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This paper is devoted to studying a type of elliptic equation that contains a varying nonlocal term. We provide a detailed analysis of the existence, non-existence, and blow-up behavior of L 2 -norm solutions for the related equation when the potential function V (x) fulfills an appropriate choice. [ABSTRACT FROM AUTHOR]

We prove the absence of the Lavrentiev gap for non-autonomous functionals ℱ ⁢ (u) ≔ ∫ Ω f ⁢ (x , D ⁢ u ⁢ (x)) ⁢ 푑 x , where the density f ⁢ (x , z) is α-Hölder continuous with respect to x ∈ Ω ⊂ ℝ n , it satisfies the (p , q) -growth conditions | z | p ⩽ f ⁢ (x , z) ⩽ L ⁢ (1 + | z | q) , where 1 < p < q < p ⁢ (n + α n) , and it can be approximated from below by suitable densities f k . [ABSTRACT FROM AUTHOR]

The amount of sequencing data for SARS-CoV-2 is several orders of magnitude larger than any virus. This will continue to grow geometrically for SARS-CoV-2, and other viruses, as many countries heavily finance genomic surveillance efforts. Hence, we need methods for processing large amounts of sequence data to allow for effective yet timely decision-making. Such data will come from heterogeneous sources: aligned, unaligned, or even unassembled raw nucleotide or amino acid sequencing reads pertaining to the whole genome or regions (e.g., spike) of interest. In this work, we propose ViralVectors, a compact feature vector generation from virome sequencing data that allows effective downstream analysis. Such generation is based on minimizers, a type of lightweight "signature" of a sequence, used traditionally in assembly and read mapping — to our knowledge, the first use minimizers in this way. We validate our approach on different types of sequencing data: (a) 2.5M SARS-CoV-2 spike sequences (to show scalability); (b) 3K Coronaviridae spike sequences (to show robustness to more genomic variability); and (c) 4K raw WGS reads sets taken from nasal-swab PCR tests (to show the ability to process unassembled reads). Our results show that ViralVectors outperforms current benchmarks in most classification and clustering tasks. Graphical Abstract showing the all steps of proposed approach. We start by collecting the sequence-based data. Then Data cleaning and preprocessing is applied. After that, we generate the feature embeddings using minimizer based approach. Then Classification and clustering algorithms are applied on the resultant data and predictions are made on the test set. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the following Kirchhoff equation where Ω ⊂ R 3 is a bounded connected domain and ∫ Ω | u | 2 d x = 1 . The results of existence and nonexistence on L 2 -norm solutions are given. Our argument shows that the blow-up behavior of L 2 -norm solution occurs, and the mass concentrates at an inner point of Ω , or the neighborhood of some boundary point. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the existence of solutions with a prescribed l 2 -norm for the following discrete Schrödinger equations, { − Δ 2 u k − 1 − f (u k) = λ u k k ∈ Z , ∑ k ∈ Z | u k | 2 = α 2 , where Δ 2 u k − 1 = u k + 1 + u k − 1 − 2 u k , f ∈ C (R) , α is a fixed constant, and λ ∈ R arises as a Lagrange multiplier. To get the solutions, we investigate the corresponding minimizing problem with the l 2 -norm constraint: E α = inf { 1 2 ∑ | Δ u k − 1 | 2 − ∑ F (u k) : ∑ | u k | 2 = α 2 }. An elaborative analysis on a minimizing sequence with respect to E α is obtained. We prove that there is a constant α 0 ≥ 0 such that there exists a global minimizer if α > α 0 , and there exists no global minimizer if α < α 0 . It seems that it is the first time to consider the solution with a prescribed l 2 -norm of the discrete Schrödinger equations. [ABSTRACT FROM AUTHOR]

