1,721,007 research outputs found
The index of isolated critical points and solutions of elliptic equations in the plane
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Symmetries in an overdetermined problem for the Green's function
Full text linkWe consider in the plane the problem of reconstructing a domain from the normal derivative of its Green's function with pole at a fixed point in the domain. By means of the theory of conformal mappings, we obtain existence, uniqueness, (non-spherical) symmetry results, and a formula relating the curvature of the boundary of the domain to the normal derivative of its Green's function
A note on Serrin's overdetermined problem
Full text linkWe consider the solution of the torsion problem
-Delta u = N in Omega, u = 0 on partial derivative Omega,
where Omega is a bounded domain in R-N.
Serrin's celebrated symmetry theorem states that, if the normal derivative u(v) is constant on partial derivative Omega, then Omega 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) <= Ct (max(Gamma i) u - min(Gamma i) u)
for some constant C-t depending on t, where r(e) and r(i) are the radii of an annulus containing partial derivative Omega and Gamma i is a surface parallel to partial derivative Omega at distance t and sufficiently close to partial derivative Omega; secondly, if in addition u(v) is constant on partial derivative Omega, show that
max(Gamma i) u - min(Gamma i) u = o(C-t) as t -> 0(+).
The estimate constructed in [6] is not sharp enough to achieve this goal. In this paper, we analyse a simple case study and show that the scheme is successful if the admissible domains Omega are ellipses
Stability in an overdetermined problem for the Green's function
Full text linkIn the plane, we consider the problem of reconstructing a domain from the normal derivative of its Green's function (with fixed pole) relative to the Dirichlet problem for the Laplace operator. By means of the theory of conformal mappings, we derive stability estimates of Holder type
The location of the hot spot in a grounded convex conductor
No full textWe investigate the location of
the (unique) hot spot in a convex heat conductor with uniform initial
temperature and with boundary grounded at zero temperature. We
present two methods to locate the hot spot: the first is based
on ideas related to the Alexandrov-Bakelmann-Pucci maximum principle and
Monge-Amp\` ere equations; the second relies on
Alexandrov's reflection principle.
We then show how such a problem can be simplified in case the conductor
is a polyhedron. Finally, we present some numerical computations
Preface: geometric properties for parabolic and elliptic PDE's
No full textPresentation of a special issue where the three authors were the Editors
Analytical results for 2-D non-rectilinear waveguides based on a Green's function
No full textWe consider the problem of wave propagation for a 2-D rectilinear
optical waveguide which presents some perturbation. We construct a
mathematical framework to study such a problem and prove the
existence of a solution for the case of small imperfections. Our
results are based on the knowledge of a Green's function for the
rectilinear case
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