Partial regularity of weak solutions to a PDE system with cubic nonlinearity

In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points.     

Degenerate elliptic operators diagonal systems and variational integrals

Using a simple quasiconformal transformation of the independent variables it is shown how some regularity results for weak solutions of quasilinear elliptic systems generalize to several cases where the ellipticity of the principal part degenerates. Similarly it is possible to study the regularity of minima of degenerate variational integrals, as well as elliptic equations and systems in unbounded domains.     

Low Regularity Error Analysis for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

This paper presents error estimates in both an energy norm and the $L^2$-norm for the weak Galerkin (WG) finite element methods for elliptic problems with low regularity solutions. The error analysis for the continuous Galerkin finite element remains same regardless of regularity. A totally different analysis is needed for discontinuous finite element methods if the elliptic regularity is lower than H-1.5. Numerical results confirm the theoretical analysis.     

Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity

The aim of this paper is to study the existence and uniqueness of weak solutions for an initial boundary problem of a fourth-order parabolic equation with variable exponent of nonlinearity. First, the authors of this paper apply Leray-Schauder’s fixed point theorem to prove the existence of solutions of the corresponding nonlinear elliptic problem and then obtain the existence of weak solutions of nonlinear parabolic problem by combining the results of the elliptic problem with Rothe’s method. In addition, the authors also discuss the regularity of weak solutions in the case of space dimension one.     

Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force

In this paper, we consider the equations of Magnetohydrodynamics with Coulomb force which is of hyperbolic–parabolic–elliptic mixed type. By constructing the approximate solutions to the modified system with an artificial pressure term added, global existence of finite energy weak solutions is established via the weak convergence method. More careful argument has been paid to overcome the new difficulty arising from the Poisson term of Coulomb force in two dimensions when the adiabatic exponent is close to one. We also investigate the large-time behavior of such weak solutions after discussing the regularity and uniqueness of solutions to the stationary problem.     

Optimal interior partial regularity¶for nonlinear elliptic systems: the method of A-harmonic approximation

We consider nonlinear elliptic systems of divergence type. We provide a new method for proving partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation. This method is applied to both homogeneous and inhomogeneous systems, in the latter case with inhomogeneity obeying the natural growth condition. Our methods extend previous partial regularity results, directly establishing the optimal H?lder exponent for the derivative of a weak solution on its regular set. We also indicate how the technique can be applied to further simplify the proof of partial regularity for quasilinear elliptic systems. Received: 22 July 1999 / Revised version: 23 May 2000     

Boundary continuity of solutions to elliptic equations with nonstandard growth

We study regularity properties of weak solutions to elliptic equations involving variable growth exponents. We prove the sufficiency of a Wiener type criterion for the regularity of boundary points. This criterion is formulated in terms of the natural capacity involving the variable growth exponent. We also prove the Hölder continuity of weak solutions up to the boundary in domains with uniformly fat complements, provided that the boundary values are Hölder continuous.     

On very weak solutions of degenerate p-harmonic equations

We establish a regularity result for very weak solutions of some degenerate elliptic PDEs. The nonnegative function which measures the degree of degeneracy of ellipticity bounds is assumed to be exponentially integrable. We find that the scale of improved regularity is logarithmic.      

一类拟线性椭圆型方程很弱解的局部正则性

本文考虑一类拟线性椭圆型方程的很弱解．使用Hodge分解等工具，得到了其局部正则性，推广了[1]之结果。     

Regularization of mixed second-order elliptic problems

It is shown that weak solutions of mixed elliptic problems are Hölder continuous of any order less than 1/2 and that they possess higher regularity in non-critical directions.     

Weighted Calderón-Zygmund Estimates for Weak Solutions of Quasi-Linear Degenerate Elliptic Equations

Potential Analysis - This paper studies regularity estimates in Sobolev spaces for weak solutions of a class of degenerate and singular quasi-linear elliptic problems of the form...     