We study a variational approach introduced by S.D. Fisher and the author in the 1970s in the context of norm minimization for differentiable mappings occurring in nonlinear elliptic boundary value problems. It may be viewed as an abstract version of the calculus of variations. A strong hypothesis, initially limiting the scope of this approach, is the assumption of a bounded minimizing sequence in the least squares formulation. In this article, we employ regularization and invariant regions to overcome this obstacle. A consequence of the framework is the convergence of approximations for regularized problems to a desired solution. The variational method is closely associated with the implicit function theorem, and it can be jointly studied, so that continuous parameter stability is naturally deduced. A significant aspect of the theory is that the reaction term in a reaction-diffusion equation can be selected to act globally as in the steady Schrödinger-Hartree equation. Local action, as in the non-equilibrium Poisson-Boltzmann equation, is also included. Both cases are studied at length prior to the development of a general theory. [ABSTRACT FROM AUTHOR]

A way to measure the lower growth rate of φ:Ω×[0,∞)→[0,∞)$$ \varphi :\Omega \times \left[0,\infty \right)\to \left[0,\infty \right) $$ is to require t↦φ(x,t)t−r$$ t\mapsto \varphi \left(x,t\right){t}^{-r} $$ to be increasing in (0,∞)$$ \left(0,\infty \right) $$. If this condition holds with r=1$$ r=1 $$, then infu∈f+W01,φ(Ω)∫Ωφ(x,|∇u|)dx$$ \underset{u\in f+{W}_0^{1,\varphi}\left(\Omega \right)}{\operatorname{inf}}{\int}_{\Omega}\varphi \left(x,|\nabla u|\right)\kern0.1em dx $$ with boundary values f∈W1,φ(Ω)$$ f\in {W}^{1,\varphi}\left(\Omega \right) $$ does not necessarily have a minimizer. However, if φ$$ \varphi $$ is replaced by φp$$ {\varphi}^p $$, then the growth condition holds with r=p>1$$ r=p>1 $$ and thus (under some additional conditions) the corresponding energy integral has a minimizer. We show that a sequence (up)$$ \left({u}_p\right) $$ of such minimizers converges when p→1+$$ p\to {1}^{+} $$ in a suitable BV$$ \mathrm{BV} $$‐type space involving generalized Orlicz growth and obtain the Γ$$ \Gamma $$‐convergence of functionals with fixed boundary values and of functionals with fidelity terms. [ABSTRACT FROM AUTHOR]

Using purely variational methods, we prove local and global higher integrability results for upper gradients of quasiminimizers of a (p , q) -Dirichlet integral with fixed boundary data, assuming it belongs to a slightly better Newtonian space. We also obtain a stability property with respect to the varying exponents p and q. The setting is a doubling metric measure space supporting a Poincaré inequality. [ABSTRACT FROM AUTHOR]

In the present paper, we study a class of discrete Schrödinger equations with concave–convex nonlinear terms via the variational method. Under certain conditions on the nonlinearities, two nontrivial solutions of the equations have been obtained by finding the minimizers on Nehari manifolds, and one of them is the ground state solution with negative energy. As we all know, there are few results on discrete systems involving concave–convex nonlinearities. [ABSTRACT FROM AUTHOR]

We show that the Weierstrass–Hilbert classical field theory can be strengthened. Namely, for each extremal field, it is true that if an extremal is an element of the field then a minimum is attained in the class of Sobolev functions with the same boundary data as for the extremal and with graphs in the set covered by the field. This result remains valid if one of the extremals is singular. If there is a field containing more than one singular extremal then each of these extremals defines the minimization problem having no solution in the class of Lipschitz functions with graphs in the set covered by the field. [ABSTRACT FROM AUTHOR]

National recommendations allow for the omission of sentinel lymph node biopsy (SLNB) and post-lumpectomy radiotherapy in women ≥ 70 y/o with early-stage, hormone-receptor positive invasive breast cancer, but these therapies remain common. Previous work demonstrates an individual's maximizing-minimizing trait—an inherent preference for more or less medical care—may influence the preference for low-value care. We recruited an equal number of women ≥ 70 yrs who were maximizers, minimizers, or neutral based on a validated measure between September 2020 and November 2020. Participants were presented a hypothetical breast cancer diagnosis before randomization to one of three follow-up messages: maximizer-tailored, minimizer-tailored, or neutral. Tailored messaging aimed to redirect maximizers and minimizers toward declining SLNB and radiotherapy. The main outcome measure was predicted probability of choosing SLNB or radiotherapy. The final analytical sample (n = 1600) was 515 maximizers (32%), 535 neutral (33%) and 550 (34%) minimizers. Higher maximizing tendency positively correlated with electing both SLNB and radiotherapy on logistic regression (P < 0.01). Any tailoring (maximizer- or minimizer-tailored) reduced preference for SLNB in maximizing and neutral women but had no effect in minimizing women. Tailoring had no impact on radiotherapy decision, except for an increased probability of minimizers electing radiotherapy when presented with maximizer-tailored messaging. Maximizing-minimizing tendencies are associated with treatment preferences among women facing a hypothetical breast cancer diagnosis. Targeted messaging may facilitate avoidance of low-value breast cancer care, particularly for SLNB. [ABSTRACT FROM AUTHOR]

