Application of the Shape Design Method to Hidden Regularity of Degenerate Hyperbolic Equations with Degenerate Boundary

Abstract: This paper investigates the well-posedness and hidden regularity of boundary-degenerate hyperbolic equations in a two-dimensional setting. The shape design method is employed, which approximates the original degenerate problem by a family of uniformly elliptic problems on truncated subdomains and passes to the limit through uniform estimates. Within this framework, the existence and uniqueness of weak solutions are established in suitable weighted Sobolev spaces. A hidden regularity estimate for the conormal derivative on the nondegenerate portion of the boundary is obtained, providing a uniform bound in terms of the natural weighted energy norms of the initial data and source term. This estimate yields the boundary trace information essential for observability and controllability of degenerate hyperbolic systems.

• The paper proves hidden regularity for weak solutions by showing that the conormal derivative on the nondegenerate boundary belongs to L²(0,T;L²(Γ)) and is bounded by the weighted norms of the initial data and forcing.
• The paper applies the shape design method to truncated domains Ωδ, obtains estimates uniform in δ through a Hardy inequality, and identifies the limiting boundary trace using weak convergence, compactness, and duality.
• The result provides essential trace information for future boundary observability and exact controllability analyses, while leaving the degenerate boundary, broader geometries, and control inequalities as open problems.

Problem setting and motivation

The paper studies the degenerate hyperbolic initial-boundary value problem

$$\partial_{tt}y - \operatorname{div}(A\nabla y) = f \quad \text{in } Q = \Omega\times(0,T), \qquad y = 0 \text{ on } \partial Q,$$

with initial data $y(0)=y^0$, $\partial_t y(0)=y^1$, on the rectangle $\Omega = (-1,1)\times(0,1)$, where the coefficient matrix is $A = \operatorname{diag}(1, x_2^\alpha)$ with $\alpha\in(0,1)$. The weight $w = x_2^\alpha$ vanishes on the boundary portion $\Gamma_2^0 = \{x_2=0\}$, so the operator is uniformly elliptic away from $\Gamma_2^0$ but degenerates there. The nondegenerate boundary portion is denoted $\Gamma = \partial\Omega\setminus\Gamma_2^0$, and the conormal derivative is $y(0)=y^0$0.

The motivation is twofold. First, boundary degeneracy invalidates the standard energy machinery, trace theorems, and multiplier identities used in boundary control of the wave equation. Second, the paper targets hidden regularity: the fact that weak hyperbolic solutions with limited interior regularity nonetheless possess a conormal derivative trace on the boundary satisfying an $y(0)=y^0$1 estimate — the essential ingredient for boundary observability and, by duality, exact controllability. In the degenerate setting the question is whether such a trace exists on $y(0)=y^0$2 and admits a uniform bound in terms of the natural weighted norms of the data. Rather than multiplier methods or Carleman estimates (both requiring delicate pointwise weighted estimates), the authors adopt the shape design method: approximate the degenerate problem by a family of uniformly elliptic problems on truncated subdomains $y(0)=y^0$3, $y(0)=y^0$4, and pass to the limit $y(0)=y^0$5 via $y(0)=y^0$6-uniform estimates.

Weighted function spaces and spectral structure

The natural energy space is the weighted Sobolev space $y(0)=y^0$7 with inner product involving $y(0)=y^0$8, and $y(0)=y^0$9 its closure of $\partial_t y(0)=y^1$0; $\partial_t y(0)=y^1$1 is defined by the condition $\partial_t y(0)=y^1$2 for $\partial_t y(0)=y^1$3, with domain $\partial_t y(0)=y^1$4. Two structural results anchor the analysis:

• Hardy-type inequality: for all $\partial_t y(0)=y^1$5 and $\partial_t y(0)=y^1$6,

$\partial_t y(0)=y^1$7

proved by a one-dimensional integration argument along the degenerate direction and optimized at $\partial_t y(0)=y^1$8. This yields the norm equivalence on $\partial_t y(0)=y^1$9 and the Poincaré-type lower bound $\Omega = (-1,1)\times(0,1)$0 for the first eigenvalue.

• Compact embedding $\Omega = (-1,1)\times(0,1)$1, proved by splitting the domain into a strip near the degeneracy (where the Hardy inequality gives smallness) and the remaining region (where classical Sobolev compactness applies).

