Seminario de Ecuaciones Diferenciales – Begoña Barrios Barrera


Event Details


Título: Monotonicity of solutions for some nonlocal elliptic problems in half-spaces


Conferenciante: Begoña Barrios Barrera (Universidad de la Laguna)


Resumen:
Along this talk we will consider classical solutions of the semilinear
fractional problem
$$\left\{
\begin{array}{ll}
(-\Delta)^s u = f(u) & \hbox{in }\mathbb{R}^N_+,\\[0.35pc]
\ \ u=0 & \hbox{on } \partial \mathbb{R}^N_+,
\end{array}
\right.$$
where $(-\Delta)^s$, $0 < s < 1$, stands for the fractional Laplacian, $N \ge 2$, $\mathbb{R}^N_+ = \{x=(x',x_N)\in \mathbb{R}^N:\ x_N>0\}$ is the half-space and $f \in C^1$ is a given function.
With no additional restriction on the function $f$, we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in $\mathbb{R}^N_+$ and verify
$$
\frac{\partial u}{\partial x_N}>0 \quad \hbox{in } \mathbb{R}^N_+.
$$
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 [1,2,3]).

This work is joint with L. Del Pezzo (UBA, Argentina), J. García-Melián (ULL) and A. Quaas (Universidad Técnica Federico Santa María, Chile).

Referencias

[1] H. Berestycki, L. Caffarelli, L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 69--94.

[2] Cortázar, M. Elgueta, J. García-Melián, Nonnegative solutions of semilinear elliptic equations in half-spaces, J. Math. Pures Appl. (2016), in press.

[3] A. Farina, B. Sciunzi, 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.


11 de octubre de 2016, 13:00, Seminario 1ª planta IEMath-GR


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