A priori estimates for general elliptic and parabolic boundary value problems over irregular domains

We investigate the realization of a myriad of general local and nonlocal inhomogeneous elliptic and parabolic boundary value problems over classes of irregular regions. We present a unified approach in which either local or nonlocal Neumann, Robin, and Wentzell boundary value problems are treated simultaneously. We establish solvability and global regularity results for both the stationary and time-dependent heat equations governed by general differential operators with unbounded measurable coefficients and various boundary conditions at once, first on a general framework, and then by presenting concrete important examples of irregular domains, Wentzell-type boundary conditions, and nonlocal maps. As a consequence, we develop a priori estimates for multiple differential equations under various situations, which are tied to a large number of applications performed over real world regions, such heat transfer, electrical conductivity, stable-like processes (probability theory), diffusion of medical sprays in the bronchial trees, and oceanography (among many others).

💡 Research Summary

The manuscript develops a unified functional‑analytic framework for a very broad class of linear elliptic and parabolic boundary‑value problems posed on highly irregular (non‑Lipschitz, fractal‑type) domains. The authors introduce a generalized weak normal derivative defined with respect to an upper d‑Ahlfors measure μ supported on the boundary Γ, which replaces the classical outward unit normal in settings where the boundary lacks a smooth structure.

The interior operator A is a second‑order, possibly non‑symmetric differential operator with essentially bounded leading coefficients αij(x) and potentially unbounded lower‑order coefficients (â, ̌a, λ). Uniform ellipticity is assumed. The boundary operator B incorporates three classical types of conditions—Neumann, Robin, and Wentzell—through a combination of a classical flux term, a boundary potential β(x), a non‑local boundary operator Lμ defined via a weakly coercive bilinear form ΛΓ, and additional non‑local maps JΩ (interior) and ΘΓ (boundary). Both JΩ and ΘΓ are linear, bounded, preserve constants, and are positive in the sense of quadratic forms; they cover examples such as Besov‑type operators and the Dirichlet‑to‑Neumann map.

The paper’s first major result concerns the stationary elliptic problem  Au = f in Ω, Bu = g on Γ, with data f∈L^p(Ω) and g∈L^q(Γ). Under the integrability thresholds p > N/2 and q > d(2+d−N)−1 (which are shown to be optimal for L∞‑estimates), any weak solution u belonging to the natural space W^2(Ω;Γ) satisfies a global supremum bound  ‖u‖{∞,Ω}+‖u‖{∞,Γ} ≤ C (‖f‖{p,Ω}+‖g‖{q,Γ}+‖u‖_{2,Ω}), and if the zero‑order coefficients λ or β are sufficiently positive, the L^2‑term can be dropped. When the integrability conditions fail, the authors still obtain L^2‑based estimates and maximum principles.

The second major contribution treats the inhomogeneous heat equation  ∂t u − Au = f(t,x) in (0,∞)×Ω, Bu = g(t,x) on (0,∞)×Γ, with initial data u(0)=u₀∈L^2(Ω). By interpreting A via the bilinear form  E_μ(u,v)=∫Ω (…) dx + Λ_Γ(u,v) + ∫Γ βuv dμ + ⟨J_Ω u,v⟩{s/2}+⟨Θ_Γ u,v⟩{r d/2}, the authors prove that the associated operator generates a compact holomorphic C₀‑semigroup on L^2(Ω) which extrapolates to all L^p‑spaces. Using a parabolic version of De Giorgi’s technique combined with Moser iteration, they derive global L∞‑estimates for mild solutions:  ‖u‖{L∞(0,T;L∞(Ω))} ≤ C (‖u₀‖{∞,Ω}+‖f‖{L^{κ₁}(0,T;L^p(Ω))}+‖g‖_{L^{κ₂}(0,T;L^q(Γ))}), provided the exponents satisfy  1/κ₁ + N/(2p) < 1, 1/κ₂ + d/(2q)(d+2−N) < 1/2. A similar bound holds for the boundary trace. These results extend the classical theory for smooth domains to a setting where both the geometry and the operators are highly irregular.

To illustrate the abstract theory, the authors present two concrete Wentzell‑type boundary models. In the first, Γ is a compact Riemannian manifold; the boundary operator L_μ becomes a Laplace‑Beltrami‑type operator with tangential derivatives, allowing unbounded lower‑order terms. In the second, Γ is the fractal boundary of the Koch snowflake, where no smooth manifold structure exists; nevertheless a well‑posed abstract Wentzell operator is constructed. Both examples demonstrate that the framework accommodates non‑symmetric interior operators and unbounded coefficients, a situation not treated in existing literature.

The paper also treats two families of non‑local maps. Besov‑type interior and boundary operators are shown to satisfy the required positivity and boundedness properties, linking the analysis to fractional diffusion and stable‑like stochastic processes. A generalized Dirichlet‑to‑Neumann map is built for domains with merely finite (N−1)‑dimensional Hausdorff measure, extending the work of Arendt and Elst. The authors verify that this map fits into the abstract setting, thereby providing the first global a priori estimates for problems involving such a non‑local boundary condition.

Methodologically, the work blends variational form techniques (continuity and weak coercivity of E_μ), semigroup theory for non‑symmetric operators, and refined regularity tools (Moser iteration, De Giorgi’s method) adapted to non‑local boundary terms. The analysis carefully tracks the dependence of constants on the geometry (through the Ahlfors measure) and on the integrability of data, yielding explicit, optimal conditions.

In summary, the paper makes four substantive contributions: (1) a generalized normal derivative and trace theory for non‑Lipschitz domains; (2) a unified operator model that simultaneously handles non‑symmetric, possibly unbounded coefficients and a wide spectrum of local and non‑local boundary conditions; (3) global L∞ a priori estimates for both elliptic and parabolic problems under optimal integrability assumptions; and (4) concrete realizations of Wentzell and Dirichlet‑to‑Neumann boundary operators on fractal and manifold boundaries. These results open new avenues for the analysis of heat conduction, electrical conductivity, diffusion in complex biological structures, and oceanographic models where the underlying spatial domains are far from smooth.