On W1,q Estimate for Weak Solutions to a Class of Divergence Elliptic Equations

Local W^{1,q} estimates for weak solutions to a class of equations in divergence form D_i(a_{ij}(x)|Du|^{p-2D_ju) = 0 are obtained, where q > p is given. These estimates are very important in obtaining higher regularity for the weak solutions to elliptic equations.     

Almost-everywhere regularity results for solutions of non linear elliptic systems

We prove almost-everywhere regularity of weak solutions of non linear elliptic systems of arbitrary order.Dedicated to Hans Lewy and Charles B. Morrey, Jr.     

Local boundedness of solutions to quasilinear elliptic systems

The mathematical analysis to achieve everywhere regularity in the interior of weak solutions to nonlinear elliptic systems usually starts from their local boundedness. Having in mind De Giorgi’s counterexamples, some structure conditions must be imposed to treat systems of partial differential equations. On the contrary, in the scalar case of a general elliptic single equation a well established theory of regularity exists. In this paper we propose a unified approach to local boundedness of weak solutions to a class of quasilinear elliptic systems, with a structure condition inspired by Ladyzhenskaya–Ural’tseva’s work for linear systems, as well as valid for the general scalar case. Our growth assumptions on the nonlinear quantities involved are new and general enough to include anisotropic systems with sharp exponents and the p, q-growth case.     

Regularity Results for the Generalized Beltrami System

Abstract For the generalized Beltrami system with two characteristic matrices, we deal with the regularity of its very weak solutions in the Sobolev class (1 < r < n). By changing the generalized Beltrami system into a class of a divergent elliptic system with nonhomogeneous items, we obtain that each of its very weak solutions is essentially a classical weak solution of a usual Sobolev class. Furthermore, we also establish a higher regularity of its weak solution if the regularity hypotheses of two characteristic matrices are improved. Supported by the National Natural Science Foundation of China (49805005) and by the research foundation of Northern Jiaotong University (2002SM061)     

Global existence of solutions to a parabolic–elliptic chemotaxis system with critical degenerate diffusion

This paper is devoted to the analysis of nonnegative solutions for a degenerate parabolic–elliptic Patlak–Keller–Segel system with critical nonlinear diffusion in a bounded domain with homogeneous Neumann boundary conditions. Our aim is to prove the existence of a global weak solution under a smallness condition on the mass of the initial data, thereby completing previous results on finite blow-up for large masses. Under some higher regularity condition on solutions, the uniqueness of solutions is proved by using a classical duality technique.     

Global Regularity of Solutions to Systems of Reaction–Diffusion with Sub-Quadratic Growth in Any Dimension

This paper is devoted to the study of the regularity of solutions to some systems of reaction–diffusion equations, with reaction terms having a subquadratic growth. We show the global boundedness and regularity of solutions, without smallness assumptions, in any dimension N. The proof is based on blow-up techniques. The natural entropy of the system plays a crucial role in the analysis. It allows us to use of De Giorgi type methods introduced for elliptic regularity with rough coefficients. In spite these systems are entropy supercritical, it is possible to control the hypothetical blow-ups, in the critical scaling, via a very weak norm. Analogies with the Navier–Stokes equation are briefly discussed in the introduction.     

Some nonlinear elliptic systems with right-hand side integrable data with respect to the distance to the boundary

In this paper, we study the very weak solutions to some nonlinear elliptic systems with right-hand side integrable data with respect to the distance to the boundary. Firstly, we study the existence of the approximate solutions. Secondly, a priori estimates are given in the framework of weighted spaces. Finally, we prove the existence, uniqueness and regularity of the very weak solutions.     

Regularity forn-harmonic maps

Here we obtain everywhere regularity of weak solutions of some nonlinear elliptic systems with borderline growth, includingn-harmonic maps between manifolds or map with constant volumes. Other results in this paper include regularity up to the boundary and a removability theorem for isolated singularities.     

Optimal partial regularity of second order nonlinear elliptic systems with Dini continuous coefficients for the superquadratic case

We consider the regularity for weak solutions of second order nonlinear elliptic systems with Dini continuous coefficients for the superquadratic case under natural growth condition, and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particular, the proof yields directly the optimal Hölder exponent for the derivative of the weak solutions on the regular set.