Theoretical analysis reveals Steklov spectral determination across convex polygons, indicating that boundary eigenvalues distinguish corners from smooth geometries.
Key Points
To determine whether the geometric shape of convex polygons is uniquely identified by their Steklov spectra and to evaluate whether the spectrum can detect boundary corners.
Applied an algebraic framework based on characteristic polynomials to analyze Steklov eigenvalue spectra.
Evaluated geometric uniqueness across triangles, specific convex quadrilaterals (rectangles, parallelograms, kites), regular n-gons, and smoothly bounded simply-connected domains.
Proved that almost all triangles are uniquely determined by their Steklov spectra within the class of all triangles.
Demonstrated that rectangles, parallelograms, and kites are spectrally determined up to at most three geometric possibilities, with regular n-gons uniquely identified within selected polygon classes.
Established that triangles and quadrilaterals are spectrally distinguished from smoothly bounded domains, confirming that the Steklov spectrum successfully detects corners and restricts edge lengths in higher-order n-gons.