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The present paper is concerned with the infimum of the norm of potentials for Sturm–Liouville eigenvalue problems with Dirichlet boundary condition such that the first two eigenvalues are known. The explicit quantity of the infimum is given by the two eigenvalues.

In this article, we establish a Lyapunov-type inequality for the following extremal Pucci’s equation:

$$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}{\mathcal{M}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC}}^{+}(D^{2}u)+b(x)|Du|+a(x)u=0 & \text{in}~\unicode[STIX]{x1D6FA},\\ u=0 & \text{on}~\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$

where

$\unicode[STIX]{x1D6FA}$ is a smooth bounded domain in

$\mathbb{R}^{N}$,

$N\geq 2$. This work generalizes the well-known works on the Lyapunov inequality for extremal Pucci’s equations with gradient nonlinearity.

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Optimal inverse problems of potentials for two given eigenvalues of Sturm–Liouville problems

LYAPUNOV-TYPE INEQUALITY FOR EXTREMAL PUCCI’S EQUATIONS

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2 results

Optimal inverse problems of potentials for two given eigenvalues of Sturm–Liouville problems

LYAPUNOV-TYPE INEQUALITY FOR EXTREMAL PUCCI’S EQUATIONS