Regularity Results Research Articles

Formula omitted] partial regularity result for elliptic systems with discontinuous coefficients and Orlicz growth

This paper is devoted to the study of C0,α-regularity for weak solutions to elliptic systems in divergence form. The model of our interest is−div(μ(x)A(x,Du))=B(x,Du), where the coefficient μ is discontinuous and x∈Ω, with Ω a bounded open subset of Rn. In particular, we assume the matrix A satisfies a G growth condition in Orlicz spaces, and the inhomogeneity B satisfies a controllable, or a critical, growth.

On two-loop effective action of 2d sigma model

In this paper, we study two-loop contribution to the effective action of a two-dimensional non-linear sigma model. We derive a new formula, which can be applicable to a regularization of general type. As examples, we obtain known results for the dimensional regularization and investigate new types of a cutoff one. Also, we discuss non-local contributions and restrictions on the regularization.

Propagation of regularity for solutions to a Kawahara equation in two spatial dimensions

We prove a propagation of regularity result for solutions of an IVP for a Kawahara equation in two spatial dimensions. We show that if the initial data has sufficient smoothness and belongs to an appropriate Sobolev space when restricted to a half-plane {(x,y)|ax+by≥c0} for some a>0, b,c0∈R, then the solution u(t) lies in the same Sobolev space when restricted to the half-plane {(x,y)|ax+by≥c} for any real number c and any t such that 0<t≤T where T is the existence time of the solution. In addition, we prove a propagation of regularity result for initial data belonging to an anisotropic Sobolev space on a half-plane. These propagation of regularity results show that regularity of the initial data is transferred to the solution with infinite speed.

Short‐time existence for the network flow

AbstractThis paper contains a new proof of the short‐time existence for the flow by curvature of a network of curves in the plane. Appearing initially in metallurgy and as a model for the evolution of grain boundaries, this flow was later treated by Brakke using varifold methods. There is good reason to treat this problem by a direct PDE approach, but doing so requires one to deal with the singular nature of the PDE at the vertices of the network. This was handled in cases of increasing generality by Bronsard‐Reitich, Mantegazza‐Novaga‐Tortorelli and eventually, in the most general case of irregular networks by Ilmanen‐Neves‐Schulze. Although the present paper proves a result similar to the one in Ilmanen et al., the method here provides substantially more detailed information about how an irregular network “resolves” into a regular one. Either approach relies on the existence of self‐similar expanding solutions found in Mazzeo and Saez. As a precursor to the main theorem, we also prove an unexpected regularity result for the mixed Cauchy‐Dirichlet boundary problem for the linear heat equation on a manifold with boundary.

A sharp Lp-regularity result for second-order stochastic partial differential equations with unbounded and fully degenerate leading coefficients

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE)(0.1)du=(aij(ω,t)uxixj+f)dt+(σik(ω,t)uxi+gk)dwtk,u(0,x)=u0, where {wtk:k=1,2,⋯} is a sequence of independent Brownian motions. The coefficients are merely measurable in (ω,t) and can be unbounded and fully degenerate, that is, coefficients aij, σik merely satisfy(0.2)(αij(ω,t))d×d:=(aij(ω,t)−12∑k=1∞σik(ω,t)σjk(ω,t))≥0. In this article, we prove that there exists a unique solution u to (0.1), and(0.3)‖uxx‖Hpγ(τ,δ)≤N(d,p)(‖u0‖Bpγ+2(1−1/p)+‖f‖Hpγ(τ,δ1−p)+‖gx‖Hpγ(τ,|σ|pδ1−p,l2)p+‖gx‖Hpγ(τ,δ1−p/2,l2)), where p≥2, γ∈R, τ is an arbitrary stopping time, δ(ω,t) is the smallest eigenvalue of αij(ω,t), Hpγ(τ,δ) is a weighted stochastic Sobolev space, and Bpγ+2(1−1/p) is a stochastic Besov space.

