123 research outputs found
Uniform Lipschitz regularity of flat segregated interfaces in a singularly perturbed problem
For the singularly perturbed system we prove that
flat interfaces are uniformly Lipschitz. As a byproduct of the proof we also
obtain the optimal lower bound near the flat interfaces,
Comment: 11 page
Some remarks on the structure of finite Morse index solutions to the Allen-Cahn equation in
For a solution of the Allen-Cahn equation in , under the
natural linear growth energy bound, we show that the blowing down limit is
unique. Furthermore, if the solution has finite Morse index, the blowing down
limit satisfies the multiplicity one property.Comment: 13 page
On Serrin's overdetermined problem and a conjecture of Berestycki, Caffarelli and Nirenberg
This paper concerns rigidity results to Serrin's overdetermined problem in an
epigraph We prove that up to isometry the epigraph must be an half
space and that the solution must be one-dimensional, provided that one of
the following assumptions are satisfied: either ; or is
globally Lipschitz, or and in
. In view of the counterexample constructed in \cite{DPW} in dimensions
this result is optimal. This partially answers a conjecture of
Berestycki, Caffarelli and Nirenberg \cite{BCN}.Comment: comments are welcom
On the uniqueness of solutions of an nonlocal elliptic system
We consider the following elliptic system with fractional laplacian
-(-\Delta)^su=uv^2,\ \ -(-\Delta)^sv=vu^2,\ \ u,v>0 \ \mbox{on}\ \R^n, where
and is the -Lapalcian. We first prove that all
positive solutions must have polynomial bound. Then we use the Almgren
monotonicity formula to perform a blown-down analysis to -harmonic
functions. Finally we use the method of moving planes to prove the uniqueness
of the one dimensional profile, up to translation and scaling.Comment: 35 page
On solutions with polynomial growth to an autonomous nonlinear elliptic problem
We study the following nonlinear elliptic problem [-\Delta u =F^{'} (u) in
{\mathbb R}^n] where is a periodic function. Moser (1986) showed that
for any minimal and nonself-intersecting solution, there exist and such that [(*) | u- \alpha \cdot x | \leq C.] He also
showed the existence of solutions with any prescribed . In this note, we first prove that any solution satisfying (*) with
nonzero vector must be one dimensional. Then we show that in , for any positive integer there exists a solution with
polynomial growth .Comment: all comments are welcom
Finite Morse index implies finite ends
We prove that finite Morse index solutions to the Allen-Cahn equation in
have {\bf finitely many ends} and {\bf linear energy growth}. The main
tool is a {\bf curvature decay estimate} on level sets of these finite Morse
index solutions, which in turn is reduced to a problem on the uniform second
order regularity of clustering interfaces for the singularly perturbed
Allen-Cahn equation in . Using an indirect blow-up technique, in the
spirit of the classical Colding-Minicozzi theory in minimal surfaces, we show
that the {\bf obstruction} to the uniform second order regularity of clustering
interfaces in is associated to the existence of nontrivial entire
solutions to a (finite or infinite) {\bf Toda system} in . For finite
Morse index solutions in , we show that this obstruction does not exist
by using information on stable solutions of the Toda system.Comment: 66 page
Global minimizers of the Allen-Cahn equation in dimension
We prove the existence of global minimizers of Allen-Cahn equation in
dimensions and above. More precisely, given any strictly area-minimizing
Lawson's cones, there are global minimizers whose nodal sets are asymptotic to
the cones. As a consequence of Jerison-Monneau's program we establish the
existence of many counter-examples to the De Giorgi's conjecture different from
the Bombierie-De Giorgi-Giusti minimal graph.Comment: 21 page
On one phase free boundary problem in
We construct a smooth axially symmetric solution to the classical one phase
free boundary problem in . Its free boundary is of
\textquotedblleft catenoid\textquotedblright\ type. This is a higher
dimensional analogy of the Hauswirth-Helein-Pacard solution in (\cite{Pacard}). The existence of such solution is conjectured in \cite
[Remark 2.4]{Pacard}.Comment: 42 page
A new proof of Savin's theorem on Allen-Cahn equations
In this paper we establish an improvement of tilt-excess decay estimate for
the Allen-Cahn equation, and use this to give a new proof of Savin's theorem on
the uniform regularity of flat level sets, which then implies
the one dimensional symmetry of minimizers in for .
This generalizes Allard's -regularity theorem for stationary
varifolds to the setting of Allen-Cahn equations.Comment: 63 pages, author's final version. To appear in J. Eur. Math. So
A generalized one phase Stefan problem as a vanishing viscosity limit
We study the vanishing viscosity limit of a nonlinear diffusion equation
describing chemical reaction interface or the spatial segregation interface of
competing species, where the diffusion rate for the negative part of the
solution converges to zero. As in the standard one phase Stefan problem, we
prove that the positive part of the solution converges uniformly to the
solution of a generalized one phase Stefan problem. This information is then
employed to determine the limiting equation for the negative part, which is an
ordinary differential equation.Comment: 18 pages, title changed, accepted for publication in Portugaliae
Mathematic
- …