Fine regularity for elliptic systems with discontinuous ingredients

We propose results on interior Morrey, BMO and H?lder regularity for the strong solutions to linear elliptic systems of order 2b with discontinuous coefficients and right-hand sides belonging to the Morrey space L^p,λ. Received: 20 October 2004     

Solvability of nonlinear elliptic systems with measure

We prove the solvability of nonlinear elliptic systems in spaces dual to the Morrey spaces. As a main consequence, we establish that, under certain restrictions on the modulus of ellipticity of a system, systems with measure are solvable.     

Regularity for p-harmonic equations with right-hand side in Orlicz-Zygmund classes

We establish regularity results for solutions of some degenerate elliptic PDEs, with right-hand side in a suitable Orlicz-Zygmund class. The nonnegative function which measures the degree of degeneracy of the ellipticity bounds is assumed to be exponentially integrable. We find that the scale of improved regularity is logarithmic and we indicate its exact dependence on the degree of the degeneracy of the problem.     

Regularity modulus of arbitrarily perturbed linear inequality systems

We aim to quantify the stability of systems of (possibly infinitely many) linear inequalities under arbitrary perturbations of the data. Our focus is on the Aubin property (also called pseudo-Lipschitz) of the solution set mapping, or, equivalently, on the metric regularity of its inverse mapping. The main goal is to determine the regularity modulus of the latter mapping exclusively in terms of the system's data. In our context, both, the right- and the left-hand side of the system are subject to possible perturbations. This fact entails notable differences with respect to previous developments in the framework of linear systems with perturbations of the right-hand side. In these previous studies, the feasible set mapping is sublinear (which is not our current case) and the well-known Radius Theorem constitutes a useful tool for determining the modulus. In our current setting we do not have an explicit expression for the radius of metric regularity, and we have to tackle the modulus directly. As an application we approach, under appropriate assumptions, the regularity modulus for a semi-infinite system associated with the Lagrangian dual of an ordinary nonlinear programming problem.     

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.     

On the improvement of summability of generalized solutions of the Dirichlet problem for nonlinear equations of the fourth order with strengthened ellipticity

We consider the Dirichlet problem for a class of nonlinear divergent equations of the fourth order characterized by the condition of strengthened ellipticity imposed on their coefficients. The main result of the present paper shows how the summability of generalized solutions of the given problem improves, depending on the variation in the exponent of summability of the right-hand side of the equation beginning with a certain critical value. The exponent of summability that guarantees the boundedness of solutions is determined more exactly. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1511–1524, November, 2006.     

Optimal interior partial regularity for nonlinear elliptic systems with Dini continuous coefficient

In this article, we consider nonlinear elliptic systems of divergence type with Dini continuous coefficients. The authors use a new method introduced by Duzaar and Grotowski, to prove partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation and directly establish the optimal Hölder exponent for the derivative of a weak solution on its regular set.     

Morrey space regularity for weak solutions of Stokes systems with VMO coefficients

We prove that for weak solutions (u, p) of Stokes system with symmetric elliptic coefficients matrix A whose entries are bounded and VMO functions and with right-hand side f in Morrey space L ^2,μ their symmetric gradients Eu and p belong to the same Morrey space L ^2,μ .     

On the asymptotic stability of discontinuous systems analysed via the averaging method

The averaging method is one of the most powerful methods used to analyse differential equations appearing in the study of nonlinear problems. The idea behind the averaging method is to replace the original equation by an averaged equation with simple structure and close solutions. A large number of practical problems lead to differential equations with discontinuous right-hand sides. In a rigorous theory of such systems, developed by Filippov, solutions of a differential equation with discontinuous right-hand side are regarded as being solutions to a special differential inclusion with upper semi-continuous right-hand side. The averaging method was studied for such inclusions by many authors using different and rather restrictive conditions on the regularity of the averaged inclusion. In this paper we prove natural extensions of Bogolyubov’s first theorem and the Samoilenko-Stanzhitskii theorem to differential inclusions with an upper semi-continuous right-hand side. We prove that the solution set of the original differential inclusion is contained in a neighbourhood of the solution set of the averaged one. The extension of Bogolyubov’s theorem concerns finite time intervals, while the extension of the Samoilenko-Stanzhitskii theorem deals with solutions defined on the infinite interval. The averaged inclusion is defined as a special upper limit and no additional condition on its regularity is required.     

