Global Pointwise Research Articles

Quasilinear Lane–Emden type systems with sub-natural growth terms

Global pointwise estimates are obtained for quasilinear Lane–Emden-type systems involving measures in the “sublinear growth” rate. We give necessary and sufficient conditions for existence expressed in terms of Wolff’s potential. Our approach is based on recent advances due to Kilpeläinen and Malý in the potential theory. This method enables us to treat several problems, such as equations involving general quasilinear operators and fractional Laplacian.

Regularity theory and Green’s function for elliptic equations with lower order terms in unbounded domains

We consider elliptic operators in divergence form with lower order terms of the form Lu = -{{textrm{div}}}(A cdot nabla u + b u ) - c cdot nabla u - du, in an open set Omega subset mathbb {R}^n, n ge 3, with possibly infinite Lebesgue measure. We assume that the n times n matrix A is uniformly elliptic with real, merely bounded and possibly non-symmetric coefficients, and either b, c in L^{n,infty }_{text {loc}}({Omega }) and d in L_{text {loc}}^{frac{n}{2}, infty }(Omega ), or |b|^2, |c|^2, |d| in mathcal {K}_{text {loc}}(Omega ), where mathcal {K}_{text {loc}}(Omega ) stands for the local Stummel–Kato class. Let {mathcal {K}_{text {Dini}}}(Omega ) be a variant of mathcal {K}(Omega ) satisfying a Carleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory for solutions of Lu = f - {{textrm{div}}}g, where f and |g|^2 in {mathcal {K}_{text {Dini}}}(Omega ) if, for q in [n, infty ), any of the following assumptions holds: (i) |b|^2, |d| in {mathcal {K}_{text {Dini}}}(Omega ) and either c in L^{n,q}_{text {loc}}(Omega ) or |c|^2 in mathcal {K}_{text {loc}}(Omega ); (ii) {{textrm{div}}}b +d le 0 and either b+c in L^{n,q}_{text {loc}}(Omega ) or |b+c|^2 in mathcal {K}_{text {loc}}(Omega ); (iii) -{{textrm{div}}}c + d le 0 and |b+c|^2 in {mathcal {K}_{text {Dini}}}(Omega ). We also prove a Wiener-type criterion for boundary regularity. Assuming global conditions on the coefficients, we show that the variational Dirichlet problem is well-posed and, assuming -{{textrm{div}}}c +d le 0, we construct the Green’s function associated with L satisfying quantitative estimates. Under the additional hypothesis |b+c|^2 in mathcal {K}'(Omega ), we show that it satisfies global pointwise bounds and also construct the Green’s function associated with the formal adjoint operator of L. An important feature of our results is that all the estimates are scale invariant and independent of Omega , while we do not assume smallness of the norms of the coefficients or coercivity of the associated bilinear form.

Pointwise error estimate of the compact difference methods for the fourth‐order parabolic equations with the third Neumann boundary conditions

Various boundary value conditions have been endowed for the fourth‐order differential equations. In the current work, a class of the fourth‐order parabolic equations with the third Neumann boundary conditions is concerned, where the values of the second and third spatial derivatives of the unknown function are given at the boundary. A novel average is defined to get the compact approximation near the boundary. Then a compact difference scheme is derived by using the weighted average and the method of order reduction. Due to the special and highly accurate discretization at the boundary, the related terms have to be handled skillfully during the analysis. By the energy method, the unique solvability, unconditional stability, and convergence of the derived compact difference scheme are strictly proved. The analytical difficulties caused by the boundary approximation are successfully overcome. As far as we know, this is the first time that the global pointwise fourth‐order convergence in space of the difference approach for this problem is achieved. In addition, the generalization to solve the case with a space‐dependent reaction coefficient is discussed. Finally, two numerical examples are computed to verify the accuracy of proposed numerical schemes.

