On an Eighth Order Overdetermined Elliptic Boundary Value Problem
نویسندگان
چکیده
We consider the overdetermined boundary value problem for the 4-harmonic operator, Δ4 = Δ(Δ3) , and show that if the solution of the problem exists, then the domain must be an open N -ball (N 2) . As a consequence of overdetermined problems mean value results are obtained for harmonic, biharmonic, triharmonic and 4-harmonic functions. Mathematics subject classification (2010): 35J25, 35P15, 35B50.
منابع مشابه
Locally-corrected spectral methods and overdetermined elliptic systems
We present fast locally-corrected spectral methods for linear constant-coefficient elliptic systems of partial differential equations in d-dimensional periodic geometry. First, arbitrary second-order elliptic systems are converted to overdetermined first-order systems. Overdetermination preserves ellipticity, while first-order systems eliminate mixed derivatives, resolve convection-diffusion co...
متن کاملOn partially and globally overdetermined problems of elliptic type
We consider some elliptic PDEs with Dirichlet and Neumann data prescribed on some portion of the boundary of the domain and we obtain rigidity results that give a classification of the solution and of the domain. In particular, we find mild conditions under which a partially overdetermined problem is, in fact, globally overdetermined: this enables to use several classical results in order to cl...
متن کاملFree and constrained equilibrium states in a variational problem on a surface
We study the equilibrium states for an energy functional with a parametric force field on a region of a surface. Consideration of free equilibrium states is based on Lyusternik - Schnirelman's and Skrypnik's variational methods. Consideration of equilibrium states under a constraint of geometrical character is based on an analog of Skrypnik's method, described in [P. Vyridis, {it Bifurcation in...
متن کاملAn Overdetermined Problem for an Elliptic Equation
We consider the following overdetermined boundary value problem: ∆u+ λu+ μ = 0 in Ω, u = 0 on ∂Ω and ∂u/∂n = c on ∂Ω, where c 6= 0, λ and μ are real constants and Ω ⊂ R is a smooth bounded convex open set. We first show that it may happen that the problem has no solution. Then we study the existence of solutions for a wide class of domains. 2010 Mathematics Subject Classification: 35J05, 35R30.
متن کاملA numerical approach to solve eighth order boundary value problems by Haar wavelet collocation method
In this paper a robust and accurate algorithm based on Haar wavelet collocation method (HWCM) is proposed for solving eighth order boundary value problems. We used the Haar direct method for calculating multiple integrals of Haar functions. To illustrate the efficiency and accuracy of the concerned method, few examples are considered which arise in the mathematical modeling of fluid dynamics an...
متن کامل