Regularity of the solution of the Cauchy problem for a higher-order parabolic equation

We prove the unique solvability of the Cauchy problem in a weighted Hölder space for a linear parabolic equation of order 2m under the condition that the lower coefficients and the right-hand side of the equation can have certain growth when approaching the plane that is the support of the initial data, while the higher coefficients do not necessarily satisfy the Dini condition near this plane.We construct a smoothness scale of solutions of the Cauchy problem in the corresponding weighted Hölder spaces.     

On stability and continuity of bounded solutions of degenerate complex Monge-Ampère equations over compact Kähler manifolds

We obtain a stability estimate for the degenerate complex Monge-Ampère operator which generalizes a result of Ko?odziej (2003) [12]. In particular, we obtain the optimal stability exponent and also treat the case when the right-hand side is a general Borel measure satisfying certain regularity conditions. Moreover, our result holds for functions plurisubharmonic with respect to a big form, thus generalizing the Kähler form setting in Ko?odziej (2003) [12]. Independently, we also provide more detail for the proof in Zhang (2006) [18] on continuity of the solution with respect to a special big form when the right-hand side is Lp-measure with p>1.     

Lipschitz Properties in Variable Exponent Problems via Relative Rearrangement

The author first studies the Lipschitz properties of the monotone and relative rearrangement mappings in variable exponent Lebesgue spaces completing the result given in [9]. This paper is ended by establishing the Lipschitz properties for quasilinear problems with variable exponent when the right-hand side is in some dual spaces of a suitable Sobolev space associated to variable exponent.     

θ-型C-Z算子在加权变指数Morrey空间上的有界性

在满足一定的正则性假设条件下,建立了θ-型Calderón-Zygmund算子T_θ在一类变指数Lebesgue空间上的加权有界性.进一步得到了T_θ在加权变指数Herz空间和Herz-Morrey空间上的有界性.另外,还证明了相应的交换子[b,T_θ]在广义加权变指数Morrey空间上是有界的.     

Quasilinear problems with singularities

We solve here some quasilinear problems with a sum of Dirac masses at the right-hand side. For that purpose, we prove a regularity theorem for nonlinear systems of the Hodge-de Rham type, and we generalize de Giorgi's notion of perimeter to subsets of compacts manifolds.     

Singular integral operators in Morrey spaces and interior regularity of solutions to systems of linear PDE’s

We obtain boundedness in Morrey spaces of singular integral operators with Calderón-Zygmund type kernel of mixed homogeneity. These estimates are used for the study of the interior regularity of the solutions of linear elliptic/parabolic systems. The proved Poincaré-type inequality permits to describe the Hölder, Morrey, and BMO regularity of the lower-order derivatives of the solutions.     

Lipschitz regularity for some asymptotically subquadratic problems

We establish a local Lipschitz regularity result for local minimizers of variational integrals under the assumption that the integrand becomes appropriately elliptic at infinity. The exponent that measures the ellipticity of the integrand is assumed to be less than two.     

Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients

In this paper we are interested in pointwise regularity of solutions to elliptic equations. In a first result, we prove that if the modulus of mean oscillation of Δu at the origin is Dini (in L ^p average), then the origin is a Lebesgue point of continuity (still in L ^p average) for the second derivatives D ² u. We extend this pointwise regularity result to the obstacle problem for the Laplace equation with Dini right hand side at the origin. Under these assumptions, we prove that the solution to the obstacle problem has a Taylor expansion up to the order 2 (in the L ^p average). Moreover we get a quantitative estimate of the error in this Taylor expansion for regular points of the free boundary. In the case where the right hand side is moreover double Dini at the origin, we also get a quantitative estimate of the error for singular points of the free boundary. Our method of proof is based on some decay estimates obtained by contradiction, using blow-up arguments and Liouville Theorems. In the case of singular points, our method uses moreover a refined monotonicity formula.      

On variational problems with obstacles and integral constraints for vector-valued functions

The authors prove existence and regularity for vectorvalued solutions of n-dimensional variational problems with boundary conditions, integral constraints, and obstacles as side conditions. Main emphasis is given to the regularity proof in the case n=2 generalizing a well known technique due to C. B. Morrey. In addition, a regularity result is stated for the general n-dimensional case.Dedicated to Hans Lewy and Charles B. Morrey, Jr.     

Method of limiting equations for the stability analysis of equations with infinite delay in the Carathéodory conditions: I

We consider systems of nonautonomous nonlinear differential equations with infinite delay. We introduce Carathéodory type conditions for the right-hand side in an equation, which permit one, on the one hand, to cover a fairly broad class of systems and, on the other hand, include the right-hand side in a compact function space and construct the so-called limiting equations. In the investigation, we use the construction of admissible spaces with fading memory, which permits one to obtain constructive results for the class of equations under study.