Abstract We study the gradient regularity of solutions to measure data elliptic systems with Uhlenbeck-type structure and Orlicz growth. For any bounded Borel measure, pointwise estimates for the gradient of solutions are provided in terms of the truncated Riesz potential. This allows us to show a precise transfer of regularity from data to solutions on various scales.

Abstract In this paper, we establish maximal function estimates, Lebesgue differentiation theory, Calderón–Zygmund decompositions, and John–Nirenberg inequalities for translation invariant Hausdorff contents. We further identify a key structural component of these results – a packing condition satisfied by these Hausdorff contents which compensates for the non-linearity of the capacitary integrals. We prove that for any outer capacity, this packing condition is satisfied if and only if the capacity is equivalent to its induced Hausdorff content. Finally, we use this equivalence to extend the preceding theory to general outer capacities which are assumed to satisfy this packing condition.

Abstract We consider a sequence, { M k } k ∈ ℕ {\{M_{k}\}_{k\in\mathbb{N}}} , of bubble converging minimal hypersurfaces, or H-CMC hypersurfaces, in compact Riemannian manifolds without boundary, of dimension 4 ≤ n + 1 ≤ 7 {4\leq n+1\leq 7} . We prove the following upper semicontinuity of index plus nullity: lim sup k → ∞ ( ind ( M k ) + nul ( M k ) ) ≤ ∑ i = 1 l co ( m ) i ( anl - ind ( co ( M ∞ i ) ) + anl - nul ( co ( M ∞ i ) ) ) + ∑ j = 1 J ind ( Σ j ) + nul ω Σ j , R ( Σ j ) \limsup_{k\rightarrow\infty}(\operatorname{ind}(M_{k})+\operatorname{nul}(M_{k% }))\leq\sum_{i=1}^{l}\operatorname{co}(m)_{i}(\operatorname{anl-ind}(% \operatorname{co}(M^{i}_{\infty}))+\operatorname{anl-nul}(\operatorname{co}(M^% {i}_{\infty})))+\sum_{j=1}^{J}\operatorname{ind}(\Sigma^{j})+\operatorname{nul% }_{\omega_{\Sigma^{j},R}}(\Sigma^{j}) for such a bubble converging sequence M k → ( ⋃ i = 1 l M ∞ i , Σ 1 , … , Σ J ) {M_{k}\rightarrow(\bigcup_{i=1}^{l}M_{\infty}^{i},\Sigma^{1},\ldots,\Sigma^{J})} , where co ( m ) i ∈ ℤ ≥ 1 {\operatorname{co}(m)_{i}\in\mathbb{Z}_{\geq 1}} is a notion of multiplicity of the convergence to the connected component M ∞ i {M_{\infty}^{i}} , and Σ 1 , … , Σ J {\Sigma^{1},\ldots,\Sigma^{J}} are the bubbles. This complements the previously known lower semicontinuity of index obtained in [R. Buzano and B. Sharp, Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area, Trans. Amer. Math. Soc. 370 2018, 6, 4373–4399] and [T. Bourni, B. Sharp and G. Tinaglia, CMC hypersurfaces with bounded Morse index, J. Reine Angew. Math. 786 2022, 175–203]. The strategy of our proof is to analyse a weighted eigenvalue problem along our sequence of degenerating hypersurfaces, { M k } k ∈ ℕ {\{M_{k}\}_{k\in\mathbb{N}}} . This strategy is inspired by the recent work [F. Da Lio, M. Gianocca and T. Rivière, Morse index stability for critical points to conformally invariant Lagrangians, preprint 2022, https://arxiv.org/abs/2212.03124]. A key aspect of our proof is making use of a Lorentz–Sobolev inequality to study the behaviour of these weighted eigenfunctions on the neck regions along the sequence, as well as the index and nullity of our non-compact bubbles Σ 1 , … , Σ J {\Sigma^{1},\ldots,\Sigma^{J}} .

Abstract We consider a model to describe stable configurations in epitaxial growth of crystals in the two-dimensional case, and in the regime of linearized elasticity. The novelty is that the model also takes into consideration the adatom density on the surface of the film. These are behind the main mechanisms of crystal growth and formation of islands (or quantum dots). The main result of the paper is the integral representation of the relaxed energy.

Abstract We obtain a weak formulation of the stationarity condition for the half Dirichlet energy, which can be expressed in terms of a fractional quantity, related to the trace of the Hopf differential of the harmonic extension of the original map. As an application we show that conformal harmonic maps from the disc are precisely the harmonic extensions of stationary points of the half Dirichlet energy on the circle. We also derive a Noether theorem and a Pohozaev identity for stationary points of the half Dirichlet energy.

Abstract We consider a gauge-invariant Ginzburg–Landau functional (also known as Abelian Yang–Mills–Higgs model), on Hermitian line bundles over closed Riemannian manifolds of dimension n ≥ 3 {n\geq 3} . Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the London limit. After a convenient choice of the gauge, we show compactness of finite-energy critical points in Sobolev norms. Moreover, thanks to a suitable monotonicity formula, we prove that the energy densities of critical points, rescaled by the logarithm of the coupling parameter, converge to the weight measure of a stationary, rectifiable varifold of codimension 2.

Abstract We study a general total variation denoising model with weighted L 1 {L^{1}} fidelity, where the regularizing term is a non-local variation induced by a suitable (non-integrable) kernel K, and the approximation term is given by the L 1 {L^{1}} norm with respect to a non-singular measure with positively lower-bounded L ∞ {L^{\infty}} density. We provide a detailed analysis of the space of non-local BV \mathrm{BV} functions with finite total K-variation, with special emphasis on compactness, Lusin-type estimates, Sobolev embeddings and isoperimetric and monotonicity properties of the K-variation and the associated K-perimeter. Finally, we deal with the theory of Cheeger sets in this non-local setting and we apply it to the study of the fidelity in our model.

Abstract The paper is concerned with Lane–Emden and Brezis–Nirenberg problems involving the affine p-Laplace nonlocal operator Δ p 𝒜 {\Delta_{p}^{\cal A}} , which has been introduced in [J. Haddad, C. H. Jiménez and M. Montenegro, From affine Poincaré inequalities to affine spectral inequalities, Adv. Math. 386 2021, Article ID 107808] driven by the affine L p {L^{p}} energy ℰ p , Ω {{\cal E}_{p,\Omega}} from convex geometry due to [E. Lutwak, D. Yang and G. Zhang, Sharp affine L p L_{p} Sobolev inequalities, J. Differential Geom. 62 2002, 1, 17–38]. We are particularly interested in the existence and nonexistence of positive C 1 {C^{1}} solutions of least energy type. Part of the main difficulties are caused by the absence of convexity of ℰ p , Ω {{\cal E}_{p,\Omega}} and by the comparison ℰ p , Ω ⁢ ( u ) ≤ ∥ u ∥ W 0 1 , p ⁢ ( Ω ) {{\cal E}_{p,\Omega}(u)\leq\|u\|_{W^{1,p}_{0}(\Omega)}} generally strict.