We study minimizers of the Dirichlet φ‐energy integral with generalized Orlicz growth. We prove the Kellogg property, the set of irregular points has zero capacity, and give characterizations of semiregular boundary points. The results are new ever for the special cases double phase and Orlicz growth. [ABSTRACT FROM AUTHOR]

In this article, we study the Schrödinger–Newton equations − Δ u + 2 π φ u = μ u + | u | p u , i n R 2 , Δ φ = u 2 , i n R 2. We prove the existence of positive spherically symmetric decreasing solutions for p ∈ (0 , 2) by constrained minimization methods. For p = 2, by the Gagliardo–Nirenberg inequality, an elaborate estimate implies similar existence results. We also give the regularity, exponential decay and blow-up behavior of these solutions. [ABSTRACT FROM AUTHOR]

The extended Painlevé P.D.E. system Δ y − x 1 y − 2 | y | 2 y = 0 , (x 1 , ... , x n) ∈ R n , y : R n → R m , is obtained by multiplying by − x 1 the linear term of the Ginzburg-Landau equation Δ η = | η | 2 η − η , η : R n → R m. The two dimensional model n = m = 2 describes in the theory of light-matter interaction in liquid crystals, the orientation of the molecules at the boundary of the illuminated region. On the other hand, the one dimensional model reduces to the second Painlevé O.D.E. y ″ − x y − 2 y 3 = 0 , x ∈ R , which has been extensively studied, due to its importance for applications. The solutions of the extended Painlevé P.D.E. share some characteristics both with the Ginzburg-Landau equation and the second Painlevé O.D.E. The scope of this paper is to construct vortex-filament solutions y : R n → R n − 1 (∀ n ≥ 3) of the extended Painlevé equation. These solutions have in every hyperplane x 1 = Const. , a profile similar to the standard vortices η : R n − 1 → R n − 1 of the Ginzburg-Landau equation, but their amplitude is determined by the Hastings-McLeod solution h of the second Painlevé O.D.E. evaluated at x 1. [ABSTRACT FROM AUTHOR]

Using a variational approach we study interior regularity for quasiminimizers of a (p, q)-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincaré inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally Hölder continuous and they satisfy Harnack inequality, the strong maximum principle and Liouville's Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for Hölder continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we consider (p, q)-minimizers and we give an estimate for their oscillation at boundary points. [ABSTRACT FROM AUTHOR]

Consider the recovery of the weight function in distributed time-fractional diffusion system using the interior measurement, which arises in some ultra-slow diffusion phenomena. Due to the nonlinear and nonlocal dependance of the measurement data on the weight function, such an inverse problem is novel and important. Based on the regularities of the direct problems for general diffusion equations with distributed time fractional derivative shown in this paper, we establish the theoretical framework for the optimization version of the inverse problem, including existence of the minimizer, the differentiability of the cost functional, as well as the gradient of the cost functional, in suitable functional spaces. Then an iteration process of descent type is realized efficiently for minimizing the regularizing cost functional in terms of the gradient type iteration, with numerical examples showing the validity of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this article we consider a variational problem related to a quasilinear singular problem and obtain a nonexistence result in a metric measure space with a doubling measure and a Poincaré inequality. Our method is purely variational and to the best of our knowledge, this is the first work concerning singular problems in a general metric setting. [ABSTRACT FROM AUTHOR]

