Lp Theory Research Articles

SL(n) contravariant function-valued valuations on polytopes

We present a complete classification of SL(n) contravariant, C(Rn∖{o})-valued valuations on polytopes, without any additional assumptions. It extends the previous results of the second author Li (2020) [10] which have a good connection with the Lp and Orlicz Brunn-Minkowski theory. Additionally, our results deduce a complete classification of SL(n) contravariant symmetric-tensor-valued valuations on polytopes.

2D Linear Canonical Transforms on Lp and Applications

As Fourier transformations of Lp functions are the mathematical basis of various applications, it is necessary to develop Lp theory for 2D-LCT before any further rigorous mathematical investigation of such transformations. In this paper, we study this Lp theory for 1≤p<∞. By defining an appropriate convolution, we obtain a result about the inverse of 2D-LCT on L1(R2). Together with the Plancherel identity and Hausdorff–Young inequality, we establish Lp(R2) multiplier theory and Littlewood–Paley theorems associated with the 2D-LCT. As applications, we demonstrate the recovery of the L1(R2) signal function by simulation. Moreover, we present a real-life application of such a theory of 2D-LCT by encrypting and decrypting real images.

An ℓp theory of PCA and spectral clustering

Principal Component Analysis (PCA) is a powerful tool in statistics and machine learning. While existing study of PCA focuses on the recovery of principal components and their associated eigenvalues, there are few precise characterizations of individual principal component scores that yield low-dimensional embedding of samples. That hinders the analysis of various spectral methods. In this paper, we first develop an ℓp perturbation theory for a hollowed version of PCA in Hilbert spaces which provably improves upon the vanilla PCA in the presence of heteroscedastic noises. Through a novel ℓp analysis of eigenvectors, we investigate entrywise behaviors of principal component score vectors and show that they can be approximated by linear functionals of the Gram matrix in ℓp norm, which includes ℓ2 and ℓ∞ as special cases. For sub-Gaussian mixture models, the choice of p giving optimal bounds depends on the signal-to-noise ratio, which further yields optimality guarantees for spectral clustering. For contextual community detection, the ℓp theory leads to simple spectral algorithms that achieve the information threshold for exact recovery and the optimal misclassification rate.

The Lp-Lq decay estimate for the multidimensional compressible flow with free surface in the exterior domain

The aim of this paper is to develop the general Lp theory for the barotropic compressible Navier-Stokes equations with the free boundary condition in the exterior domain in RN (N≥3). By the spectral analysis, we obtain the classical Lp-Lq decay estimate for the linearized model problem (with variable coefficients) in view of the partial Lagrangian transformation.

Lp theory for Boussinesq system with Dirichlet boundary conditions

ABSTRACTWe consider the stationary Boussinesq system with non-homogeneous Dirichlet boundary conditions in a bounded domain of class with a possibly disconnected boundary. We prove the existence of weak solutions in , strong solutions in and very weak solutions in of the stationary Boussinesq system by assuming that the fluxes of the velocity are sufficiently small. Finally, as it is expected, we obtain the uniqueness of the solution by considering small data.

Apolipoprotein(a) is the Product of a Pseudogene: Implications for the Pathophysiology of Lipoprotein(a).

Apolipoprotein(a) [apo(a)] is an apolipoprotein unique to lipoprotein(a) [Lp(a)]. Although it has no known function, Lp(a) is a risk factor for accelerated atherothrombosis. We hypothesize that LPA, the gene which encodes apo(a), is a heretofore unrecognized unprocessed pseudogene created by duplication of PLG, the gene which encodes plasminogen. Unprocessed pseudogenes are genes which were created by duplication of functional genes and subsequently lost function after acquiring various mutations. This hypothesis explains many of the unusual features of Lp(a) and apo(a). Also, this hypothesis has implications for the therapy of elevated Lp(a) and atherothrombosis theory. Because apo(a) is functionless, the diseases associated with elevated levels of Lp(a) are due to its impact on blood viscosity.

Boundary estimates in homogenization of systems of elasticity on Lipschitz domains: Lp theory

The primary purpose of this paper is to investigate a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization in Lipschitz domains. As a consequence, for d≥4, we prove that the Lp Neumann and Lp Dirichlet boundary value problems for systems of second order linear elasticity are uniquely solvable for 2(d−1)d+1−δ<p<2+δ and 2−δ<p<2(d−1)d−3+δ respectively.

