Eigenvalue problems in

In this article we characterize the L mathvariant="normal">∞ \mathrm {L}^\infty eigenvalue problem associated to Rayleigh quotient alttext="double-vertical-bar nabla u double-vertical-bar Subscript Sub infinity Baseline slash fence="false" stretchy="false">‖ mathvariant="normal">∇<!-- ∇ <mml:mi>u class="MJX-TeXAtom-CLOSE"> stretchy="true">/ class="MJX-TeXAtom-OPEN"> encoding="application/x-tex">\left .{\|\nabla u\|_{\mathrm {L}^\infty }}\middle /{\|u\|_\infty }\right . and relate it a divergence-form PDE, similarly what is known for p"> p {L}^p problems alttext="p"> encoding="application/x-tex">p -Laplacian alttext="p greater-than &gt; encoding="application/x-tex">p&gt;\infty . Contrary existing methods, which study -problems as limits of right-arrow stretchy="false">→<!-- → encoding="application/x-tex">p\to \infty , develop novel framework analyzing limiting directly using convex analysis geometric measure theory. For this, derive fine characterization subdifferential Lipschitz-constant-functional alttext="u from bar stretchy="false">↦<!-- ↦ encoding="application/x-tex">u\mapsto \|\nabla } We show that takes form alttext="lamda nu equals minus d i v left-parenthesis tau right-parenthesis"> λ<!-- λ <mml:mi>ν<!-- ν <mml:mo>= −<!-- − <mml:mi>div ⁡<!-- ⁡ stretchy="false">( τ<!-- τ stretchy="false">) encoding="application/x-tex">\lambda \nu =-\operatorname {div}(\tau \nabla _\tau u) where alttext="nu"> encoding="application/x-tex">\nu alttext="tau"> encoding="application/x-tex">\tau are non-negative measures concentrated alttext="StartAbsoluteValue EndAbsoluteValue"> stretchy="false">| encoding="application/x-tex">|u| respectively encoding="application/x-tex">|\nabla u| maximal, alttext="nabla u"> encoding="application/x-tex">\nabla u tangential gradient alttext="u"> encoding="application/x-tex">u with respect Lastly, investigate dual whose minimizers solve an optimal transport generalized Kantorovich–Rubinstein norm. Our results apply all stationary points quotient, including ground states, harmonic potentials, distance functions, etc., generalize in literature.