We consider a sequence of convex integral functionals Fs : W1,p(Ωs) → ℝ and a sequence of weakly lower semicontinuous and, in general, nonintegral functionals Gs : W1,p(Ωs) → ℝ, where {Ωs} is a sequence of domains in ℝn contained in a bounded domain Ω ⊂ ℝn (n ⩾ 2) p > 1. Along with this, we consider a sequence of closed convex sets Vs = {v ∈ W1,p(Ωs) : Ms(v) ⩽ 0 a.e. in Ωs}, where Ms is a mapping from W1,p(Ωs) to the set of all functions defined on Ωs. We establish conditions under which minimizers and minimum values of the functionals Fs +Gs on the sets Vs converge to a minimizer and the minimum value of a functional on the set V = {v ∈ W1,p(Ω) : M(v) ⩽ 0 a.e. in Ω}, where M is a mapping from W1,p(Ω) to the set of all functions defined on Ω. [ABSTRACT FROM AUTHOR]

Even though intensifiers have received a good deal of attention over the past few decades, downtoners, comprising diminishers and minimizers, have remained by and large a neglected category (but cf. Brinton, this issue). Among downtoners, the adverb little or a little stands out as the most frequent item. It is multifunctional and serves as a diminishing and minimizing intensifier and also in non-degree uses as a quantifier, frequentative, and durative. Therefore, the present paper is devoted to the structural and functional profile of (a) little in Late Modern English speech-related data. The data source is the socio-pragmatically annotated Old Bailey Corpus (OBC, version 2.0), which allows, among other things, the investigation of the usage of the item among different speaker groups. Our research charts the semantic and formal uses of adverbial little. Downtoner uses outnumber non-degree uses in the data, and diminishing uses are more common than minimizing uses. The formal realization is predominantly a little, with very rare determinerless or modified instances, such as very little. Little modifies a wide range of "targets," but most frequently adjectives and prepositional phrases, focusing on human states and circumstantial detail. With regard to variation and change, adverbial little declines in use over the 200 years and is used more commonly by speakers from the lower social ranks and by the lay, non-professional participants in the courtroom. [ABSTRACT FROM AUTHOR]

We give regularity results for minimizers of two special cases of polyconvex functionals. Under some structural assumptions on the energy density, we prove that minimizers are either bounded, or have suitable integrability properties, by using the classical Stampacchia Lemma. [ABSTRACT FROM AUTHOR]

We prove a continuous embedding that allows us to obtain a boundary trace imbedding result for anisotropic Musielak-Orlicz spaces, which we then apply to obtain an existence result for Neumann problems with nonlinearities on the boundary associated to some anisotropic nonlinear elliptic equations in Musielak-Orlicz spaces constructed from Musielak-Orlicz functions on which and on their conjugates we do not assume the Δ2-condition. The uniqueness of weak solutions is also studied. [ABSTRACT FROM AUTHOR]

For a finitely irreducible countable Markov shift and a potential with summable variations, we provide a condition on the associated pressure function which ensures that Bowen's Gibbs state, the equilibrium state, and the minimizer of the level-2 large deviations rate function are all unique and they coincide. From this, we deduce that the set of periodic points weighted with the potential equidistributes with respect to the Gibbs-equilibrium state as the periods tend to infinity. Applications are given to the Gauss map, and the Bowen-Series map associated with a finitely generated free Fuchsian group with parabolic elements. [ABSTRACT FROM AUTHOR]

In this paper, we describe domain walls appearing in a thin, nematic liquid crystal sample subject to an external field with intensity close to the Fréedericksz transition threshold. Using the gradient theory of the phase transition adapted to this situation, we show that depending on the parameters of the system, domain walls occur in the bistable region or at the border between the bistable and the monostable region. [ABSTRACT FROM AUTHOR]

