Regularity and energy of hyperbolic boundary value problems on non-timelike hypersurfaces with lower order terms

TL;DR

This paper extends the analysis of second order hyperbolic equations to non-timelike hypersurfaces, establishing regularity, energy estimates, and conservation properties, with implications for boundary value problems in mathematical physics.

Contribution

It generalizes energy concepts and regularity results for hyperbolic equations to non-timelike hypersurfaces, including new estimates and conservation laws.

Findings

Solutions have $H^1$ regularity on piecewise $C^1$-smooth hypersurfaces.

Energy differences between hypersurfaces and initial plane are bounded and conserved under certain conditions.

Normal derivatives of solutions satisfy $L^2$ estimates.

Abstract

We study second order hyperbolic equations with initial conditions, a nonhomogeneous Dirichlet boundary condition and a source term. We prove the solution possesses $H^1$ regularity on any piecewise $C^1$-smooth non-timelike hypersurfaces. We generalize the notion of energy to these hypersurfaces, and establish an estimate of the difference between square roots of energies on these hypersurfaces and on the initial plane where the time $t = 0$. The energy is shown to be conserved when the source term and the boundary datum are both zero. We also obtain an $L^2$ estimate for the normal derivative of the solution. We establish these results for $C^2$-smooth solutions first by using multiplier methods, then we go back to the original setting using approximation.