Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian

Details Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian

2010-04-16

arxiv.org/abs/1004.2866v1

Mathematics

Analysis of PDEs

Scientific paper

We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation $(-\Delta)^{1/2} u=f(u)$ in $\re^n$. Our energy estimates hold for every nonlinearity $f$ and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable. As a consequence, in dimension $n=3$, we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-\Delta u=f(u)$ in $\re^n$.

Affiliated with

Also associated with

No associations

Rating

Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-624372