Global pointwise estimates of positive solutions to sublinear equations

Bilateral pointwise estimates are provided for positive solutions u u to the sublinear integral equation u = G ( σ u q ) + f in Ω , \begin{equation*} u = \mathbf {G}(\sigma u^q) + f \quad \text {in } \ \Omega , \end{equation*} for 0 > q > 1 0 > q > 1 , where σ ≥ 0 \sigma \ge 0 is a measurable function or a Radon measure, f ≥ 0 f \ge 0 , and G \mathbf {G} is the integral operator associated with a positive kernel G G on Ω × Ω \Omega \times \Omega . The main results, which include the existence criteria and uniqueness of solutions, hold true for quasimetric, or quasimetrically modifiable kernels G G . As a consequence, bilateral estimates are obtained, along with existence and uniqueness, for positive solutions u u , possibly unbounded, to sublinear elliptic equations involving the fractional Laplacian, ( − Δ ) α 2 u = σ u q + μ in Ω , u = 0 in Ω c , \begin{equation*} (-\Delta )^{\frac {\alpha }{2}} u = \sigma u^q + \mu \quad \text {in}\quad \Omega , \quad u=0 \quad \text {in}\,\, \Omega ^c, \end{equation*} where 0 > q > 1 0>q>1 , and μ , σ ≥ 0 \mu , \sigma \ge 0 are measurable functions, or Radon measures, on a bounded uniform domain Ω ⊂ R n \Omega \subset \mathbb {R}^n for 0 > α ≤ 2 0 > \alpha \le 2 , or on the entire space R n \mathbb {R}^n , a ball or half-space, for 0 > α > n 0 > \alpha >n .

A Comparison Estimate for Singular p-Laplace Equations and Its Consequences

Comparison estimates are an important technical device in the study of regularity problems for quasilinear possibly degenerate elliptic and parabolic equations. Such tools have been employed indispensably in many papers of Mingione, Duzaar-Mingione, and Kuusi-Mingione, etc. on certain measure datum problems to obtain pointwise bounds for solutions and their full or fractional derivatives in terms of appropriate linear or nonlinear potentials. However, a comparison estimate for $p$-Laplace type elliptic equations with measure data is still unavailable in the strongly singular case $1< p\leq \frac{3n-2}{2n-1}$, where $n\geq 2$ is the dimension of the ambient space. This issue will be completely resolved in this work by proving a comparison estimate in a slightly larger range $1<p<3/2$. Applications include a `sublinear' Poincar\'e type inequality, pointwise bounds for solutions and their derivatives by Wolff's and Riesz's potentials, respectively. Some global pointwise and weighted estimates are also obtained for bounded domains, which enable us to treat a quasilinear Riccati type equation with possibly sublinear growth in the gradient.

Monotone Inclusions, Acceleration, and Closed-Loop Control

We propose and analyze a new dynamical system with a closed-loop control law in a Hilbert space [Formula: see text], aiming to shed light on the acceleration phenomenon for monotone inclusion problems, which unifies a broad class of optimization, saddle point, and variational inequality (VI) problems under a single framework. Given an operator [Formula: see text] that is maximal monotone, we propose a closed-loop control system that is governed by the operator [Formula: see text], where a feedback law [Formula: see text] is tuned by the resolution of the algebraic equation [Formula: see text] for some [Formula: see text]. Our first contribution is to prove the existence and uniqueness of a global solution via the Cauchy–Lipschitz theorem. We present a simple Lyapunov function for establishing the weak convergence of trajectories via the Opial lemma and strong convergence results under additional conditions. We then prove a global ergodic convergence rate of [Formula: see text] in terms of a gap function and a global pointwise convergence rate of [Formula: see text] in terms of a residue function. Local linear convergence is established in terms of a distance function under an error bound condition. Further, we provide an algorithmic framework based on the implicit discretization of our system in a Euclidean setting, generalizing the large-step hybrid proximal extragradient framework. Even though the discrete-time analysis is a simplification and generalization of existing analyses for a bounded domain, it is largely motivated by the aforementioned continuous-time analysis, illustrating the fundamental role that the closed-loop control plays in acceleration in monotone inclusion. A highlight of our analysis is a new result concerning [Formula: see text]-order tensor algorithms for monotone inclusion problems, complementing the recent analysis for saddle point and VI problems. Funding: This work was supported in part by the Mathematical Data Science Program of the Office of Naval Research [Grant N00014-18-1-2764] and by the Vannevar Bush Faculty Fellowship Program [Grant N00014-21-1-2941].

Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

For the Stokes system in the half space, Kang [Math. Ann. 331 (2005), pp. 87-109] showed that a solution generated by a compactly supported, Hölder continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in an earlier paper. We also examine the Stokes flows generated by dipole bumps’ boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary.

The Cauchy problem for the fast p-Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast p-Laplacian evolution equation ut=Δpu on the whole Euclidean space, in the so-called “good fast diffusion range” 2NN+1<p<2. It is well-known that non-negative solutions behave for large times as B, the Barenblatt (or fundamental) solution, which has an explicit expression.We prove the so-called Global Harnack Principle (GHP), that is, precise global pointwise upper and lower estimates of nonnegative solutions in terms of B. This can be considered the nonlinear counterpart of the celebrated Gaussian estimates for the linear heat equation.We characterize the maximal (hence optimal) class of initial data such that the GHP holds, by means of an integral tail condition, easy to check. The GHP is then used as a tool to analyze the fine asymptotic behaviour for large times. For initial data that satisfy the same integral condition, we prove that the corresponding solutions behave like the Barenblatt with the same mass, uniformly in relative error.When the integral tail condition is not satisfied we show that both the GHP and the uniform convergence in relative error, do not hold anymore, and we provide also explicit counterexamples. We then prove a “generalized GHP”, that is, pointwise upper and lower bounds in terms of explicit profiles with a tail different from B. Finally, we derive sharp global quantitative upper bounds of the modulus of the gradient of the solution, and, when data are radially decreasing, we show uniform convergence in relative error for the gradients.To the best of our knowledge, analogous issues for the linear heat equation p=2, do not possess such clear answers, only partial results are known.

Global estimates for the fundamental solution of homogeneous Hörmander operators

Let $${\mathcal {L}}=\sum _{j=1}^{m}X_{j}^{2}$$ be a Hörmander sum of squares of vector fields in $${\mathbb {R}}^{n}$$ , where any $$X_{j}$$ is homogeneous of degree 1 with respect to a family of non-isotropic dilations in $${\mathbb {R}}^{n}$$ . Then, $${\mathcal {L}}$$ is known to admit a global fundamental solution $$\Gamma (x;y)$$ that can be represented as the integral of a fundamental solution of a sublaplacian operator on a lifting space $${\mathbb {R}}^{n}\times {\mathbb {R}}^{p}$$ , equipped with a Carnot group structure. The aim of this paper is to prove global pointwise (upper and lower) estimates of $$\Gamma$$ , in terms of the Carnot–Carathéodory distance induced by $$X=\{X_{1},\ldots ,X_{m}\}$$ on $${\mathbb {R}}^{n}$$ , as well as global pointwise (upper) estimates for the X-derivatives of any order of $$\Gamma$$ , together with suitable integral representations of these derivatives. The least dimensional case $$n=2$$ presents several peculiarities which are also investigated. Applications to the potential theory for $${\mathcal {L}}$$ and to singular-integral estimates for the kernel $$X_{i}X_{j}\Gamma$$ are also provided. Finally, most of the results about $$\Gamma$$ are extended to the case of Hörmander operators with drift $$\sum _{j=1}^{m}X_{j}^{2}+X_{0}$$ , where $$X_{0}$$ is 2-homogeneous and $$X_{1},\ldots ,X_{m}$$ are 1-homogeneous.

A low-cost post-processing technique improves weather forecasts around the world

Computer-generated weather forecasts divide the Earth’s surface into gridboxes, each currently spanning about 400 km2, and predict one value per gridbox. If weather varies markedly within a gridbox, forecasts for specific sites inevitably fail. Here we present a statistical post-processing method for ensemble forecasts that accounts for the degree of variation within each gridbox, bias on the gridbox scale, and the weather dependence of each. When applying this post-processing, skill improves substantially across the globe; for extreme rainfall, for example, useful forecasts extend 5 days ahead, compared to less than 1 day without post-processing. Skill improvements are attributed to creation of huge calibration datasets by aggregating, globally rather than locally, forecast-observation differences wherever and whenever the observed “weather type” was similar. A strong focus on meteorological understanding also contributes. We suggest that applications for our methodology include improved flash flood warnings, physics-related insights into model weaknesses and global pointwise re-analyses.