Background: Seeding is usually the initial step of high-throughput sequence aligners. Two popular seeding strategies are fixed-size seeding (k-mers, minimizers) and variable-size seeding (MEMs, SMEMs, maximal spanning seeds). The former strategy supports fast seed computation, while the latter one benefits from a high seed uniqueness. Algorithmic bridges between instances of both seeding strategies are of interest for combining their respective advantages. Results: We introduce an efficient strategy for computing MEMs out of fixed-size seeds (k-mers or minimizers). In contrast to previously proposed extend-purge strategies, our merge-extend strategy prevents the creation and filtering of duplicate MEMs. Further, we describe techniques for extracting SMEMs or maximal spanning seeds out of MEMs. A comprehensive benchmarking shows the applicability, strengths, shortcomings and computational requirements of all discussed seeding techniques. Additionally, we report the effects of seed occurrence filters in the context of these techniques. Aside from our novel algorithmic approaches, we analyze hierarchies within fixed-size and variable-size seeding along with a mapping between instances of both seeding strategies. Conclusion: Benchmarking shows that our proposed merge-extend strategy for MEM computation outperforms previous extend-purge strategies in the context of PacBio reads. The observed superiority grows with increasing read size and read quality. Further, the presented filters for extracting SMEMs or maximal spanning seeds out of MEMs outperform FMD-index based extension techniques. All code used for benchmarking is available via GitHub at https://github.com/ITBE-Lab/seed-evaluation. [ABSTRACT FROM AUTHOR]

We deal with maps u : Ω ⊂ R n → R n minimizing variational integrals ∫ Ω [ | D u (x) | p + h (det D u (x)) ] d x , where 2 ≤ p < n , h : (0 , + ∞) → [ 0 , + ∞) is convex and blows when det D u → 0 + : lim t → 0 + h (t) = + ∞. If such a blow up is a power of | ln ⁡ (t) | , then we derive regularity for the minimizer u = (u 1 , ... , u n). We are able also to deal with integrals containing all the minors: ∫ Ω [ | D u (x) | p + ∑ s = 2 n − 1 | adj s (D u (x)) | q s + h (det D u (x)) ] d x with q s ≥ 1. [ABSTRACT FROM AUTHOR]

The goal of this paper is to provide both a description and an explanation of the combination of minimizers (ligeramente 'slightly') with gradable adjectives in Spanish. According to Kennedy & McNally (2005) these elements are degree items that are sensitive to the scalar structure of adjectives and are combined with closed scale, minimum standard adjectives. Unexpected combinations, according to this semantics, are considered as cases of coercion. In this paper we propose that minimizers create derived adjectives. They are modifiers of the adjective's granularity, which allow the selection of the standard of comparison to take into account a greater number of degree distinctions. From this proposal, this article shows that unexpected combinations of ligeramente with gradable adjectives, such as un cine ligeramente lleno 'a slightly crowded cinema', can be explained without the need to propose that a coercion process affecting the scalar structure of the adjective takes place. [ABSTRACT FROM AUTHOR]

We consider a control problem given by a mathematical model of the temperature control in industrial hothouses. The model is based on one-dimensional parabolic equations with variable coefficients. The optimal control is defined as a minimizer of a quadratic cost functional. We describe qualitative properties of this minimizer, study the structure of the set of accessible temperature functions, and prove the dense controllability for some set of control functions. [ABSTRACT FROM AUTHOR]

We prove C 1 , ν -regularity for local minimizers of the multi-phase energy: w ↦ ∫ Ω | D w | p + a (x) | D w | q + b (x) | D w | s d x , under sharp assumptions relating the couples (p , q) and (p , s) to the Hölder exponents of the modulating coefficients a (⋅) and b (⋅) , respectively. [ABSTRACT FROM AUTHOR]

We study the Dirichlet '-energy integral with Sobolev boundary values. The function' has generalized Orlicz growth. Special cases include variable exponent and double phase growths. We show that minimizers are regular at the boundary provided a weak capacity fatness condition is satisfied. This condition is satisfies for instance if the boundary is Lipschitz. The results are new even for Orlicz spaces. [ABSTRACT FROM AUTHOR]