The Neumann problem for the fractional Laplacian: regularity up to the boundary

We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if $u$ is a weak solution of $(-\Delta)^s u=f$ in $\Omega$, $\mathcal N_s u=0$ in $\Omega^c$, then $u$ is $C^\alpha$ up tp the boundary for some $\alpha>0$. Moreover, in case $s>\frac12$, we then show that $u\in C^{2s-1+\alpha}(\overline\Omega)$. To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections. Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary regularity result. Prior to our results, the interior regularity for these Neumann problems was well understood, but near the boundary even the continuity of solutions was open.

Asymptotics for Time-Fractional Venttsel’ Problems in Fractal Domains

In this study, we consider fractional-in-time Venttsel’ problems in fractal domains of the Koch type. Well-posedness and regularity results are given. In view of numerical approximation, we consider the associated approximating pre-fractal problems. Our main result is the convergence of the solutions of such problems towards the solution of the fractional-in-time Venttsel’ problem in the corresponding fractal domain. This is achieved via the convergence (in the Mosco–Kuwae–Shioya sense) of the approximating energy forms in varying Hilbert spaces.

Global Well-Posedness and Regularity of Stochastic 3D Burgers Equation with Multiplicative Noise

By utilizing the so-called Doss–Sussman transformation, we link our stochastic 3D Burgers equation with linear multiplicative noise to a random 3D Burger equation. With the help of techniques from partial differential equations (PDEs) and probability, we establish the global well-posedness of stochastic 3D Burgers with the diffusion coefficient being constant. Next, by developing a solution which is orthogonal with the gradient of coefficient of the noise, we extend the global well-posedness to a more general case in which the diffusion coefficient is spatial dependent, i.e., it is a function of the spatial variable. Our results and methodology pave a way to extend some regularity results of stochastic 1D Burgers equation to stochastic 3D Burgers equations.

Two‐Phase Rubber–Plastic Matrices’ Stabilization of Organic Room‐Temperature Phosphorescence Afterglows Better than Plastic Matrix

The rigid yet polar polymers as matrices for organic‐doped room‐temperature phosphorescence (RTP) polymers are widely reported, but nonpolar rigid plastics such as polystyrene (PS) are thought to be ineffective matrix and rarely attempted. Herein, it is reported that PS–polyisoprene–PS (SIS) elastomer and rubber phase‐containing high‐impact PS (HIPS) can stabilize brighter and longer‐lived organic RTP than PS as matrix; moreover, photoactivation time for RTP production is also greatly shortened. Three N‐arylcarbazole derivatives are employed as dopants and afford the same regular results, and the afterglow lifetime of RTP elastomer is up to 1.22 s. Since the general rubbers such as polyisoprene are ineffective doping matrices and PS is not good matrix, the interface phase of polyphase polymers plays an important role in promoting and stabilizing triplet‐state emission. Based on the different confined environments in various phase regions for organic dopants, a new stabilizing RTP mechanism is discussed to understand the unique stabilizing RTP effect of multiphase matrices. This study not only develops a high‐performance RTP elastomer but also discloses a fresh strategy for enhancing RTP of organic‐doped polymers.

BMO varepsilon -regularity results for solutions to Legendre–Hadamard elliptic systems

We will establish an varepsilon -regularity result for weak solutions to Legendre–Hadamard elliptic systems, under the a-priori assumption that the gradient nabla u is small in textrm{BMO}. Focusing on the case of Euler–Lagrange systems to simplify the exposition, regularity results will be obtained up to the boundary, and global consequences will be explored. Extensions to general quasilinear elliptic systems and higher-order integrands is also discussed.

Nomogram predicting early recurrence defined by the minimum P value approach for colorectal liver metastasis patients receiving colorectal cancer resection with simultaneous liver metastasis resection: development and validation.

