Semicontinuity Result Research Articles

Semi-continuity of conductors, and ramification bound of nearby cycles

Abstract For a constructible étale sheaf on a smooth variety of positive characteristic ramified along an effective divisor, the largest slope in Abbes and Saito’s ramification theory of the sheaf gives a divisor with rational coefficients called the conductor divisor. In this article, we prove decreasing properties of the conductor divisor after pull-backs. The main ingredient behind is the construction of étale sheaves with pure ramifications. As applications, we first prove a lower semi-continuity property for conductors of étale sheaves on relative curves in the equal characteristic case, which supplement Deligne and Laumon’s lower semi-continuity property of Swan conductors ([33]) and is also an ℓ {\ell} -adic analogue of André’s semi-continuity result of Poincaré–Katz ranks for meromorphic connections on complex relative curves. ([6]). Secondly, we give a ramification bound for the nearby cycle complex of an étale sheaf ramified along the special fiber of a regular scheme semi-stable over an equal characteristic henselian trait, which extends a main result in a joint work with Teyssier ([20]) and answers a conjecture of Leal ([35]) in a geometric situation.

Lower semicontinuity inGSBDfor nonautonomous surface integrals

We provide a sufficient condition for lower semicontinuity of nonautonomous noncoercive surface energies defined on the space ofGSBDpfunctions, whose dependence on thex-variable isW1,1or evenBV: the notion ofnonautonomous symmetric joint convexity, which extends the analogous definition devised for autonomous integrands in Friedrichet al.[J. Funct. Anal.280(2021) 108929] where the conservativeness of the approximating vector fields is assumed. This condition allows to extend to our setting a nonautonomous chain formula inSBVobtained in Ambrosioet al.[Manuscr. Math.140(2013) 461–480], and this is a key tool in the proof of the lower semicontinuity result. This new joint convexity can be checked explicitly for some classes of surface energies arising from variational models of fractures in inhomogeneous materials.

A new space of generalised functions with bounded variation motivated by fracture mechanics

We introduce a new space of generalised functions with bounded variation to prove the existence of a solution to a minimum problem that arises in the variational approach to fracture mechanics in elastoplastic materials. We study the fine properties of the functions belonging to this space and prove a compactness result. In order to use the Direct Method of the Calculus of Variations we prove a lower semicontinuity result for the functional occurring in this minimum problem. Moreover, we adapt a nontrivial argument introduced by Friedrich to show that every minimizing sequence can be modified to obtain a new minimizing sequence that satisfies the hypotheses of our compactness result.

Quasiconvexity, Null Lagrangians, and Hardy Space Integrability Under Constant Rank Constraints

We present a systematic treatment of the theory of Compensated Compactness under Murat’s constant rank assumption. We give a short proof of a sharp weak lower semicontinuity result for signed integrands, extending aspects of the results of Fonseca–Müller. The null Lagrangians are an important class of signed integrands, since they are the weakly continuous functions. We show that they are precisely the compensated compactness quantities with Hardy space integrability, thus proposing an answer to a question raised by Coifman–Lions–Meyer–Semmes. Finally we provide an effective way of computing the null Lagrangians associated with a given operator.

$$\Gamma $$-convergence of polyconvex functionals involving s-fractional gradients to their local counterparts

In this paper we study localization properties of the Riesz s-fractional gradient \(D^s u\) of a vectorial function u as \(s \nearrow 1\). The natural space to work with s-fractional gradients is the Bessel space \(H^{s,p}\) for \(0< s < 1\) and \(1< p < \infty \). This space converges, in a precise sense, to the Sobolev space \(W^{1,p}\) when \(s \nearrow 1\). We prove that the s-fractional gradient \(D^s u\) of a function u in \(W^{1,p}\) converges strongly to the classical gradient Du. We also show a weak compactness result in \(W^{1,p}\) for sequences of functions \(u_s\) with bounded \(L^p\) norm of \(D^s u_s\) as \(s \nearrow 1\). Moreover, the weak convergence of \(D^s u_s\) in \(L^p\) implies the weak continuity of its minors, which allows us to prove a semicontinuity result of polyconvex functionals involving s-fractional gradients defined in \(H^{s,p}\) to their local counterparts defined in \(W^{1,p}\). The full \(\Gamma \)-convergence of the functionals is achieved only for the case \(p>n\).

Continuous Wong–Zakai Approximations of Random Attractors for Quasi-linear Equations with Nonlinear Noise

We consider a family of random quasi-linear equations driven by nonlinear Wong–Zakai noise and parameterized by the non-zero size $$\lambda $$ of noise. After proving the existence of a random attractor $$A_\lambda (\omega )$$ in the square Lebesgue space, we then show that there is a residual dense subset of the space of nonzero real numbers such that, under the Hausdorff metric, the map $$\lambda \rightarrow A_\lambda (\theta _s\omega )$$ is continuous at all points of the residual dense set, where $$\theta _s$$ is a group of self-transformations on the probability space. We also prove that as $$\lambda \rightarrow \pm \infty $$ the random attractor converges upper-semicontinuously to the global attractor of the deterministic quasi-linear equation. The upper semi-continuity result is new for nonlinear noise, while, the lower semi-continuity result is new even for linear noise. The theory of Baire category is the main tool used to prove the residual continuity.

