ABSTRACT FROM AUTHOR]