Gradient regularity for widely degenerate elliptic partial differential equations

Michael Strunk ()
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Michael Strunk: Universität Salzburg

Partial Differential Equations and Applications, 2025, vol. 6, issue 5, 1-51

Abstract: Abstract In this paper, we investigate the regularity of weak solutions $$u:\Omega \rightarrow {\mathbb {R}}$$ u : Ω → R to elliptic equations of the type $$\begin{aligned} \textrm{div}\, \nabla {\mathcal {F}}(x,Du) = f\qquad \text {in }\Omega , \end{aligned}$$ div ∇ F ( x , D u ) = f in Ω , whose ellipticity degenerates in a fixed bounded and convex set $$E\subset {\mathbb {R}}^n$$ E ⊂ R n with $$0\in \textrm{Int}\, E$$ 0 ∈ Int E . Here, $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n denotes a bounded domain, and $${\mathcal {F}}:\Omega \times {\mathbb {R}}^n \rightarrow {\mathbb {R}}_{\ge 0}$$ F : Ω × R n → R ≥ 0 is a function with the properties: for any $$x\in \Omega $$ x ∈ Ω , the mapping $$\xi \mapsto {\mathcal {F}}(x,\xi )$$ ξ ↦ F ( x , ξ ) is regular outside E and vanishes entirely within this set. Additionally, we assume $$f\in L^{n+\sigma }(\Omega )$$ f ∈ L n + σ ( Ω ) for some $$\sigma > 0$$ σ > 0 , representing an arbitrary datum. Our main result establishes the regularity $$\begin{aligned} {\mathcal {K}}(Du)\in C^0(\Omega ) \end{aligned}$$ K ( D u ) ∈ C 0 ( Ω ) for any continuous function $${\mathcal {K}}\in C^0({\mathbb {R}}^n)$$ K ∈ C 0 ( R n ) vanishing on E.

Keywords: Widely degenerate elliptic PDEs; Weak solutions; Gradient regularity; 35B65; 35G20; 35J70 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-025-00349-8

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