We investigate a probabilistic model for routeing of messages in relay-augmented multihop ad-hoc networks, where each transmitter sends one message to the origin. Given the (random) transmitter locations, we weight the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectories with low interference (measured in terms of signal-to-interference ratio) and trajectory families with little congestion (measured in terms of the number of pairs of hops using the same relay). Under the resulting Gibbs measure, the system targets the best compromise between entropy, interference and congestion for a common welfare, instead of an optimization of the individual trajectories. In the limit of high spatial density of users, we describe the totality of all the message trajectories in terms of empirical measures. Employing large deviations arguments, we derive a characteristic variational formula for the limiting free energy and analyse the minimizer(s) of the formula, which describe the most likely shapes of the trajectory flow. The empirical measures of the message trajectories well describe the interference, but not the congestion; the latter requires introducing an additional empirical measure. Our results remain valid under replacing the two penalization terms by more general functionals of these two empirical measures. [ABSTRACT FROM AUTHOR]

We consider the weighted anisotropic integral functionals ..., where Ω ⊂ R n (n > 2) is a bounded open set, u: Ω ⊂ Rn → R, f: Ω × Rn → [0, +∞) is a Carathéodory function which satisfies the nonstandard growth condition ..., where c > 0 is a constant, 1 < pi < qi < n, i = 1, 2, ..., n, νi is the positive weighted function on Ω and .... By using the weighted anisotropic Sobolev inequality and the iteration Lemma, we prove the higher integrability for the minimizer u of I(u) when the boundary datum has the higher integrability. We also obtain the global boundednesses of exponential form and L ∞(Ω) for the minimizer, respectively. Furthermore, similar results for the minimizer of the obstacle problem to I(u) are given. [ABSTRACT FROM AUTHOR]

Abstract We study local quasiminimizers of the Dirichlet energy under generalized growth conditions. Special cases include standard, variable exponent and double phase growths. We show that the gradient of a local quasiminimizer has local higher integrability. [ABSTRACT FROM AUTHOR]

Abstract We establish the higher differentiability and the higher integrability for the gradient of vectorial minimizers of integral functionals with (p , q) -growth conditions. We assume that the non-homogeneous densities are uniformly convex and have a radial structure, with respect to the gradient variable, only at infinity. The results are obtained under a possibly discontinuous dependence on the spatial variable of the integrand. [ABSTRACT FROM AUTHOR]

We consider polyconvex functionals of the Calculus of Variations defined on maps from the three-dimensional Euclidean space into itself. Counterexamples show that minimizers need not to be bounded. We find conditions on the structure of the functional, which force minimizers to be locally bounded. [ABSTRACT FROM AUTHOR]

We prove the existence of minimal heteroclinic orbits for a class of fourth order O.D.E. systems with variational structure. In our general set-up, the set of equilibria of these systems is a union of manifolds, and the heteroclinic orbits connect two disjoint components of this set. [ABSTRACT FROM AUTHOR]

Results on the convergence of minimizers and minimum values of integral and more general functionals Js: W1,p(Ωs) → ℝ on the sets Us(hs) = {v ∈ W1,p(Ωs): hs(v) ≤ 0 a.e. in Ωs}, where p > 1, {Ωs} is a sequence of domains contained in a bounded domain Ω of ℝn (n > 2), and {hs} is a sequence of functions on ℝ, are announced. [ABSTRACT FROM AUTHOR]

In the paper [1] “Boundary regularity of minimizers of p ( x ) -energy functionals”, some modifications are needed. 1. The exponent p 2 = p 2 ( 2 R ) in the statement of Theorem 2.6 should be p 2 ( ρ ) . According to this correction, we should modify the proof of Theorem 3.2. 2. In Theorem 1.1, the domain Ω is assumed to have the Lipschitz boundary ∂Ω. However, we need to assume that ∂Ω is in the class C 1 . [ABSTRACT FROM AUTHOR]

In this paper we show that the classical field theory ofWeierstrass-Hilbert can be strengthen on applying direct methods. Concretely, given a field of extremals and an extremal that is an element of the field, we can show that the latter gives minimum in the class of Lipschitz functions with the same boundary data and with the graphs in the set covered by the field. We suggest the two proofs: a modern one (exploiting Tonelli's Theorem on lower semicontinuity of integral functionals with respect to the weak convergence of admissible functions in W ) and the one based only on arguments available already in the 19th century. [ABSTRACT FROM AUTHOR]