Pointwise gradient bounds for a class of very singular quasilinear elliptic equations

A pointwise gradient bound for weak solutions to Dirichlet problem for quasilinear elliptic equations \begin{document}$ -\mathrm{div}(\mathbb{A}(x,\nabla u)) = \mu $\end{document} is established via Wolff type potentials. It is worthwhile to note that the model case of \begin{document}$ \mathbb{A} $\end{document} here is the non-degenerate \begin{document}$ p $\end{document} -Laplacian operator. The central objective is to extend the pointwise regularity results in [Q.-H. Nguyen, N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equations with measure data, J. Funct. Anal. 278(5) (2020), 108391] to the very singular case \begin{document}$ 1 , where the data \begin{document}$ \mu $\end{document} on right-hand side is assumed belonging to some classes that close to \begin{document}$ L^1 $\end{document} . Moreover, a global pointwise estimate for gradient of weak solutions to such problem is also obtained under the additional assumption that \begin{document}$ \Omega $\end{document} is sufficiently flat in the Reifenberg sense.

Green Functions for the Pressure of Stokes Systems

Abstract We study Green functions for the pressure of stationary Stokes systems in a (possibly unbounded) domain $\Omega \subset \mathbb{R}^d$, where $d\ge 2$. We construct the Green function when coefficients are merely measurable in one direction and have Dini mean oscillation in the other directions and $\Omega $ is such that the divergence equation is solvable there. We also establish global pointwise bounds for the Green function and its derivatives when coefficients have Dini mean oscillation and $\Omega $ has a $C^{1,\textrm{Dini}}$ boundary. Green functions for the flow velocity of Stokes systems are also considered.

Quasilinear elliptic equations with exponential nonlinearity and measure data

Let be either a bounded domain or the entire space. This paper concerns existence and global pointwise estimates in terms of Wolff potentials of solutions to quasilinear and Hessian equations of Lane‐Emden type: under minimal assumptions on σ and ω, where σ is an A∞ weight and ω is a nonnegative Borel measure, and with γ&gt;0, β ≥ 1. Our methods are based on systematic application of Wolff potentials and good‐λ inequality for fractional maximal potential.

Hörmander vector fields equipped with dilations: lifting, Lie-group construction, applications

Let X = {X1, ... ,Xm} be a set of Hormander vector fields in Rn, where any Xj is homogeneous of degree 1 with respect to a family of non-isotropic dilations in Rn. If N is the dimension of Lie{X}, we can either lift X to a system of generators of a higher dimensional Carnot group on Rn (if N>n, or we can equip Rn with a Carnot group structure with Lie algebra equal to Lie{X} (if N=n). We shall deduce these facts via a local-to-global procedure (available in the homogeneous setting), starting from more general results on the lifting of finite-dimensional Lie algebras of vector fields. The use of the Baker-Campbell-Hausdorff Theorem is crucial. Due to homogeneity, the lifting procedure is simpler than Rothschild-Stein's lifting technique. We finally provide applications to the study of the fundamental solution Gamma for the Hormander sum of squares Σj=1..m Xj2, including global pointwise estimates of Gamma and of its X-derivatives in terms of the Carnot-Caratheodory distance induced by X on Rn.

Pointwise gradient estimates for a class of singular quasilinear equations with measure data

Local and global pointwise gradient estimates are obtained for solutions to the quasilinear elliptic equation with measure data −div(A(x,∇u))=μ in a bounded and possibly nonsmooth domain Ω in Rn. Here div(A(x,∇u)) is modeled after the p-Laplacian. Our results extend earlier known results to the singular case in which 3n−22n−1<p≤2−1n.

Differentiation of skin incision and laparoscopic trocar insertion via quantifying transient bradycardia measured by electrocardiogram.

