1,857 research outputs found
The classical overdetermined Serrin problem
In this survey we consider the classical overdetermined problem which was
studied by Serrin in 1971. The original proof relies on Alexandrov's moving
plane method, maximum principles, and a refinement of Hopf's boundary point
Lemma. Since then other approaches to the same problem have been devised. Among
them we consider the one due to Weinberger which strikes for the elementary
arguments used and became very popular. Then we discuss also a duality approach
involving harmonic functions, a shape derivative approach and a purely integral
approach, all of them not relying on maximum principle. For each one we
consider pros and cons as well as some generalizations
On a P\'olya functional for rhombi, isosceles triangles, and thinning convex sets
Let be an open convex set in with finite width, and
let be the torsion function for , i.e. the solution of
. An upper bound is obtained for the product
of , where
is the bottom of the spectrum of the Dirichlet Laplacian
acting in . The upper bound is sharp in the limit of a thinning
sequence of convex sets. For planar rhombi and isosceles triangles with area
, it is shown that , and that this bound is sharp.Comment: 12 pages, 4 figure
Shape optimization for monge-ampére equations via domain derivative
In this note we prove that, if Ω is a smooth, strictly convex, open set in R n (n ≥ 2) with given measure, the L 1 norm of the convex solution to the Dirichlet problem detD 2u = 1 in , u = 0 on δΩ, is minimum whenever is an ellipsoid
The Neumann eigenvalue problem for the -Laplacian
The first nontrivial eigenfunction of the Neumann eigenvalue problem for the
-Laplacian, suitable normalized, converges as goes to to a
viscosity solution of an eigenvalue problem for the -Laplacian. We show
among other things that the limit of the eigenvalue, at least for convex sets,
is in fact the first nonzero eigenvalue of the limiting problem. We then derive
a number of consequences, which are nonlinear analogues of well-known
inequalities for the linear (2-)Laplacian.Comment: Corrected few typos. Corollary 5 adde
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