1,346 research outputs found
Counterexamples to Symmetry for Partially Overdetermined Elliptic Problems
We exhibit several counterexamples showing that the famous Serrin's symmetry
result for semilinear elliptic overdetermined problems may not hold for
partially overdetermined problems, that is when both Dirichlet and Neumann
boundary conditions are prescribed only on part of the boundary. Our
counterexamples enlighten subsequent positive symmetry results obtained by the
first two authors for such partially overdetermined systems and justify their
assumptions as well
Some properties of solutions to weakly hypoelliptic equations
A linear different operator L is called weakly hypoelliptic if any local
solution u of Lu=0 is smooth. We allow for systems, that is, the coefficients
may be matrices, not necessarily of square size. This is a huge class of
important operators which cover all elliptic, overdetermined elliptic,
subelliptic and parabolic equations.
We extend several classical theorems from complex analysis to solutions of
any weakly hypoelliptic equation: the Montel theorem providing convergent
subsequences, the Vitali theorem ensuring convergence of a given sequence and
Riemann's first removable singularity theorem. In the case of constant
coefficients we show that Liouville's theorem holds, any bounded solution must
be constant and any L^p-solution must vanish.Comment: published version (up to cosmetic issues
Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds
Our work proposes a unified approach to three different topics in a general
Riemannian setting: splitting theorems, symmetry results and overdetermined
elliptic problems. By the existence of a stable solution to the semilinear
equation on a Riemannian manifold with non-negative Ricci
curvature, we are able to classify both the solution and the manifold. We also
discuss the classification of monotone (with respect to the direction of some
Killing vector field) solutions, in the spirit of a conjecture of De Giorgi,
and the rigidity features for overdetermined elliptic problems on submanifolds
with boundary
On algebraic integrability of the deformed elliptic Calogero--Moser problem
Algebraic integrability of the elliptic Calogero--Moser quantum problem
related to the deformed root systems \pbf{A_{2}(2)} is proved. Explicit
formulae for integrals are found
- …