Sesión Ecuaciones Diferenciales y Probabilidad

Diciembre 12, 16:30 ~ 16:50

Qualitative properties of solutions of some nonlocal elliptic problems in half spaces

Barrios, Begoña

Along this talk we will consider classical solutions of the semilinear fractional problem $$(P)=\left\{ \begin{array}{ll} (-\Delta)^s u = f(u) & \hbox{in }\R^N_+,\\[0.35pc] \ \ u=0 & \hbox{on } \partial \R^N_+, \end{array} \right.$$ where $N\ge 2$, $\R^N_+=\{x=(x',x_N)\in \R^N:\ x_N>0\}$ is the half-space, $f$ is a given function and $(-\Delta)^s$, $00 \quad \hbox{in } \R^N_+. $ This is in contrast with previously known results for the local case $s=1$ where nonnegative solutions which are not positive do exist and the monotonicity property above is not known to hold in general even for positive solutions when $f(0)<0$ (see for instance \cite{one, FS}). } \item{If $f$ is a locally Lipschitz nonlinearity we prove, using a complete characterization of one-dimensional solutions, that if $u$ is a bounded solution with $\rho:=\sup_{\mathbb{R}^N}u$ verifying $f(\rho)=0$, then, as in the classical case (\cite{BCN93}), $u$ is necessarily one-dimensional.} \end{itemize} The result presented in this talk can be found in \cite{B1, B2}. \begin{thebibliography}{99} \footnotesize \bibitem{B1} B. Barrios, L. Del Pezzo, J. García-Melián, A. Quaas, {\em Monotonicity of solutions for some nonlocal elliptic problems in half-spaces} Calc. Var. Partial Differential Equations {\bf 56} (2017), no. 2, 56-39 \bibitem{B2} B. Barrios, L. Del Pezzo, J. García-Melián, A. Quaas, {\em Symmetry results in the half space for a semi-linear fractional Laplace equation through a one-dimensional analysis}, Submitted. arXiv:1704.02597 \bibitem{BCN93} H. Berestycki, L. Caffarelli, L. Nirenberg, {\em Symmetry for elliptic equations in a half space}, RMA Res. Notes Appl. Math. {\bf 29} (1993), 27-42. \bibitem{one} C. Cortázar, M. Elgueta, J. García-Melián, {\em Nonnegative solutions of semilinear elliptic equations in half-spaces}, J. Math. Pures Appl. (2016), in press. \bibitem{FS} A. Farina, B. Sciunzi, {\em Qualitative properties and classification of nonnegative solutions to $-\Delta u = f(u)$ in unbounded domains when $f(0) < 0$}, Rev. Mat. Iberoam. (2016), in press. \end{thebibliography}

Autores: Barrios, Begoña / Del Pezzo, Leandro / García-Melián, Jorge J. / Quaas, Alexander.