Regularity lemmas in a Banach space setting

Szemerédi's regularity lemma is a fundamental tool in extremal graph theory, theoretical computer science and combinatorial number theory. Lovász and Szegedy [Lovász, L., and B. Szegedy, Szemerédi's Lemma for the analyst, Geom. Funct. Anal. 17 (2007), 252–270] gave a Hilbert space interpretation of the lemma and an interpretation in terms of compactness of the space of graph limits. In this paper we prove several compactness results in a Banach space setting, generalising results of Lovász and Szegedy [Lovász, L., and B. Szegedy, Szemerédi's Lemma for the analyst, Geom. Funct. Anal. 17 (2007), 252–270] as well as a result of Borgs, Chayes, Cohn and Zhao [Borgs, C., J.T. Chayes, H. Cohn, and Y. Zhao, An Lp theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions, arXiv preprint arXiv:1401.2906 (2014)].

BMO estimate to A-harmonic systems with discontinuous coefficients

In this note, we prove a global BMO estimate of the gradient of each weak solution to a class of nonlinear degenerate elliptic system with leading coefficients satisfying vanishing means oscillations. This conclusion perfects nonlinear Calderón–Zygmund theory in the limiting case to certain degenerate elliptic system. As an application of the BMO estimate, we give another proof to its Lp theory with 1<p<∞ to linear elliptic system with VMO leading coefficients by Stampacchia’s interpolation and duality.

Higher regularity of the free boundary in the elliptic Signorini problem

In this paper we study the higher regularity of the free boundary for the elliptic Signorini problem. By using a partial hodograph–Legendre transformation we show that the regular part of the free boundary is real analytic. The first complication in the study is the invertibility of the hodograph transform (which is only C0,1/2) which can be overcome by studying the precise asymptotic behavior of the solutions near regular free boundary points. The second and main complication in the study is that the equation satisfied by the Legendre transform is degenerate. However, the equation has a subelliptic structure and can be viewed as a perturbation of the Baouendi–Grushin operator. By using the Lp theory available for that operator, we can bootstrap the regularity of the Legendre transform up to real analyticity, which implies the real analyticity of the free boundary.

Qualitative analysis of a time‐delayed free boundary problem for tumor growth under the action of external inhibitors

In this paper, a time‐delayed free boundary problem for tumor growth under the action of external inhibitors is studied. It is assumed that the process of proliferation is delayed compared with apoptosis. By Lp theory of parabolic equations, the Banach fixed point theorem and the continuation theorem, the existence and uniqueness of a global solution is proved. The asymptotic behavior of the solution is also studied. The proof uses the comparison principle and the iteration method. Copyright © 2014 John Wiley &amp; Sons, Ltd.

An Lp theory for stationary radiative transfer

We present a self-contained analysis of the stationary radiative transfer equation in weighted spaces. The use of weighted spaces allows us to derive uniform a-priori estimates for under minimal assumptions on the parameters. By constructing an explicit example, we show that our estimates are sharp and cannot be improved in general. Better estimates are, however, derived under additional assumptions on the parameters. We also present estimates for derivatives and traces of the solution and formulate a natural energy space, for which the data-to-solution map becomes an isomorphism. As a side result, we are able to prove uniform convergence of the source iteration for all without the assumption of positive absorption that is frequently used in the literature.

Global Well-Posedness for the Generalized Large-Scale Semigeostrophic Equations

We prove existence and uniqueness of global classical solutions to the generalized large-scale semigeostrophic equations with periodic boundary conditions. This family of Hamiltonian balance models for rapidly rotating shallow water includes the L1 model derived by R. Salmon in 1985 and its 2006 generalization by the second author. The results are, under the physical restriction that the initial potential vorticity is positive, as strong as those available for the Euler equations of ideal fluid flow in two dimensions. Moreover, we identify a special case in which the velocity field is two derivatives smoother in Sobolev space as compared to the general case. Our results are based on careful estimates which show that, although the potential vorticity inversion is nonlinear, bounds on the potential vorticity inversion operator remain linear in derivatives of the potential vorticity. This permits the adaptation of an argument based on elliptic Lp theory, proposed by Yudovich in 1963 for proving existence and uniqueness of weak solutions for the two-dimensional Euler equations, to our particular nonlinear situation.

