The extremal solution for the fractional Laplacian
By Xavier Ros-Oton, J. Serra. 2013
151 citations recorded by the source corpus.
We study the extremal solution for the problem $$(-\Delta )^s u=\lambda f(u)$$(-Δ)su=λf(u) in $$\Omega $$Ω, $$u\equiv 0$$u≡0 in $$\mathbb R ^n\setminus \Omega $$Rn∖Ω, where $$\lambda >0$$λ>0 is a parameter and $$s\in (0,1)$$s∈(0,1). We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions $$n<4s$$n<4s. We also show that, for exponential and power-like nonlinearities, the extremal solution is bounded whenever $$n<10s$$n<10s. In the limit $$s\uparrow 1$$s↑1, $$n<10$$n<10 is optimal. In addition, we show that the extremal solution is $$H^s(\mathbb R ^n)$$Hs(Rn) in any dimension whenever the domain is convex. To obtain some of these results we need $$L^q$$Lq estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with $$L^p$$Lp data. We prove optimal $$L^q$$Lq and $$C^\beta $$Cβ estimates, depending on the value of $$p$$p. These estimates follow from classical embedding results for the Riesz potential in $$\mathbb R ^n$$Rn. Finally, to prove the $$H^s$$Hs regularity of the extremal solution we need an $$L^\infty $$L∞ estimate near the boundary of convex domains, which we obtain via the moving planes method. For it, we use a maximum principle in small domains for integro-differential operators with decreasing kernels.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2ce46a8fd22438c013fffea4199b30bee4e75d12
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In Applied Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations (2015, background): “, [31], [21]) to the nonnegative solution |u|, we infer that |u| > 0 or u ≡ 0 in Ω.”