These two facts give the operator $\Omega = (-1,1)\times(0,1)$2 a discrete spectrum $\Omega = (-1,1)\times(0,1)$3 with an orthonormal eigenbasis $\Omega = (-1,1)\times(0,1)$4 of $\Omega = (-1,1)\times(0,1)$5, and a spectral characterization: $\Omega = (-1,1)\times(0,1)$6 belongs to $\Omega = (-1,1)\times(0,1)$7 exactly when $\Omega = (-1,1)\times(0,1)$8, with $\Omega = (-1,1)\times(0,1)$9 and equality of norms. This diagonalization underpins both the Galerkin well-posedness argument and the higher-regularity estimates.

Well-posedness and preliminary trace estimates

The main well-posedness theorem establishes, for $A = \operatorname{diag}(1, x_2^\alpha)$0, $A = \operatorname{diag}(1, x_2^\alpha)$1, $A = \operatorname{diag}(1, x_2^\alpha)$2, a unique weak solution in $A = \operatorname{diag}(1, x_2^\alpha)$3, via a Galerkin scheme in the eigenbasis with energy and Gronwall estimates. Crucially, the energy estimate already contains a boundary trace term:

$A = \operatorname{diag}(1, x_2^\alpha)$4

with $A = \operatorname{diag}(1, x_2^\alpha)$5 depending only on $A = \operatorname{diag}(1, x_2^\alpha)$6 and $A = \operatorname{diag}(1, x_2^\alpha)$7. The proof of this trace term uses a cutoff multiplier $A = \operatorname{diag}(1, x_2^\alpha)$8 (with $A = \operatorname{diag}(1, x_2^\alpha)$9 localized at distance $\alpha\in(0,1)$0 from the degenerate edge, $\alpha\in(0,1)$1) and an integration-by-parts identity; the degenerate boundary $\alpha\in(0,1)$2 is cut away, so the argument only controls the trace on $\alpha\in(0,1)$3 at positive distance from the degeneracy. Under the stronger assumptions $\alpha\in(0,1)$4, $\alpha\in(0,1)$5, $\alpha\in(0,1)$6, an improved estimate holds with $\alpha\in(0,1)$7, $\alpha\in(0,1)$8, $\alpha\in(0,1)$9, local $w = x_2^\alpha$0 bounds, and the trace of $w = x_2^\alpha$1; the $w = x_2^\alpha$2 bound is obtained by cutting off with a test function vanishing near the degeneracy and applying classical interior elliptic regularity on the resulting uniformly elliptic problem. A remark extends the local $w = x_2^\alpha$3 estimate to $w = x_2^\alpha$4 for any $w = x_2^\alpha$5, with a constant $w = x_2^\alpha$6 that may blow up as $w = x_2^\alpha$7 — a point the authors state explicitly.

An equivalent transposition formulation of the weak solution is also established, which is what allows the shape design convergence argument to identify limits.

Shape design approximation and uniform estimates

On each truncated domain $w = x_2^\alpha$8, the authors consider the uniformly elliptic problem with operator $w = x_2^\alpha$9 and data $\Gamma_2^0 = \{x_2=0\}$0. Since $\Gamma_2^0 = \{x_2=0\}$1 is bounded above and below on $\Gamma_2^0 = \{x_2=0\}$2, one has $\Gamma_2^0 = \{x_2=0\}$3, and the entire well-posedness and trace theory of the previous section transfers verbatim; the key observation is that the constants in the energy, spectral, and trace estimates are independent of $\Gamma_2^0 = \{x_2=0\}$4, because they depend only on the Hardy constant $\Gamma_2^0 = \{x_2=0\}$5, which is $\Gamma_2^0 = \{x_2=0\}$6-uniform. The trace estimate on the full lateral boundary $\Gamma_2^0 = \{x_2=0\}$7 is derived by a cutoff multiplier $\Gamma_2^0 = \{x_2=0\}$8 and the same integration-by-parts identity.