Simultaneous resections have been increasingly performed for colorectal liver metastasis patients. However, studies explored risk stratification for these patients are scarce. Among which, a clear definition of early recurrence remains controversial and models for predicting early recurrence in these patients are lacking. Colorectal liver metastasis patients who developed recurrence followed by simultaneous resection were enrolled. Early recurrence was determined by the minimum P value method, and patients were divided into an early recurrence group and late recurrence group. Standard clinical data were collected from each patient including demographics features, preoperative laboratory tests and postoperative regular follow-up results. All the data were accessed by clinicians and recorded accordingly. The nomogram for early recurrence was constructed in the training cohort and validated externally in the test cohort. The optimal value of early recurrence by the minimum P value method was 13 months. A total of 323 patients were included in the training cohort, of which 241 (74.6%) experienced early recurrence. Seventy-one patients were included in the test cohort, of which 49 (69.0%) experienced early recurrence. Significantly worse post-recurrence survival (median 27.0 vs. 52.8 months, P=0.00083) and overall survival (median 33.8 vs. 70.9 months, P<0.0001) were observed in patients with early recurrence in the training cohort. Positive lymph node metastases (P=0.003), tumour burden scores ≥4.09 (P=0.001), preoperative neutrophil-to-lymphocyte ratios ≥1.44 (P=0.006), preoperative blood urea nitrogen levels ≥3.55 µmol/L (P=0.017) and postoperative complications (P=0.042) were independently associated with early recurrence, and all these predictors were included in the nomogram. The nomogram for predicting early recurrence had a receiver operating characteristic curve of 0.720 in the training cohort and a receiver operating characteristic curve of 0.740 in the test cohort. The Hosmer-Lemeshow test and calibration curves showed acceptable model calibration in the training set (P=0.7612) and in the test set (P=0.8671). The decision curve analysis results for the training cohort and test cohort also indicated that the nomogram showed good clinical applicability. Our findings provide clinicians with new insights into accurate risk stratification for colorectal liver metastasis patients receiving simultaneous resection and contributing to the management of patients.

Global regularity for a class of fully nonlinear PDEs with unbalanced variable degeneracy

AbstractWe establish the existence and sharp global regularity results (, and estimates) for a class of fully nonlinear elliptic Partial Differential Equations (PDEs) with unbalanced variable degeneracy. In a precise way, the degeneracy law of the model switches between two different kinds of degenerate elliptic operators of variable order, according to the null set of a modulating function . The model case in question is given by for a bounded, regular, and open set , and appropriate continuous data , , and . Such sharp regularity estimates generalize and improve, to some extent, earlier ones via geometric treatments. Our results are consequences of geometric tangential methods and make use of compactness, localized oscillating, and scaling techniques. In the end, our findings are applied in the study of a wide class of nonlinear models.

Improved Models for Diabetes Prediction by Integrating PCA Technique

Diabetes is an incurable, chronic disease indicated by hyperglycemia. It may lead to severe complications and is consequently one of the top ten causes of mortality in recent years. Fortunately, early detection can significantly aid in disease management. Based on their regular medical checkup results, people can make a preliminary evaluation of their risk of having the condition with the help of machine learning techniques. In this study, diabetes is predicted using Extreme Gradient Boosting (XGB), Support Vector Machine (SVM), Random Forest (RF), Logistic Regression (LR), Decision Tree (DT), and K-Nearest Neighbor Algorithm (KNN), ranked in descending order of accuracy. XGB comes out on top with an accuracy of 87.29%. The dimensionality reduction approach Principal Component Analysis (PCA) is conducted, resulting in a maximum accuracy gain of 1.27%. The study's enhanced diabetes prediction models allow people to more accurately gauge their risk of developing the condition. More potential patients would be notified to get checked, enhancing the disease's early detection rate.

Existence and stability of pullback exponential attractors for a nonautonomous semilinear evolution equation of second order

We consider a nonautonomous semilinear evolution problem that models some sort of propagation problem in nonlinear elastic rods and nonlinear ion-acoustic waves. We investigate the existence and stability of a family of pullback exponential attractors for our problem under suitable growth and dissipativeness conditions. Moreover, we also prove the upper and lower semicontinuity of this family of pullback exponential attractors at time zero. As a particular case, we obtain the existence of the pullback attractor in an appropriate space, we prove its upper semicontinuity and, lastly, we obtain a regularity result of this pullback attractor.

Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016). To this end, we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. This concept is then used in order to prove regularization by noise for stochastic equations on the plane. The statement of regularization by noise is formulated in terms of the regularity of the local time associated to the perturbing stochastic field. For this, we provide two quantified example: a fractional Brownian sheet and the sum of two one-parameter fractional Brownian motions. As a further illustration of our regularization results, we also prove well-posedness of a 1D non-linear wave equation with a noisy boundary given by fractional Brownian motions. A discussion of open problems and further investigations is provided.

