[Paper Review] Existence of $p$ harmonic multiple valued maps into a separable Hilbert space

Philippe Bouafia, Thierry De Pauw+1 more|arXiv (Cornell University)|Feb 14, 2014

Fixed Point Theorems Analysis12 references3 citations

TL;DR

This paper establishes the existence of p-harmonic multiple-valued maps from the unit ball in R^m into a separable Hilbert space, using an intrinsic approach to Sobolev theory for Q-valued maps. It proves the Dirichlet problem for p-energy minimizers has a solution via compactness, trace theory, and lower semicontinuity in the Sobolev class W^{1}_p(U; Q_Q(ℓ₂)).

ABSTRACT

The general existence of $p$-Dirichlet energy minimizing maps into $Q_Q(l_2)$ is obtained.

Motivation & Objective

• To establish the existence of p-harmonic multiple-valued maps into an infinite-dimensional separable Hilbert space, where classical embeddings like Almgren’s are unavailable.
• To develop an intrinsic Sobolev theory for Q-valued maps with values in ℓ₂, including definitions of L_p and W^{1}_p spaces, trace, and Poincaré inequality.
• To prove the Dirichlet problem for p-energy minimizers has a solution in the Sobolev class W^{1}_p(U; Q_Q(ℓ₂)) with prescribed boundary data.
• To extend the theory of p-harmonic maps beyond finite-dimensional targets by avoiding reliance on bi-Lipschitz embeddings in infinite dimensions.
• To ensure lower semicontinuity of the p-energy and differentiability a.e. for Sobolev Q-valued maps using Rademacher-type results and approximation techniques.

Proposed method

• Define the L_p and W^{1}_p spaces of Q-valued maps f: U → Q_Q(ℓ₂) using limits of Lipschitz maps under the L_p seminorm d_p(f₁,f₂) = (∫_U G(f₁,f₂)^p dℒ^m)^{1/p}.
• Introduce the p-energy 𝒟_p^p(f;U) via relaxation to ensure lower semicontinuity under L_p convergence.
• Use the nearest point projection P: ℓ₂ → a compact convex set C ⊆ ℓ₂ to control values of maps and ensure compactness in the Sobolev class.
• Apply the Rellich-type compactness theorem (Theorem 4.8.2) to extract convergent subsequences from bounded sequences in W^{1}_p(U; Q_Q(ℓ₂)).
• Leverage the Poincaré inequality (Theorem 4.6.2) and Luzin-type approximation (Proposition 4.6.3(1)) to establish a.e. differentiability of Sobolev maps.
• Prove trace theory and extension theorems (Theorem 4.5.1) to ensure boundary data can be extended and preserved under minimization.

Experimental results

Research questions

• RQ1 Can the existence of p-harmonic Q-valued maps be established in infinite-dimensional Hilbert spaces when Almgren’s bi-Lipschitz embedding is not available?
• RQ2 Does the p-energy functional on W^{1}_p(U; Q_Q(ℓ₂)) admit a minimizer for given boundary data?
• RQ3 Is the p-energy lower semicontinuous under L_p convergence of Q-valued maps?
• RQ4 Can Sobolev Q-valued maps into ℓ₂ be approximated by Lipschitz maps in the W^{1}_p norm, and does this imply a.e. differentiability?
• RQ5 Does the trace operator for Q-valued maps in W^{1}_p(U; Q_Q(ℓ₂)) preserve boundary data under projection and compactness arguments?

Key findings

• The Dirichlet problem for p-harmonic Q-valued maps into a separable Hilbert space ℓ₂ admits a solution: for any Lipschitz boundary data g: ∂U → Q_Q(ℓ₂), there exists f ∈ W^{1}_p(U; Q_Q(ℓ₂)) with 𝒯(f) = g minimizing the p-energy.
• The p-energy functional is lower semicontinuous with respect to L_p convergence, as guaranteed by Proposition 4.4.1 and Corollary 4.6.4.
• Sobolev Q-valued maps f ∈ W^{1}_p(U; Q_Q(ℓ₂)) are differentiable almost everywhere, a result derived from Rademacher-type differentiability (Theorem 2.5.8) and approximation by Lipschitz maps.
• The Poincaré inequality (Theorem 4.6.2) holds in the infinite-dimensional setting, enabling stronger approximation results and compactness in the Sobolev class.
• The Rellich-type compactness theorem (Theorem 4.8.2) ensures that bounded sequences in W^{1}_p(U; Q_Q(ℓ₂)) have convergent subsequences in L_p, provided values are contained in a compact set.
• The use of nearest point projection P: ℓ₂ → a compact convex set C ensures that minimizing sequences can be recentered into a compact range, enabling extraction of weak limits in the Sobolev class.