Search results for "overdetermined problems"
showing 8 items of 8 documents
Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration
2016
A positive solution of a homogeneous Dirichlet boundary value problem or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of its level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. We show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls $${B_{{r_e}}}$$ and $${B_{{r_i}}}$$ , with the difference r e -r i (linearly) controlled by a suitable norm of the deviation…
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.
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.
Symmetry of minimizers with a level surface parallel to the boundary
2015
We consider the functional $$I_\Omega(v) = \int_\Omega [f(|Dv|) - v] dx,$$ where $\Omega$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, G. Crasta [Cr1] has shown that if $I_\Omega$ admits a minimizer in $W_0^{1,1}(\Omega)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball. With some restrictions on $f$, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general settings, in particular to functionals that are not differenti…
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.
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.
The method of moving planes: a quantitative approach
2018
We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some recent results related to quantitative studies of the method of moving planes, where quantitative approximate symmetry results are obtained.
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…