Infinite dimensional tensor variational inequalities with applications to an economic equilibrium problem

In this paper, we present a general oligopolistic market equilibrium model in which each firm produces several commodities in a time interval. To this aim, we introduce tensor variational inequalities in Hilbert spaces which are a powerful tool to analyse the model. Indeed we characterize the equilibrium condition by means of a suitable time-dependent tensor variational inequality. In addition, we prove some existence and regularity results and a numerical scheme to compute the solution. Finally we provide a numerical example.

Regularity analysis for SEEs with multiplicative fBms and strong convergence for a fully discrete scheme

Abstract One of the open problems in the study of stochastic differential equations is regularity analysis and approximations to stochastic partial differential equations driven by multiplicative fractional Brownian motions (fBms), especially for the case $H\in (0,\frac {1}{2})$. In this paper, we address this problem by considering a class of stochastic evolution equations (SEEs) driven by multiplicative fBms. We analyze the well-posedness and regularity of mild solutions to such equations with $H\in (0,\frac {1}{2})$ and $H\in (\frac {1}{2},1)$ under the Lipschitz conditions and linear growth conditions. The two cases are treated separately. Compared with the standard Brownian motion case, the main difficulty is that fBm is neither a Markov process nor a semimartingale such that some classical stochastic calculus theories are unavailable. As a consequence, we need to explore some new strategies to complete the existence and uniqueness and regularity analysis of the solutions. Especially for the case $H\in (0,\frac {1}{2})$, we utilize some delicate techniques to overcome the difficulties from the singularity of the covariance of fBms. In addition, we give a fully discrete scheme for such equations, carried out by the spectral Galerkin method in space and a time-stepping method in time. The obtained regularity results of the equations help us to examine the strong convergence of the discrete scheme. In final, several numerical examples are done to substantiate the theoretical findings.

On weak solutions for a nonlocal model with nonlocal damping term

In this paper, we consider a linear bond-based peridynamic nonlocal evolution problem with the nonlocal viscoelastic damping term, the existence, uniqueness, and continuous dependence upon datum of a weak solution are proved by Galerkin methods. Together with energy inequality, we establish the regularity results of weak solutions in time. In addition, we also briefly analyze the limit behavior of the weak solution as δ→0, and find that its limit function solves the corresponding classical local evolution problem exactly in the sense of distributions. Under more stronger regularity conditions for solutions, the solution of the nonlocal evolution problem strongly converges to the solution of the corresponding classical local problem only in the interior of domain Ω.

Quasilinear elliptic and parabolic systems with nondiagonal principal matrices and strong nonlinearities in the gradient. Solvability and regularity problems

We consider nondiagonal elliptic and parabolic systems of equations with strongly nonlinear terms in the gradient. We review and comment known solvability and regularity results and describe the last author’s results in this field.

Analysis and Hermite Spectral Approximation of Diffusive-Viscous Wave Equations in Unbounded Domains Arising in Geophysics

The diffusive-viscous wave equation (DVWE) is widely used in seismic exploration since it can explain frequency-dependent seismic reflections in a reservoir with hydrocarbons. Most of the existing numerical approximations for the DVWE are based on domain truncation with ad hoc boundary conditions. However, this would generate artificial reflections as well as truncation errors. To this end, we directly consider the DVWE in unbounded domains. We first show the existence, uniqueness, and regularity of the solution of the DVWE. We then develop a Hermite spectral Galerkin scheme and derive the corresponding error estimate showing that the Hermite spectral Galerkin approximation delivers a spectral rate of convergence provided sufficiently smooth solutions. Several numerical experiments with constant and discontinuous coefficients are provided to verify the theoretical result and to demonstrate the effectiveness of the proposed method. In particular, We verify the error estimate for both smooth and non-smooth source terms and initial conditions. In view of the error estimate and the regularity result, we show the sharpness of the convergence rate in terms of the regularity of the source term. We also show that the artificial reflection does not occur by using the present method.