We prove partial regularity of minimizers u for functionals of the following type A ( u ) = ∫ Ω [ ( A i j α β ( x , u ) D α u i D β u j ) p ( x ) / 2 + g ( x , u , D u ) ] d x , assuming that A i j α β ( x , u ) and p ( x ) are sufficiently smooth and that p ( x ) > 1 . We prove that u ∈ C 0 , α ( Ω 0 ) for some α ∈ ( 0 , 1 ) and an open set Ω 0 ⊂ Ω with H m − γ 1 ( Ω − Ω 0 ) = 0 , where H s denotes the s -dimensional Hausdorff measure and γ 1 = inf Ω p ( x ) . [ABSTRACT FROM AUTHOR]

We analyze the Lawrence–Doniach model for three-dimensional highly anisotropic superconductors with layered structure. For such a superconductor occupying a bounded generalized cylinder in ℝ³ with equally spaced parallel layers, we assume an applied magnetic field that is perpendicular to the layers with intensity hex ∼ ∣ ln ∈ ∣ as ∈ → 0, where ∈ is the reciprocal of the Ginzburg–Landau parameter. We prove Gamma-convergence of the Lawrence–Doniach energy as ∈ and the interlayer distance s tend to zero, under the additional assumption that the layers are weakly coupled (i.e., s ⪢ ∈). [ABSTRACT FROM AUTHOR]

Non-convex functionals have shown sharper results in signal reconstruction as compared to convex ones, although the existence of a minimum has not been established in general. This paper addresses the study of a general class of either convex or non-convex functionals for denoising signals which combines two general terms for fitting and smoothing purposes, respectively. The first one measures how close a signal is to the original noisy signal. The second term aims at removing noise while preserving some expected characteristics in the true signal such as edges and fine details. A theoretical proof of the existence of a minimum for functionals of this class is presented. The main merit of this result is to show the existence of minimizer for a large family of non-convex functionals. [ABSTRACT FROM AUTHOR]

In this article, we give sufficient conditions for the convergence of minimizers and minimum values of integral and more general functionals on sets of functions defined by bilateral constraints in a sequence of domains Ω s contained in a bounded domain Ω of R n ( n ⩾ 2 ). We study the case where the lower constraint is zero and the upper constraint is an arbitrary nonnegative measurable function on Ω . The statements of our main results include the condition of the Γ -convergence of the functionals (defined on the spaces W 1 , p ( Ω s ) ) to a functional defined on W 1 , p ( Ω ) and the condition of the strong connectedness of the spaces W 1 , p ( Ω s ) with the space W 1 , p ( Ω ) , where p > 1 . At the same time, because of the specificity of the imposed constraints, the exhaustion condition of the domain Ω by the domains Ω s and the proposed requirement on the behavior of the integrands of the principal components of the considered functionals are also important for our convergence results. [ABSTRACT FROM AUTHOR]

It is widely acknowledged that polarity sensitivity pertains to the lexical nature of NPI minimizers, phrases denoting a minimal quantity, extent or degree. This paper, however, proposes that so-called 'negative polarity' of Mandarin Chinese minimizers ( yi-CL-N 'one.CL+N' and yidian-N 'one.point N') is not lexically determined, but is facilitated by utilizing the existing lian... dou 'including... all' EVEN construction. Specifically, total negation is decomposed into a scalar operator lian, which evokes a set of order ranked alternatives determined in context, and the maximizing/universal operator dou 'all' that quantifies over the contextual alternatives plus the focused minimizer, which is placed at the end of the scale. The scalar minimizer syntactically scopes over the negation to represent the logic of ∀¬. This paper further distinguishes minimizers from lexical NPI renhe 'any' with respect to (i) scoping out of the negation for the former, (ii) being irrelevant to the non-veridical licensing conditions that otherwise license any and NPI- renhe, (iii) a clausemate relation between dou and negation, and (iv) the lack of intervention effects of strong quantifiers between the minimizers and negation. The study lends further support to the claim that scalar EVEN is construed with minimizers. A comparison of Chinese minimizers with those in Hindi and Japanese has an implication for varieties of coding polarity ranging from purely lexical to syntactical means crosslinguistically. [ABSTRACT FROM AUTHOR]

