[Paper Review] Global higher integrability for minimisers of convex functionals with (p,q)-growth

Lukas Koch|arXiv (Cornell University)|Oct 29, 2020

Nonlinear Partial Differential EquationsMathematics49 references21 citations

TL;DR

This paper establishes global W¹,q(Ω, ℝᵐ) regularity for minimisers of convex functionals with (p,q)-growth and α-Hölder continuous dependence on x, under the condition q < (n+α)p/n. Using a global difference quotient method and a relaxation argument, it proves higher integrability and higher differentiability for both pointwise and relaxed minimisers, extending local regularity results to the global boundary setting without geometric assumptions on the domain or boundary data.

ABSTRACT

We prove global $W^{1,q}(\Omega,\mathbb{R}^m)$-regularity for minimisers of convex functionals of the form $\mathscr{F}(u)=\int_\Omega F(x,Du)\mathrm{d} x$. $W^{1,q}(\Omega,\mathbb{R}^m)$ regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on $F(x,z)$ are a uniform $\alpha$-H\"older continuity assumption in $x$ and controlled $(p,q)$-growth conditions in $z$ with $q<\frac{(n+\alpha)p}{n}$.

Motivation & Objective

• To establish global W¹,q(Ω, ℝᵐ) regularity for minimisers of convex functionals with (p,q)-growth and α-Hölder continuous dependence on the spatial variable x.
• To extend local higher integrability results to the global setting, including up to the boundary of a Lipschitz domain.
• To prove regularity for both pointwise minimisers and relaxed minimisers (in the sense of weak limits in W¹,q), without geometric assumptions on the domain or boundary data.
• To provide a global analogue of local W¹,q_loc regularity results under (p,q)-growth and Hölder continuity in x, under the critical threshold q < (n+α)p/n.
• To demonstrate that the Lavrentiev phenomenon does not obstruct global W¹,q regularity under the stated conditions, via a relaxation argument and a-priori estimates.

Proposed method

• Adapts the global difference quotient method from [43] to handle non-autonomous functionals with (p,q)-growth and α-Hölder continuity in x.
• Establishes a-priori estimates for regularised q-growth functionals using difference quotients, then passes to the limit to recover regularity for the original minimiser.
• Uses a relaxation framework to define relaxed minimisers as weak limits in W¹,p of sequences in W¹,q, ensuring the relaxed functional is well-defined and lower semicontinuous.
• Applies a variant of the dominated convergence theorem to pass from approximating sequences to the limit minimiser, relying on the condition (H4) to control the growth of F(x, Du*φ_ε).
• Implements a truncation and approximation argument to ensure the minimiser satisfies the necessary integrability and convergence properties.
• Verifies that the key assumptions (H1)–(H3) and (H4) are preserved under the relaxation process and for the approximating sequences.

Experimental results

Research questions

• RQ1 Can global W¹,q regularity be established for minimisers of convex (p,q)-growth functionals with α-Hölder continuous dependence on x, without geometric assumptions on the domain?
• RQ2 Does the relaxed minimiser framework allow for global higher integrability when the original functional exhibits the Lavrentiev phenomenon?
• RQ3 What is the sharp threshold for q in terms of p, n, and α such that global W¹,q regularity holds under (p,q)-growth and Hölder continuity in x?
• RQ4 How does the condition (H4), which ensures local infimum control, contribute to the global regularity proof?
• RQ5 Can the results be extended to functionals with variable exponent p(x) or anisotropic growth, under suitable Hölder continuity?

Key findings

• Global W¹,q(Ω, ℝᵐ) regularity is established for relaxed minimisers of convex (p,q)-growth functionals under the condition q < (n+α)p/n, provided F(x,z) satisfies (H1)–(H3) and (H4).
• For any 0 ≤ β < α, the W¹, np/(n−β) norm of the minimiser is bounded by a power γ of the functional value, boundary data norm, and external force norm: ||u||_W¹, np/(n−β) ≲ (1 + F(u) + ||g||_W¹+α,q + ||f||_Lq′)^γ.
• The proof relies on a-priori estimates derived via difference quotients applied globally, using a technique from [43] to handle boundary terms.
• The Lavrentiev phenomenon does not prevent global W¹,q regularity, as shown via a relaxation argument and convergence of approximating sequences in W¹,p and W¹,q.
• The results extend to functionals with variable exponent p(x) or anisotropic growth (e.g., p_i(x)-growth), provided the corresponding assumptions (H1.1) or (H1.2) hold.
• Examples such as the double-phase functional, anisotropic p(x)-Laplacian, and more general anisotropic functionals satisfy the assumptions and thus admit global W¹,q regularity.