The convergence theorem is the technical core. With $\Gamma_2^0 = \{x_2=0\}$9 the zero extension of $\Gamma_2^0$0 to $\Gamma_2^0$1 (which preserves all weighted norms), the results are:

1. For $\Gamma_2^0$2, $\Gamma_2^0$3, $\Gamma_2^0$4, up to a subsequence, $\Gamma_2^0$5 weakly in $\Gamma_2^0$6 and $\Gamma_2^0$7 weakly in $\Gamma_2^0$8; by compactness, $\Gamma_2^0$9 strongly in $\Gamma = \partial\Omega\setminus\Gamma_2^0$0.
2. If moreover $\Gamma = \partial\Omega\setminus\Gamma_2^0$1 and $\Gamma = \partial\Omega\setminus\Gamma_2^0$2, then for each fixed $\Gamma = \partial\Omega\setminus\Gamma_2^0$3,

$\Gamma = \partial\Omega\setminus\Gamma_2^0$4

and the uniform bound

$\Gamma = \partial\Omega\setminus\Gamma_2^0$5

The identification argument is a duality computation: testing the truncated equation with a function $\Gamma = \partial\Omega\setminus\Gamma_2^0$6 compactly supported in $\Gamma = \partial\Omega\setminus\Gamma_2^0$7, passing to the limit using weak convergence of $\Gamma = \partial\Omega\setminus\Gamma_2^0$8 and the compactness of the embedding of $\Gamma = \partial\Omega\setminus\Gamma_2^0$9 into $y(0)=y^0$00, and comparing against the same identity for the original solution. The authors are careful to note two caveats. First, the strong convergence of normal derivatives holds on $y(0)=y^0$01 for fixed $y(0)=y^0$02, not necessarily on all of $y(0)=y^0$03; the estimate on all of $y(0)=y^0$04 follows instead from the $y(0)=y^0$05-uniform trace bound. Second, the higher-regularity constant $y(0)=y^0$06 in the truncated $y(0)=y^0$07 estimate may diverge as $y(0)=y^0$08, so the convergence argument relies on the low-regularity uniform estimates rather than on uniform $y(0)=y^0$09 bounds.

Full hidden regularity by density

The final step removes the smoothness assumptions on the data. Approximating $y(0)=y^0$10 strongly in the natural norms by data in $y(0)=y^0$11, applying the uniform trace estimate to each approximate solution, and using the Lipschitz dependence of the trace on the data (via the cutoff-multiplier estimate on $y(0)=y^0$12, $y(0)=y^0$13, with constant independent of the approximation index), the authors conclude:

Hidden regularity estimate. For every weak solution with $y(0)=y^0$14, $y(0)=y^0$15, $y(0)=y^0$16,

$y(0)=y^0$17

with $y(0)=y^0$18 depending only on $y(0)=y^0$19 and $y(0)=y^0$20. The conormal trace on the nondegenerate boundary is thus well defined and continuously controlled by the weighted energy norm of the data — precisely the estimate required to run multiplier-based observability arguments for boundary controllability of degenerate hyperbolic systems. A remark emphasizes that this hidden regularity cannot be obtained directly from the well-posedness theorem alone; it genuinely requires the shape design approximation.

Limitations and open questions

The result is established for a specific two-dimensional model: rectangular domain, diagonal coefficient with power-type degeneracy $y(0)=y^0$21, $y(0)=y^0$22, and homogeneous Dirichlet conditions. The authors assert that the framework extends to time-dependent coefficients, nonlinearities, and more general degeneracy structures in higher dimensions, but no such extension is carried out here. The trace estimate concerns only the nondegenerate boundary $y(0)=y^0$23; whether a meaningful conormal trace exists on the degenerate portion $y(0)=y^0$24, and at what weighted rate, is not addressed. As noted above, strong convergence of normal derivatives $y(0)=y^0$25 on the full boundary is not established, and the constant in the local $y(0)=y^0$26 estimate may blow up as the truncation approaches the degeneracy. The connection to actual observability inequalities — the stated motivation — is asserted rather than proved; deriving the corresponding boundary observability estimate and controllability result from the hidden regularity theorem remains open.

Conclusion

The paper establishes well-posedness of a boundary-degenerate hyperbolic equation in weighted Sobolev spaces and, via the shape design method of truncated uniformly elliptic approximations with $y(0)=y^0$27-uniform estimates, proves a hidden regularity estimate bounding the conormal derivative trace on the nondegenerate boundary by the weighted energy norms of the data. The main technical contributions are the $y(0)=y^0$28-independence of the energy, spectral, and trace constants — traceable to the uniform Hardy constant — and the duality-based identification of the limiting normal derivative. The estimate supplies the boundary trace information prerequisite for multiplier-based observability and controllability analysis of degenerate hyperbolic systems, though the control-theoretic consequences and extensions beyond the model geometry remain to be developed.