A Relaxation Result in the Vectorial Setting and Power Law Approximation for Supremal Functionals

We provide relaxation for not lower semicontinuous supremal functionals defined on vectorial Lipschitz functions, where the Borel level convex density depends only on the gradient. The connection with indicator functionals is also enlightened, thus extending previous lower semicontinuity results in that framework. Finally, we discuss the power law approximation of supremal functionals, with nonnegative, coercive densities having explicit dependence also on the spatial variable, and satisfying minimal measurability assumptions.

The metric geometry of singularity types

Abstract Let X be a compact Kähler manifold. Given a big cohomology class { θ } {\{\theta\}} , there is a natural equivalence relation on the space of θ-psh functions giving rise to 𝒮 ⁢ ( X , θ ) {\mathcal{S}(X,\theta)} , the space of singularity types of potentials. We introduce a natural pseudo-metric d 𝒮 {d_{\mathcal{S}}} on 𝒮 ⁢ ( X , θ ) {\mathcal{S}(X,\theta)} that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence of positive mass we show that this metric space is complete. As applications, we show that solutions to a family of complex Monge–Ampère equations with varying singularity type converge as governed by the d 𝒮 {d_{\mathcal{S}}} -topology, and we obtain a semicontinuity result for multiplier ideal sheaves associated to singularity types, extending the scope of previous results from the local context.

Existence of varifold minimizers for the multiphase Canham–Helfrich functional

We address the minimization of the Canham–Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers is here assumed. Correspondingly, the problem is reformulated and solved in the weaker frame of oriented curvature varifolds. We present a lower semicontinuity result and prove existence of single- and multiphase minimizers under area and enclosed-volume constrains. Additionally, we discuss regularity of minimizers and establish lower and upper diameter bounds.

Lower semicontinuity of the ADM mass in dimensions two through seven

The semicontinuity phenomenon of the ADM mass under pointed (i.e., local) convergence of asymptotically flat metrics is of interest because of its connections to nonnegative scalar curvature, the positive mass theorem, and Bartnik's mass-minimization problem in general relativity. In this paper, we extend a previously known semicontinuity result in dimension three for $C^2$ pointed convergence to higher dimensions, up through seven, using recent work of S. McCormick and P. Miao (which itself builds on the Riemannian Penrose inequality of H. Bray and D. Lee). For a technical reason, we restrict to the case in which the limit space is asymptotically Schwarzschild. In a separate result, we show that semicontinuity holds under weighted, rather than pointed, $C^2$ convergence, in all dimensions $n \geq 3$, with a simpler proof independent of the positive mass theorem. Finally, we also address the two-dimensional case for pointed convergence, in which the asymptotic cone angle assumes the role of the ADM mass.

Assessing the stability of anaerobic co-digestion in the context of a WWTP with thermal pre-treatment of sewage sludge. Case study in Chile

In this study, beverage wastewater (BW) and extended aeration sewage sludge were assessed as co-substrates with both mixed sludge and thermally hydrolyzed sludge as substrates in an anaerobic codigestion (AcoD) process. The methodology is an integrated methodology that comprises a batch test (BMP) and semicontinuous reactor along with microbial diversity analysis. The results showed that in batch conditions, neither significant synergistic nor antagonistic effects were observed on the maximum potential (P) due to the cosubstrates presence. However, synergistic effects on a kinetic parameter (Rm) were observed when BW was used as cosubstrate. Process instability was observed in semi-continuous mode with cosubstrate BW, which was not predicted by the BMP test. As demonstrated in this study, the BMP approach is also not suitable for predicting continuous performance in real systems; however, it can be used to screen an array of substrates. In semi-continuous mode, one can expect to reach values between 50 and 60% of the biogas potential and 20–40% of the maximum production rate achieved in batch tests. The agreement between batch and semi-continuous results is also dependent on the type of inoculum used. The microbial diversity of the system was more influenced by the reactor stability than the presence of cosubstrates. To predict the performance (and stability) of real anaerobic digesters, dynamic models should be employed.

Liftings, Young Measures, and Lower Semicontinuity

This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs (uj, Duj)j for {(u_j)_jsubset {rm BV}(Omega;mathbb{R}^m)} under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functionalF:u↦∫Ωf(x,u(x),∇u(x))dx,u∈W1,1(Ω;Rm),Ω⊂Rdopen,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\mathcal{F}\\colon u\\mapsto\\int_\\Omega f(x,u(x),\\nabla u(x))\\, {\\rm d}x, \\quad u \\in {\\rm W}^{1,1}(\\Omega;\\mathbb{R}^m),\\quad\\Omega\\subset\\mathbb{R}^d {\\rm open,}$$\\end{document}to the space {{rm BV}(Omega;mathbb{R}^m)}. Lower semicontinuity results of this type were first obtained by Fonseca and Müller (Arch Ration Mech Anal 123:1–49, 1993) and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that f be Carathéodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising {mathcal{F}} in the x and u variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure.

Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations

We establish backward compactness of a pullback random attractor (PRA) from a backward limit-set compact cocycle. We prove that backward compactness of a PRA is a necessary and sufficient condition such that its time-fibers are upper semi-continuous at negative infinity, and obtain the minimal compact limit-set. Also, we obtain the maximal limit-set in the lower semi-continuity sense. These results greatly reinforce the upper semi-continuity result at the finite time, given by Cui et al. (2018).In the application part, we show existence, backward compactness and longtime robustness of a PRA for the non-autonomous stochastic magneto-hydrodynamics equations perturbed by a backward tempered force. Moreover, we prove that the PRA does not change on different attracted universes, this fact allows us to establish the finite-time robustness of the PRA if the force is tempered only.

A Reshetnyak-type lower semicontinuity result for linearised elasto-plasticity coupled with damage in $$W^{1,n}$$

In this paper we prove a lower semicontinuity result of Reshetnyak type for a class of functionals which appear in models for small-strain elasto-plasticity coupled with damage. To do so we characterise the limit of measures \(\alpha _k\,{\mathrm {E}}u_k\) with respect to the weak convergence \(\alpha _k\rightharpoonup \alpha \) in \(W^{1,n}(\Omega )\) and the weak\(^*\) convergence \(u_k{\mathop {\rightharpoonup }\limits ^{*}}u\) in \(BD(\Omega )\), \({\mathrm {E}}\) denoting the symmetrised gradient. A concentration compactness argument shows that the limit has the form \(\alpha \,{\mathrm {E}}u+\eta \), with \(\eta \) supported on an at most countable set.

Existence and upper semi-continuity of pullback attractors of a p-Laplacian equation with delay

In this paper, we study the dynamics of a non-autonomous p-Laplacian equation with delay. We first prove the existence and uniqueness of solutions for this equation by using the Galerkin approximations. Then, applying the Sobolev compactness theorem, Ascoli-Azerla theorem, and energy method, we show the existence of pullback attractors for the associated process in space C([−h, 0]; Lp) as well as in space C([−h, 0]; L2). The upper semi-continuity result of the pullback attractors is also established here.

A lower semicontinuity result for a free discontinuity functional with a boundary term

We study the lower semicontinuity in GSBVp(Ω;Rm) of a free discontinuity functional F(u) that can be written as the sum of a crack term, depending only on the jump set Su, and of a boundary term, depending on the trace of u on ∂Ω. We give necessary and sufficient conditions on the integrands for the lower semicontinuity of F. Moreover, we prove a relaxation result, which shows that the lower semicontinuous envelope of F can still be represented as the sum of two integrals on Su and ∂Ω, respectively.

Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations on an unbounded domain

The paper contributes box-counting dimensions and upper semicontinuities of random bi-spatial attractors for stochastic degenerate parabolic equations on the whole Euclid space. Under some weak assumptions for the force and the nonlinearity, we first prove the existence of a unique (L2,D01,2∩Lq)-random attractor for any q∈[2,(p−2)I+2], where p−1 is the order of the nonlinearity and I is a given integer such that the force is (I+1)-times integrable. By using truncation and splitting techniques, and also induction methods, we then prove that a priori estimate is uniform with respect to the density of noise, which leads to the upper semicontinuity result of the obtained attractors as the density tends to a constant (including zero) under the topology of the terminative space. Furthermore, we give a new framework to discuss the bound of Lq-box-counting dimensions of random attractors for SPDE on an unbounded domain.

Lower semicontinuity for an integral functional in BV

We prove a lower semicontinuity result for a functional of linear growth initially defined by $$\begin{aligned} \int _{\Omega }F\left( \frac{dDu}{d\mu }\right) \,d\mu \end{aligned}$$ for \(u\in \mathrm {BV}(\Omega ;{\mathbb R}^N)\) with \(Du\ll \mu \). The positive Radon measure \(\mu \) is only assumed to satisfy \(\mathcal L^n\ll \mu \).

Lower Semicontinuity Results in Parametric Multivalued Weak Vector Equilibrium Problems and Applications

ABSTRACTThis article gives new sufficient conditions for the lower semicontinuity of the solution mapping of a parametric multivalued weak vector equilibrium problem with moving cones. A scalarizing approach, based on the signed distance function of Hiriart Urruty is used to discuss this lower semicontinuity property. The main results of the article are obtained under some assumptions different from those introduced earlier by previous linear and nonlinear scalarizing approaches. Some applications to the study of connectedness of weak solution sets of multivalued vector equilibrium problems are given.

Lower semi-continuity for non-coercive polyconvex integrals in the limit case

Lower semi-continuity results for polyconvex functionals of the calculus of variations along sequences of maps u: Ω ⊂ ℝn → ℝm in W1,m, 2 ⩽ m⩽ n, weakly converging in W1,m-1, are established. In addition, for m = n + 1, we also consider the autonomous case for weakly converging maps in W1,n-1.