Biological aggregations such as insect swarms and bird flocks may arise from a combination of social interactions and environmental cues. We focus on nonlocal continuum equations, which are often used to model aggregations, and yet which pose significant analytical and computational challenges. Beginning with a particular nonlocal aggregation model [C. M. Topaz, A. L. Bertozzi, and M. A. Lewis, Bull. Math. Bio., 68 (2006), pp. 1601-1623], we derive the minimal well-posed longwave approximation, which is a degenerate Cahn-Hilliard equation. Energy minimizers of this reduced, local model retain many salient features of those of the nonlocal model, especially for large populations and away from an aggregation's boundaries. Using the Cahn-Hilliard model as a testbed, we investigate the degree to which an external potential modeling food sources can be used to suppress peak population density, which is essential for controlling locust outbreaks. A random distribution of food sources tends to increase peak density above its intrinsic value, while a periodic pattern of food sources can decrease it. [ABSTRACT FROM AUTHOR]

We prove partial regularity for vector-valued minimizers u of the variational integral ∫ [ f ( x , u , D u ) + g ( x , u ) ] d x , where f is strictly quasiconvex, of polynomial growth and continuous, but where g is only a bounded Carathéodory function. We present an elementary proof for the special case of strict convexity and quadratic growth of f ( x , u , ·). [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the regularity on the boundary ∂ Ω of a bounded open set Ω ⊂ R m for minimizers u for p ( x ) -energy functionals of the following type E ( u ; Ω ) : = ∫ Ω ( g α β ( x ) G i j ( u ) D α u i ( x ) D β u j ( x ) ) p ( x ) / 2 d x where ( g α β ( x ) ) and ( G i j ( u ) ) are symmetric positive definite matrices whose entries are continuous functions and p ( x ) ≥ 2 is a continuous function. The authors prove that such minimizers u have no singular points on the boundary. [ABSTRACT FROM AUTHOR]

In this paper, systems of quasilinear elliptic equations are investigated, which involve critical homogeneous nonlinearities and deferent Hardy-type terms. By variational methods and careful analysis, positive minimizers of the related best Sobolev constants are found and the existence of positive solutions to the systems is verified. The results are new even in the case p = 2 . [ABSTRACT FROM AUTHOR]

We deal with variational integrals ∫ Ω f ( x , D u ( x ) ) d x and we consider a minimizer u : Ω ⊂ R n → R among all functions that agree on the boundary ∂ Ω with some fixed boundary value u ∗ . We assume that the boundary datum u ∗ makes the density f ( x , D u ∗ ( x ) ) more integrable and we prove that the minimizer u enjoy higher integrability. [ABSTRACT FROM AUTHOR]

In this paper, we establish Liouville type theorem for boundedness solutions with finite Morse index of the following mixed boundary value problems: $-\Delta u=|u|^{p-1}u$ in $\mathbb{R}^{N}_{+}$, $\frac{\partial u}{\partial\nu}=|u|^{q-1}u$ on $\Gamma_{1}$, $\frac{\partial u}{\partial\nu}=0$ on $\Gamma_{0}$, and $-\Delta u=|u|^{p-1}u$ in $\mathbb{R}^{N}_{+}$, $\frac{\partial u}{\partial\nu}=|u|^{q-1}u$ on $\Gamma_{1}$, $u=0$ on $\Gamma_{0}$, where $\mathbb{R}^{N}_{+} =\{x\in\mathbb{R}^{N}:x_{N}>0\}$, $\Gamma_{1}=\{x\in \mathbb{R}^{N}:x_{N}=0,x_{1}<0\}$ and $\Gamma_{0}=\{x\in\mathbb{R}^{N}:x_{N}=0,x_{1}>0\}$. The exponents p, q satisfy the conditions in Theorem 1.1. [ABSTRACT FROM AUTHOR]

In this paper, we study systems of elliptic equations involving critical nonlinearities and different Hardy-type terms. By variational methods, the existence of minimizers to Rayleigh quotients and ground state solutions to the systems is proved for large ranges of parameters. [ABSTRACT FROM AUTHOR]