An $L^p$-Theory for Time-Periodic Mixed-Order Partial Differential Equations under General Boundary Conditions

Technical Summary: An $L^p$-Theory for Time-Periodic Mixed-Order Partial Differential Equations Under General Boundary Conditions

Problem Statement
The paper addresses the well-posedness of time-periodic boundary value problems associated with partial differential equations (PDEs) and systems of mixed order. While a comprehensive well-posedness theory exists for elliptic and parabolic operators (e.g., via Agmon-Douglis-Nirenberg or Lopatinski˘ı–Shapiro conditions), such a theory has been largely unavailable for operators with genuinely mixed scaling properties. These operators, where competing scalings are encoded geometrically by a Newton polygon, often arise in complex physical models. Previous work by the authors established an $L^2$-theory for a restricted class of such operators, but it was noted that classical Lopatinski˘ı–Shapiro conditions are generally insufficient for well-posedness in this setting. The current work aims to develop a comprehensive $L^p$-theory ($p \in (1, \infty)$) for a broad class of mixed-order boundary value problems on the time-periodic half-space $\mathbb{T} \times \mathbb{R}^n_+$, accommodating general boundary operators that may involve both spatial and temporal derivatives.

Methodology
The authors construct the theory through three interconnected pillars:

1. Unified Function Space Scales:
   The authors extend the theory of Newton polygon spaces by introducing a unified scale of potential spaces generated by order functions. Unlike classical Newton polygons, which are restricted to convex sets in the $(|\xi|, \tau)$-plane, order functions allow for a more flexible algebraic framework where operations correspond to ordinary addition rather than Minkowski sums. Crucially, the authors introduce the concept of a Newton polygon scale, which includes classical Newton polygon spaces as well as hybrid spaces arising from real interpolation (involving both Besov and Bessel-potential regularity). This abstraction is necessary because trace spaces for mixed-order operators are generally not pure Newton polygon spaces but rather such hybrid interpolation spaces.

2. Abstract Boundary Value Framework:
   The paper develops an abstract operator framework that separates the analytic properties of the differential operator from the structure of the function spaces. The core mechanism involves:

   • Defining an abstract trace tuple $(U, \dot{U}, \mathcal{U}, \text{Tr}, E)$, where $\dot{U}$ is the kernel of the trace operator $\text{Tr}$, and $E$ is a right inverse.
   • Introducing admissible operators, which factorize into components that preserve half-space support and components invertible on the kernel of the trace operator.
   • Formulating an abstract complementing boundary condition. Well-posedness is characterized by the invertibility of a "complemented operator" $(B, C)$, where $B$ is the given boundary operator and $C$ is a complementary operator constructed from the trace of the solution. This condition ensures that the combined operator is injective, thereby identifying the precise compatibility conditions required for the data.

3. Trace Theory for Mixed-Order Spaces:
   A significant technical contribution is the development of an $L^p$-trace theory for Newton polygon spaces. The authors explicitly identify the trace spaces $T^{\mu, p}_\perp(\mathbb{R}^{n-1})$ associated with restricted Newton polygon spaces. They demonstrate that for $p \neq 2$, these trace spaces are not Newton polygon spaces but can be represented as real interpolation spaces between closely related Newton polygon spaces. This characterization allows for the precise description of the data spaces and compatibility conditions.

Key Results

• General Well-Posedness Theorem (Theorem 4.36): The authors establish that for an admissible operator $P$ and boundary operators $B_i$, the boundary value problem is well-posed in $L^p$-based spaces if the associated complemented boundary matrix $BC$ satisfies an invertibility condition (specifically, that $BC$ is a mixed-order system and its determinant is non-vanishing). The theorem explicitly identifies the admissible data space, including the necessary compatibility conditions induced by the boundary operators.
• Explicit Compatibility Conditions: The framework naturally yields the compatibility conditions required for the data $(f, g)$. These conditions are expressed as the requirement that a specific linear combination of the interior data and boundary data belongs to a specific trace space, rather than imposing simple equality constraints.
• Applications:
  • Heat Equation with Dirichlet Conditions: The theory recovers higher-order regularity results, showing that for $f$ in a specific space, the boundary data $g$ must satisfy a compatibility condition involving $\partial_t g - \text{Tr}_0 f$ belonging to a specific Besov-type space, rather than requiring $f$ to vanish on the boundary.
  • Cahn-Hilliard-Gurtin System: The authors extend their previous $L^2$ results to general $p \in (1, \infty)$, establishing maximal regularity for this system with mixed-order structure.
  • Dynamic Boundary Conditions: The theory is applied to parabolic problems with dynamic boundary conditions, providing well-posedness results for systems where the boundary evolution is coupled with the interior dynamics.

Significance and Claims
The paper claims to provide a systematic and unified approach to mixed-order boundary value problems that goes beyond the classical Lopatinski˘ı–Shapiro theory. By utilizing Newton polygon scales and an abstract complementing condition, the framework:

• Substantially enlarges the class of admissible operators and boundary conditions compared to previous $L^2$ results.
• Systematically identifies the associated compatibility spaces, which emerge naturally from the abstract framework rather than being imposed ad hoc.
• Handles boundary operators involving time derivatives and non-trivial compatibility conditions among data.
• Provides a "mechanical" procedure for deriving well-posedness results: once the operator is identified as admissible and the complementing condition is verified, the well-posedness follows from the general theorem.

The authors emphasize that this theory is particularly valuable for problems where the geometry of the Newton polygon is non-triangular or where the trace spaces involve hybrid regularity scales, situations where classical anisotropic Sobolev space techniques are insufficient.