The Perron solution for elliptic equations without the maximum principle

In this article we consider the Dirichlet problem on a bounded domain $Ω\subset {\bf R}^d$ with respect to a second-order elliptic differential operator in divergence form. We do not assume a divergence condition as in the pioneering work by Stampacchia, but merely assume that $0$ is not a Dirichlet eigenvalue. The purpose of this article is to define and investigate a solution of the Dirichlet problem, which we call Perron solution, in a setting where no maximum principle is available. We characterise this solution in different ways: by approximating the domain by smooth domains from the interior, by variational properties, by the pointwise boundary behaviour at regular boundary points and by using the approximative trace. We also investigate for which boundary data the Perron solution has finite energy. Finally we show that the Perron solution is obtained as an $H^1_0$-perturbation of a continuous function on $\overline Ω$. This is new even for the Laplacian and solves an open problem.

💡 Research Summary

The paper addresses the Dirichlet problem for a very general second‑order elliptic operator in divergence form on a bounded open set Ω⊂ℝᵈ (d≥2). The coefficients a_{kl}, b_k, c_k and c₀ are merely bounded measurable, and the only structural assumption is the uniform ellipticity of the matrix (a_{kl}) and the spectral condition that 0 is not a Dirichlet eigenvalue of the operator. Under this hypothesis the operator A, restricted to H¹₀(Ω), is an isomorphism onto H⁻¹(Ω). This simple spectral condition replaces the classical divergence condition of Stampacchia and, crucially, it makes the maximum principle unnecessary.

The authors introduce a bounded linear map
 T : C(∂Ω) → C_b(Ω)
which they call the Perron solution operator. For any continuous boundary datum ϕ, the function u = Tϕ is A‑harmonic (i.e. Au = 0 in the distributional sense) and satisfies a remarkable decomposition: there exists a continuous extension Φ∈C(Ω)∩H¹_loc(Ω) with trace Φ|∂Ω = ϕ such that Φ−u ∈ H¹₀(Ω). In other words, the Perron solution is always an H¹₀‑perturbation of a continuous function that agrees with the prescribed boundary values. This decomposition is new even for the Laplacian and resolves an open problem concerning the existence of an H¹₀‑perturbation representation for continuous boundary data.

Four equivalent characterisations of the Perron solution are proved:

1. Interior Approximation – Let Ω₁⊂Ω₂⊂… be smooth bounded domains increasing to Ω. For any continuous extension Φ of ϕ, the solutions T_{Ω_n}(Φ|∂Ω_n) converge uniformly on compact subsets of Ω to Tϕ. This shows that the Perron solution can be obtained as a limit of classical Dirichlet solutions on smoother subdomains.

2. Variational (Sobolev) Description – For each ϕ∈C(∂Ω) there exists a Φ∈C(Ω)∩H¹_loc(Ω) with AΦ∈H⁻¹(Ω). Solving Av = AΦ for v∈H¹₀(Ω) (possible because A is an isomorphism on H¹₀) yields u = Φ−v = Tϕ. Consequently, Tϕ belongs to H¹(Ω) if and only if ϕ admits an H¹(Ω) continuous extension; this condition is independent of the particular elliptic operator.

3. Pointwise Boundary Behaviour (Regular Points) – A boundary point z is called A‑regular if lim_{x→z}Tϕ(x)=ϕ(z) for every continuous ϕ. The authors prove that A‑regularity coincides with the classical notion of regularity for the Laplacian; thus regularity depends only on the geometry of Ω, not on the lower‑order coefficients. Moreover, for any A‑harmonic u∈C_b(Ω) that attains the boundary data quasi‑everywhere (i.e. outside a polar set), one has u = Tϕ. This extends a well‑known result for harmonic functions to the full class of operators considered.

4. Approximate Trace and Variational Solution – Using the (d−1)‑dimensional Hausdorff measure σ on ∂Ω (assumed finite), the paper defines an “approximate trace” space Tr(Ω) ⊂ L²(∂Ω). If ϕ∈Tr(Ω) and u∈H¹(Ω) is the variational solution obtained by decomposing u = v + w with v∈H¹₀(Ω) and w∈H¹_A(Ω) (the A‑harmonic subspace), then u coincides with the Perron solution Tϕ. Conversely, if Tϕ∈H¹(Ω) then ϕ is an approximate trace of Tϕ, and the variational and Perron solutions agree. This establishes a strong link between the classical variational framework and the Perron construction even when the trace operator is only defined in a weak sense.

A substantial technical contribution is the derivation of an L^∞‑estimate for the solution without invoking Stampacchia’s divergence condition. The authors combine careful functional‑analytic arguments (Fredholm–Lax–Milgram lemma, compactness of the embedding H¹₀↪L²) with delicate approximation procedures to control constants uniformly across the approximating domains.

In summary, the paper provides a complete theory of Perron solutions for a broad class of elliptic operators without relying on the maximum principle. It shows that every continuous boundary datum yields a unique bounded A‑harmonic function, characterises this function through domain approximation, variational methods, pointwise boundary limits, and Sobolev extensions, and clarifies when the solution possesses finite energy. The results unify and extend classical Perron theory, Stampacchia’s work, and recent developments on approximative traces, offering a robust foundation for further investigations of non‑coercive elliptic problems, irregular domains, and numerical approximations.