Lyapunov-type inequalities for fractional partial differential equations

Abstract We present Lyapunov inequalities for the fractional p-Laplacian ( − Δ p ) s , 1 p ∞ , s ∈ ( 0 , 1 ) , in an open bounded set Ω ⊂ R N , under homogeneous Dirichlet boundary conditions. We discuss two cases, the case s p > N and the case s p N . We use the obtained inequalities to provide lower bounds for the first eigenvalue of the fractional p-Laplacian. To the best of our knowledge, this is the first work dealing with Lyapunov-type inequalities for fractional partial differential equations.

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