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

Shiqi Ma

TL;DR

This work extends the theory of hyperbolic initial/boundary value problems to non-timelike hypersurfaces by defining a generalized energy $\mathcal{E}(u; \Gamma_S)$ and proving $H^1$ regularity of the solution on $\Gamma_S$ for piecewise $C^1$ surfaces with $|\nabla S|_A\le 1$. Using multiplier methods, the authors derive a sharp bound for the difference between the square roots of energies on $\Gamma_S$ and at $t=0$, and establish energy conservation when the source term and boundary data vanish. The analysis handles a lower-order potential $q$, proves an $L^2$ bound on the conormal derivative, and justifies the results first for $C^2$-smooth solutions and then via an approximation scheme to general data, including compatibility and regularity adjustments. The approach unifies energy considerations on non-timelike hypersurfaces with classical hyperbolic theory and provides tools applicable to inverse problems and relativity-inspired foliations. Overall, the paper significantly broadens trace and energy concepts for hyperbolic PDEs beyond horizontal slices, with concrete quantitative estimates that depend only on the ambient coefficients and geometry.

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.

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

TL;DR

This work extends the theory of hyperbolic initial/boundary value problems to non-timelike hypersurfaces by defining a generalized energy and proving regularity of the solution on for piecewise surfaces with . Using multiplier methods, the authors derive a sharp bound for the difference between the square roots of energies on and at , and establish energy conservation when the source term and boundary data vanish. The analysis handles a lower-order potential , proves an bound on the conormal derivative, and justifies the results first for -smooth solutions and then via an approximation scheme to general data, including compatibility and regularity adjustments. The approach unifies energy considerations on non-timelike hypersurfaces with classical hyperbolic theory and provides tools applicable to inverse problems and relativity-inspired foliations. Overall, the paper significantly broadens trace and energy concepts for hyperbolic PDEs beyond horizontal slices, with concrete quantitative estimates that depend only on the ambient coefficients and geometry.

Abstract

We study second order hyperbolic equations with initial conditions, a nonhomogeneous Dirichlet boundary condition and a source term. We prove the solution possesses regularity on any piecewise -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 . The energy is shown to be conserved when the source term and the boundary datum are both zero. We also obtain an estimate for the normal derivative of the solution. We establish these results for -smooth solutions first by using multiplier methods, then we go back to the original setting using approximation.

Paper Structure

This paper contains 11 sections, 14 theorems, 148 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Motivation
  4. Some preparations
  5. The decomposition of the gradient
  6. The C2 smooth case
  7. Approximation of the solution
  8. Perturbation of the potential
  9. The compatibility issue
  10. The regularity issue
  11. Proofs of the main results

Key Result

Theorem 1.1

Given Assumption asp:nS-HR21, then in the system eq:1-HR21-eq:1con-HR21, the restriction of the solution $u$ on $\Gamma_S$ is in $H^1(\Gamma_S)$, and $\mathcal{E}(u; \Gamma_S)$ is well-defined. Moreover, there hold for some constant $C$ depending only on $A$ and the dimension $n$, and is independent of $\Omega$.

Figures (4)

  • Figure 1: An example of the non-timelike hypersurfaces $\Gamma_S$.
  • Figure 2: Left: the cylinder $Q$. Right: the part of $Q$ underneath $\Gamma_S$.
  • Figure 3: Non-timelike hypersurfaces
  • Figure 4: The observers $X$ and $Y$ stand at the origin $O$ at the time $\tau = 0$. $X$ stands still while $Y$ possesses an instantaneous velocity $v$ pointing to the right. $\{t, x_1\}$ is the instantaneous coordinate of $X$ and $\{\tilde{t}, \tilde{x}_1\}$ is that of $Y$. The time and space axes are perpendicular to each other for both $\{t, x_1\}$ and $\{\tilde{t}, \tilde{x}_1\}$, under the Minkowski metric $-\mathrm{d} t^2 + \mathrm{d} x_1^2$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Remark 2.1
  • ...and 22 more