Global solvability of the Laplace equation in weighted Sobolev spaces

We consider a non-local boundary value problem for the Laplace equation in unbounded studding the weak and strong solvability of that problem in the framework of the weighted Sobolev space $W^{1,p}_ν$, with a Muckenhoupt weight. We proved that if any weak solution belongs to the space $W_ν^{2,p}$, then it is also a strong solution and satisfies the corespding boundary conditions. It should be noted that such problems do not fit into the general theory of elliptic equations and require a special technique.

💡 Research Summary

This paper presents a detailed analysis of the global solvability for a non-local boundary value problem associated with the Laplace equation in an unbounded strip domain Π = (0, 2π) × (0, ∞). The work investigates the existence, uniqueness, and regularity of solutions within the framework of weighted Sobolev spaces, employing a novel technique that diverges from the standard theory of elliptic partial differential equations.

The problem under consideration is formally stated as Δu = 0 in Π, subject to the boundary conditions: u(0, y) = u(2π, y) for y > 0 (periodicity in x), u(x, 0) = f(x) for x ∈ (0, 2π) (Dirichlet condition on the bottom), and u_x(0, y) = h(y) for y > 0 (Neumann condition on the left edge). The authors study this problem in weighted Lebesgue spaces L_p_ν and the corresponding weighted Sobolev spaces W_ν^{m,p}, where the weight function ν belongs to the Muckenhoupt class A_p. This class of weights ensures crucial properties like the boundedness of the Hardy-Littlewood maximal operator, which is foundational for harmonic analysis in these spaces.

The core methodological innovation lies in the use of biorthonormal systems and Fourier series techniques in Banach function spaces. Specifically, the authors introduce and analyze two systems: {y_n} = {1, cos nx, x sin nx} and its dual system {ϑ_n}. A key auxiliary result (Theorem 2.4) proves that under the A_p condition on ν, the system {y_n} forms a basis in the space L_p_ν(0, 2π). This extends classical Fourier series methods beyond the Hilbert space L^2 and provides the functional-analytic foundation for representing solutions.

The main solvability result is centered on the concepts of weak and strong solutions. A weak solution is defined as a function u in W_ν^{1,p}(Π) satisfying an integral identity (3.2) derived from the problem via integration by parts and the boundary conditions. A strong solution is a function in W_ν^{2,p}(Π) that satisfies the differential equation and boundary conditions almost everywhere.

The pivotal theorem (Theorem 3.2) establishes the uniqueness of a weak solution given certain conditions on the data f and h. The proof involves representing a hypothetical weak solution u via its biorthonormal series expansion (3.7) with coefficients u^c_n(y) and u^s_n(y). By carefully choosing test functions from a dense set, the authors derive ordinary differential equations that these Fourier coefficients must satisfy. Combining these ODEs with the boundary conditions and the requirement that the solution remains bounded leads to the conclusion that all coefficients must be identically zero for the homogeneous problem (f=0, h=0), thus proving uniqueness.

The paper’s central insight is that if a weak solution possesses higher regularity, i.e., if it belongs to W_ν^{2,p}(Π), then it automatically qualifies as a strong solution and satisfies the boundary conditions pointwise almost everywhere. This establishes a bridge between the weak formulation and the classical pointwise notion of a solution in this challenging non-local and unbounded setting.

In summary, this research successfully tackles a problem outside the scope of standard elliptic theory by combining weighted space theory with biorthogonal expansions. It provides a rigorous framework for understanding weak and strong solvability for the Laplace equation in unbounded domains with non-standard boundary constraints, offering a specialized technique that could be applicable to other similar degenerate or non-local problems.