A Note on Some Overdetermined Elliptic Problem
نویسندگان
چکیده
Ω 7−→ λ1(Ω) , under the volume constraint Vol(Ω) = α (where α ∈ (0,Vol(M)) is fixed) are called extremal domains. Smooth extremal domains are characterized by the property that the eigenfunctions associated to the first eigenvalue of the Laplace-Beltrami operator have constant Neumann boundary data [2]. In other words, a smooth domain is extremal if and only if there exists a positive function u1 and a constant λ1 such that ∆gu1 + λ1 u1 = 0 ,
منابع مشابه
On 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 کاملClassification of the solutions to an overdetermined elliptic problem in the plane
we classify the solutions to an overdetermined elliptic problem in the plane in the finite connectivity case. This is achieved by establishing a one-to-one correspondence between the solutions to this problem and a certain type of minimal surfaces.
متن کاملOn an Eighth Order Overdetermined Elliptic Boundary Value Problem
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.
متن کاملOverdetermined problems with possibly degenerate ellipticity, a geometric approach
Given an open bounded connected subset Ω of R, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation −div(A(|∇u|)∇u) = 1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under fairly general assumptions on Ω and A. In particular, A...
متن کاملMaximum Principles for a Class of Nonlinear Second Order Elliptic Differential Equations
In this paper we investigate maximum principles for functionals defined on solutions to special partial differential equations of elliptic type, extending results by Payne and Philippin. We apply such maximum principles to investigate one overdetermined problem.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010