[Paper Review] Dirichlet problem associated with Dunkl Laplacian on $W$-invariant open sets

Mohamed Ben Chrouda, Khalifa El Mabrouk|arXiv (Cornell University)|Feb 7, 2014

Spectral Theory in Mathematical Physics1 references3 citations

TL;DR

This paper establishes the existence and uniqueness of solutions to the Dirichlet problem for the Dunkl Laplacian Δₖ on W-invariant bounded open sets D ⊂ ℝᵈ. Using probabilistic tools—specifically the Dunkl process and its first exit time τ_D—it proves that the solution h(x) = Eˣ[f(X_{τ_D})] is continuous on D̄ and twice differentiable in D, satisfying Δₖh = 0 in D and h = f on ∂D, provided D is regular and f is continuous on ∂D.

ABSTRACT

Combining probabilistic and analytic tools from potential theory, we investigate Dirichlet problems associated with the Dunkl Laplacian $Δ_k$. We establish, under some conditions on the open set $D\subset\R^d$, the existence of a unique continuous function $h$ in the closure of $D$, twice differentiable in $D$, such that $$ Δ_kh=0 \quad extrm{in}\;D\quad extrm{and}\quad h=f\quad extrm{on}\; \partial D. $$ We also give a probabilistic formula characterizing the solution $h$. The function $f$ is assumed to be continuous on the Euclidean boundary $\partial D$ of $D$.

Motivation & Objective

• To establish the existence and uniqueness of solutions to the Dirichlet problem associated with the Dunkl Laplacian Δₖ on W-invariant open sets.
• To characterize the solution via a probabilistic formula involving the first exit time of the Dunkl process from the domain.
• To prove that the solution is C² in the interior and continuous up to the boundary under regularity and W-invariance conditions.
• To show that the solution is harmonic with respect to the Dunkl process, leveraging balayage theory and hypoellipticity of Δₖ.
• To extend known results from the unit ball case to general W-invariant domains using the interplay between potential theory and stochastic processes.

Proposed method

• Utilizes the Dunkl process X, a càdlàg Markov process with infinitesimal generator (1/2)Δₖ, to define harmonic functions via the exit time τ_D.
• Defines the harmonic extension H_D f(x) = Eˣ[f(X_{τ_D})] for f ∈ C(∂D), which serves as the candidate solution.
• Establishes equivalence between Δₖ-harmonicity and X-harmonicity: Δₖh = 0 in D iff H_U h(x) = h(x) for all U ⋐ D.
• Applies balayage theory from Bliedtner and Hansen (1980) to prove uniqueness of the solution under regularity of D.
• Uses hypoellipticity of Δₖ in D (established in [7, 10]) to show that H_D f is C² in D when D is W-invariant and f is continuous.
• Employs L² duality via the inner product ⟨⋅,⋅⟩ₖ and integration by parts to verify that H_D f satisfies the weak form of Δₖh = 0.

Experimental results

Research questions

• RQ1 Under what conditions does the Dirichlet problem for the Dunkl Laplacian admit a unique solution on a W-invariant open set?
• RQ2 Can the solution to the Dirichlet problem for Δₖ be characterized probabilistically via the first exit distribution of the Dunkl process?
• RQ3 Is the solution h = H_D f twice differentiable in D when f is continuous on ∂D and D is regular and W-invariant?
• RQ4 Does the hypoellipticity of Δₖ in D imply that the probabilistic solution H_D f is C² in D under W-invariance?
• RQ5 How does the regularity of the boundary ∂D (e.g., cone condition) affect the existence and smoothness of the solution?

Key findings

• For any bounded W-invariant open set D that is regular for the Dunkl process and any f ∈ C(∂D), there exists a unique solution h ∈ C(𝒟̄) ∩ C²(D) to the Dirichlet problem Δₖh = 0 in D and h = f on ∂D.
• The solution is given by the probabilistic formula h(x) = Eˣ[f(X_{τ_D})], where X is the Dunkl process and τ_D is the first exit time from D.
• The solution h is X-harmonic in D, meaning H_U h(x) = h(x) for all x ∈ U and U ⋐ D, which characterizes Δₖ-harmonicity via the Markov process.
• The hypoellipticity of Δₖ in D ensures that H_D f is infinitely differentiable in D, and in particular C², when D is W-invariant.
• The condition that D is W-invariant is essential for the hypoellipticity argument; without it, the C²-regularity of H_D f remains an open question.
• The result generalizes the known Poisson integral formula for the unit ball (Maslouhi and Youssfi, 2011), showing that H_B(x, dy) = Pₖ(x,y) wₖ(y) σ(dy) on the unit ball B.