Search results for "Overdetermined system"
showing 10 items of 15 documents
Wulff shape characterizations in overdetermined anisotropic elliptic problems
2017
We study some overdetermined problems for possibly anisotropic degenerate elliptic PDEs, including the well-known Serrin's overdetermined problem, and we prove the corresponding Wulff shape characterizations by using some integral identities and just one pointwise inequality. Our techniques provide a somehow unified approach to this variety of problems.
Hölder stability for Serrin’s overdetermined problem
2015
In a bounded domain \(\varOmega \), we consider a positive solution of the problem \(\Delta u+f(u)=0\) in \(\varOmega \), \(u=0\) on \(\partial \varOmega \), where \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a locally Lipschitz continuous function. Under sufficient conditions on \(\varOmega \) (for instance, if \(\varOmega \) is convex), we show that \(\partial \varOmega \) is contained in a spherical annulus of radii \(r_i 0\) and \(\tau \in (0,1]\). Here, \([u_\nu ]_{\partial \varOmega }\) is the Lipschitz seminorm on \(\partial \varOmega \) of the normal derivative of u. This result improves to Holder stability the logarithmic estimate obtained in Aftalion et al. (Adv Differ Equ 4:907–93…
On the stability of the Serrin problem
2008
We investigate stability issues concerning the radial symmetry of solutions to Serrin's overdetermined problems. In particular, we show that, if $u$ is a solution to $\Delta u=n$ in a smooth domain $\Omega \subset \rn$, $u=0$ on $\partial\Omega$ and $|Du|$ is close to 1 on $\partial\Omega$, then $\Omega$ is close to the union of a certain number of disjoint unitary balls.
Characterization of ellipsoids through an overdetermined boundary value problem of Monge–Ampère type
2014
Abstract The study of the optimal constant in an Hessian-type Sobolev inequality leads to a fully nonlinear boundary value problem, overdetermined with non-standard boundary conditions. We show that all the solutions have ellipsoidal symmetry. In the proof we use the maximum principle applied to a suitable auxiliary function in conjunction with an entropy estimate from affine curvature flow.
A note on Serrin's overdetermined problem
2014
We consider the solution of the torsion problem $$−Δu = N \quad\mathrm{in}\quad Ω,\quad u = 0\quad\mathrm{on}\quad ∂Ω,$$ where Ω is a bounded domain in RN. ¶ Serrin's celebrated symmetry theorem states that, if the normal derivative uν is constant on ∂Ω, then Ω must be a ball. In [6], it has been conjectured that Serrin's theorem may be obtained by stability in the following way: first, for the solution u of the torsion problem prove the estimate $$r_e − r_i ≤ C_t\Bigl(\max_{\Gamma_t} u-\min_{\Gamma_t} u\Bigr)$$ for some constant Ct depending on t, where re and ri are the radii of an annulus containing ∂Ω and Γt is a surface parallel to ∂Ω at distance t and sufficiently close to ∂Ω secondly…
Serrin-Type Overdetermined Problems: an Alternative Proof
2008
We prove the symmetry of solutions to overdetermined problems for a class of fully nonlinear equations, namely the Hessian equations. In the case of the Poisson equation, our proof is alternative to the proofs proposed by Serrin (moving planes) and by Weinberger. Moreover, our proof makes no direct use of the maximum principle while it sheds light on a relation between the Serrin problem and the isoperimetric inequality.
Stability of radial symmetry for a Monge-Ampère overdetermined problem
2008
Recently the symmetry of solutions to overdetermined problems has been established for the class of Hessian operators, including the Monge-Ampère operator. In this paper we prove that the radial symmetry of the domain and of the solution to an overdetermined Dirichlet problem for the Monge-Ampère equation is stable under suitable perturbations of the data. © 2008 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag.
An overdetermined problem for the anisotropic capacity
2015
We consider an overdetermined problem for the Finsler Laplacian in the exterior of a convex domain in \({\mathbb {R}}^{N}\), establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of Reichel (Arch Ration Mech Anal 137(4):381–394, 1997), where the usual Newtonian capacity is considered, giving rise to an overdetermined problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm H. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm \(H_0\)).
TOWARD A SOLUTION OF ALLOCATION IN LIFE CYCLE INVENTORIES: THE USE OF LEAST SQUARES TECHNIQUES
2010
Purpose: The matrix method for the solution of the so-called inventory problem in LCA generally determines the inventory vector related to a specific system of processes by solving a system of linear equations. The paper proposes a new approach to deal with systems characterized by a rectangular (and thus non-invertible) coefficients matrix. The approach, based on the application of regression techniques, allows solving the system without using computational expedients such as the allocation procedure. Methods: The regression techniques used in the paper are (besides the ordinary least squares, OLS) total least squares (TLS) and data least squares (DLS). In this paper, the authors present t…
A rigidity problem on the round sphere
2015
We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.