From zero solutions with partially bounded supports to solvability and Runge approximation for partial differential equations

Technical Summary: From Zero Solutions with Partially Bounded Supports to Solvability and Runge Approximation for Partial Differential Equations

Problem Statement
The paper addresses two fundamental problems in the theory of linear partial differential operators (PDOs) with constant coefficients:

1. Solvability: Characterizing the geometric conditions under which a PDO $P(D)$ is surjective on spaces of smooth functions, distributions, or local subspaces (specifically, determining when an open set $X \subset \mathbb{R}^d$ is P-convex for supports). While classical results exist for elliptic operators and specific cases (e.g., the wave operator in two dimensions), a general geometric characterization for non-elliptic operators in arbitrary dimensions remains elusive.
2. Runge Approximation: Determining geometric criteria for the approximability of solutions to homogeneous equations on a smaller set by solutions on a larger set (Runge pairs) across various function spaces, including smooth functions, distributions, and smooth Whitney jets. This is particularly relevant for systems of equations arising in fluid dynamics.

The authors note that the existence of non-trivial "zero solutions" (solutions to $P(D)u=0$) with specific support properties is a prerequisite for establishing necessary conditions for surjectivity and for proving Runge-type approximation theorems.

Methodology
The paper employs a constructive approach centered on the existence of specific zero solutions, followed by functional analytic arguments to derive geometric characterizations.

1. Construction of Zero Solutions (Section 2):

   • The authors first establish the existence of non-trivial smooth zero solutions with partially bounded supports for a broad class of operators.
   • They utilize Gevrey classes and solve a Cauchy problem for operators of the form $P(\xi', \xi_d) = \xi_d^m + \sum Q_k(\xi')\xi_d^k$ (Theorem 2). This involves constructing solutions via series expansions in Gevrey spaces and proving convergence using estimates on the derivatives of the coefficients.
   • Using convolution techniques and Lemma 4 (properties of convolution with partially bounded supports), they extend these results to operators that are orthogonally degenerate on a linear subspace $Z$. Specifically, they prove the existence of a solution $v$ whose support is bounded in directions orthogonal to $Z$ (Theorem 9).
   • This class of operators includes all semi-elliptic operators and operators whose principal part is the wave operator (for $d \ge 2$).

2. Geometric Characterization of P-Convexity (Section 3):

   • Leveraging the constructed zero solutions, the authors derive a necessary condition for $P$-convexity: the boundary distance function $d_X$ must satisfy the minimum principle in every affine subspace parallel to the orthogonal complement of the characteristic cone ($Z^\perp$).
   • For operators orthogonally degenerate on their linear characteristic cone, this condition is shown to be sufficient.
   • The authors introduce geometric concepts such as critical points and points of inner support with respect to a subspace to reformulate the minimum principle condition. They prove that the absence of such points is equivalent to $P$-convexity for this class of operators (Corollary 21).
   • They further establish that for semi-elliptic operators, the kernel of the operator satisfies condition ($\Omega$) (a property of Fréchet spaces related to splitting theory), which implies surjectivity on spaces of vector-valued smooth functions and resolves parameter dependence problems.

3. Runge Approximation (Sections 4 & 5):

   • The existence of zero solutions with partially bounded supports is used to derive necessary conditions for Runge pairs. Specifically, if $(F_1, F_2)$ is a Runge pair, $F_2$ cannot contain a bounded connected component of $(\mathbb{R}^d \setminus F_1) \cap (x + Z^\perp)$.
   • For operators orthogonally degenerate on their characteristic cone, these conditions are shown to be sufficient (Theorem 33).
   • The framework is extended to square systems of constant coefficient PDOs. By utilizing the adjugate matrix and the determinant of the system, the authors reduce the approximation problem for systems to the scalar case (Theorem 39).
   • They apply these general results to specific systems:
     • Beltrami fields (eigenfunctions of the curl operator).
     • The 3D unsteady Stokes system.

Key Contributions and Results

• Existence of Partially Bounded Zero Solutions: The paper proves the existence of non-trivial smooth solutions with supports bounded in specific directions for any constant coefficient operator that is orthogonally degenerate on a linear subspace (Theorem 9). This generalizes previous results (e.g., Cohoon's) and applies to semi-elliptic operators and wave-type operators.
• Geometric Characterization of Solvability: For the class of operators orthogonally degenerate on their linear characteristic cone (which includes all semi-elliptic operators), the paper provides a complete geometric characterization of $P$-convexity for supports. This is expressed via the minimum principle of the boundary distance function in characteristic hyperplanes and the non-existence of critical points or points of inner support (Corollary 21).
• Condition ($\Omega$) and Parameter Dependence: The authors prove that the kernel of every surjective semi-elliptic operator satisfies condition ($\Omega$). Consequently, by the splitting theorem of Vogt and Wagner, these operators are surjective on spaces of vector-valued smooth functions, solving the parameter dependence problem for these equations.
• Runge Approximation for Systems: A general framework is developed for Runge approximation of square systems. This recovers known results for Beltrami fields and the 3D unsteady Stokes system (previously studied by Higaki/Sueur and Enciso/Peralta-Salas) via alternative methods.
• Whitney Jets and Boundary Approximation: The results are extended to spaces of smooth Whitney jets on closed sets. This yields approximation theorems up to the boundary for domains with Hölder continuous boundaries, complementing existing results for open sets.

Significance and Claims
The paper claims to extend classical geometric ideas of Hörmander regarding $P$-convexity to a large class of non-elliptic operators, specifically those orthogonally degenerate on their characteristic cones. By establishing the existence of zero solutions with partially bounded supports, the authors provide a unified tool to tackle both solvability and approximation problems.

The work is significant for:

1. Unification: It offers a general framework that covers semi-elliptic operators, wave operators, and various systems, unifying previously disparate results.
2. New Geometric Criteria: It provides explicit geometric conditions (minimum principle, absence of critical points) for solvability and approximation in dimensions where such characterizations were previously incomplete.
3. Applications to Fluid Dynamics: The results directly apply to the Stokes and Beltrami systems, offering new approximation theorems (including up to the boundary for Hölder domains) that are relevant to the study of fluid equations.
4. Functional Analysis: The proof that semi-elliptic kernels satisfy condition ($\Omega$) resolves a specific question regarding parameter dependence for these operators.

The authors emphasize that while the geometric characterization of $P$-convexity remains open for general operators in arbitrary dimensions, their results cover a "broad class" including all semi-elliptic operators, thereby significantly extending the known landscape of solvability and approximation theory.