Most surgical procedures involve structures deeper than the skin. However, the difference in surgical noxious stimulation between skin incision and laparoscopic trocar insertion is unknown. By analyzing instantaneous heart rate (IHR) calculated from the electrocardiogram, in particular the transient bradycardia in response to surgical stimuli, this study investigates surgical noxious stimuli arising from skin incision and laparoscopic trocar insertion, and their difference. Thirty-five patients undergoing laparoscopic cholecystectomy were enrolled in this prospective observational study. Sequential surgical steps including umbilical skin incision (11mm), umbilical trocar insertion (11mm), xiphoid skin incision (5mm), xiphoid trocar insertion (5mm), subcostal skin incision (3mm), and subcostal trocar insertion (3mm) were investigated. IHR was derived from electrocardiography and calculated by the modern time-varying power spectrum. Similar to the classical heart rate variability analysis, the time-varying low frequency power (tvLF), time-varying high frequency power (tvHF), and tvLF-to-tvHF ratio (tvLHR) were calculated. Prediction probability (PK) analysis and global pointwise F-test were used to compare the statistical performance between indices and the heart rate readings from the patient monitor. Analysis of IHR showed that surgical stimulus elicits a transient bradycardia, followed by the increase of heart rate. Transient bradycardia is more significant in trocar insertion than skin incision (p < 0.001 for tvHF). The IHR change quantifies differential responses to different surgical intensity. Serial PK analysis demonstrates de-sensitization in skin incision, but not in laparoscopic trocar insertion. Quantitative indices present the transient bradycardia introduced by noxious stimulation. The results indicate different effects between skin incision and trocar insertion.

Global Existence and Pointwise Estimates of Solutions to Generalized Kuramoto-Sivashinsky System in Multi-Dimensions

The Cauchy problem of the generalized Kuramoto-Sivashinsky equation in multi-dimensions (n ≥ 3) is considered. Based on Green’s function method, some ingenious energy estimates are given. Then the global existence and pointwise convergence rates of the classical solutions are established. Furthermore, the Lp convergence rate of the solution is obtained.

Green’s function for second order elliptic equations with singular lower order coefficients

We construct Green’s function for second order elliptic operators of the form in a domain and obtain pointwise bounds, as well as Lorentz space bounds. We assume that the matrix of principal coefficients is uniformly elliptic and bounded and the lower order coefficients b, c, and d belong to certain Lebesgue classes and satisfy the condition . In particular, we allow the lower order coefficients to be singular. We also obtain the global pointwise bounds for the gradient of Green’s function in the case when the mean oscillations of the coefficients and b satisfy the Dini conditions and the domain is .

MiKM: multi-step inertial Krasnosel’skiǐ–Mann algorithm and its applications

In this paper, we first introduce a multi-step inertial Krasnosel’skiǐ–Mann algorithm (MiKM) for nonexpansive operators in real Hilbert spaces. We give the convergence of the MiKM by investigating the convergence of the Krasnosel’skiǐ–Mann algorithm with perturbations. We also establish global pointwise and ergodic iteration complexity bounds of the Krasnosel’skiǐ–Mann algorithm with perturbations. Based on the MiKM, we construct some multi-step inertial splitting methods, including the multi-step inertial Douglas–Rachford splitting method (MiDRS), the multi-step inertial forward–backward splitting method, multi-step inertial backward–forward splitting method and and the multi-step inertial Davis–Yin splitting method. Numerical experiments are provided to illustrate the advantage of the MiDRS over the one-step inertial DRS and the original DRS.

Green Functions of Conormal Derivative Problems for Stationary Stokes System

We study Green functions for stationary Stokes systems satisfying the conormal derivative boundary condition. We establish existence, uniqueness, and various estimates for the Green function under the assumption that weak solutions of the Stokes system are continuous in the interior of the domain. Also, we establish the global pointwise bound for the Green function under the additional assumption that weak solutions of the conormal derivative problem for the Stokes system are locally bounded up to the boundary. We provide some examples satisfying such continuity and boundedness properties.