Analysis of a solid avascular tumor growth model with time delays in proliferation process

In this paper we study a free boundary problem modeling solid avascular tumor growth. The model is based on the reaction–diffusion dynamics and mass conservation law. The model is considered with time delays in proliferation process. The quasi-steady-state (i.e., d=0) is studied by Foryś and Bodnar [see U. Foryś, M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Modelling 37 (2003) 1201–1209]. In this paper we mainly consider the case d>0. In the case considered by Foryś and Bodnar, the model is reduced to an ordinary differential equation with time delay, but in the case d>0 the model cannot be reduced to an ordinary differential equation with time delay. By Lp theory of parabolic equations and the Banach fixed point theorem, we prove the existence and uniqueness of a local solutions and apply the continuation method to get the existence and uniqueness of a global solution. We also study the long time asymptotic behavior of the solutions under some conditions.

Research Spotlights

This issue's Research Spotlights article by Wei, Zheng, Chen, and Xia highlights the significance of charge transport and specifically centers on multiscale models capable of providing comprehensive predictions in complex settings. As the article states, charge transport is “omnipresent in nature and man-made devices” and includes a diverse range of applications from metal oxide semiconductors to fuel/solar/battery cells to “nature-made nano-bio transistors” or ion channels which effectively maintain ion concentrations within cells, the basic unit of structure and function in all living things. To appreciate the challenges associated with charge transport systems such as those found in sophisticated nano devices (mechanical, optical, electronic, and biological), the article underscores the importance of properly accounting for intricate molecular mechanisms, sophisticated material structures, chemical reactions, and potentially large spatial dimensions when these devices are embedded within large systems, e.g., fuel cells. The article then proceeds to develop variational multiscale models with a focus on addressing heterogeneous chemical and biological systems that are far from equilibrium. Numerous examples are discussed, including fuel cells, a host of nanofluidic devices, and in particular ion channels which are used to validate the variational multiscale models. Derivations are given for coupled Laplace--Beltrami (LB) and Poisson--Nernst--Planck (PNP) equations as well as coupled Laplace--Beltrami and Poisson--Boltzmann--Nernst--Planck (PBNP) equations and also Laplace--Beltrami coupled with Poisson--Nernst--Planck and Navier--Stokes equations. The authors demonstrate two important model features: (1) consistency between equilibrium LP and Poisson--Boltzmann theory and nonequilibrium LP and PNP theory at equilibrium, and (2) how the reduced equations associated with the LB and PBNP formulation can capture behavior obtained from the coupled LB-PNP equations in nonequilibrium settings. The differential geometry of surfaces as a means of separating macroscopic and microscopic domains also plays a prominent role in the paper. The article concludes with computational algorithms that implement the multiscale models and a number of interesting numerical experiments which include comparisons with experimental measurements of current-voltage curves for a specific ion channel.

Nonlinear multi-scale transforms: Lp theory

We treat the Lp theory (1≤p<∞) for univariate nonlinear subdivision schemes and multi-scale transforms based on the concept of offset invariance and nonlinear derived subdivision operators. The paper covers convergence, smoothness, and stability issues, and complements the recent survey [8], where results for Lp have been included without proof.

Quantifier elimination in the theory of Lp(Lq)-Banach lattices

We introduce the class of doubly atomless bands in Lp(Lq)-Banach lattices and show that this class is axiomatizable by positive bounded sentences in the language of Banach lattices. (Here p 6= q are fixed and in the interval 1 ≤ p, q < ∞ .) The theory of this class is complete (indeed, we show it is separably categorical) and model complete. Further, we show that it satisfies quantifier elimination if and only if the ratio p/q is not an integer. On the functional analytic side, the proof of the latter result uses a positive-coefficient version of the well known Rudin-Plotkin-Hardin extension theorems. 2000 Mathematics Subject Classification 03C65 (primary); 46B04, 46E30, 46M07 (secondary)

Solvability in Hardy space for second order elliptic equations

We obtain a priori estimates and solvability in Hardy type space in a bounded domain of Rn for second order elliptic equations with coefficients of limited smoothness. Such a result can be served as an endpoint case of the classical Lp(1 < p < ∞) theory for second order elliptic equations. Our approach is based on a standard technique of perturbation rather than that of integral representation formula.

Convex integration and the Lp theory of elliptic equations

Convex integration and the Lp theory of elliptic equations

Exterior Stokes problem in the half-space

The purpose of this work is to solve the exterior Stokes problem in the half-space \({\mathbb{R}{^n_+}}\) . We study the existence and the uniqueness of generalized solutions in weighted Lp theory with 1 < p < ∞. Moreover, we consider the case of strong solutions and very weak solutions. This paper extends the studies done in Alliot, Amrouche (Math. Methods Appl. 23:575–600, 2000) for an exterior Stokes problem in the whole space and in Amrouche, Bonzom (Exterior Problems in the Half-space, submitted) for the Laplace equation in the same geometry as here.