Partial regularity results for subelliptic systems in the Heisenberg group

We consider subelliptic systems in the Heisenberg group. We give a new proof for the smoothness of solutions of inhomogeneous systems with constant coefficients. With this result, we prove partial Hölder continuity of the horizontal gradient for non-linear systems with p-growth for p≥2 via the $\mathcal {A}We consider subelliptic systems in the Heisenberg group. We give a new proof for the smoothness of solutions of inhomogeneous systems with constant coefficients. With this result, we prove partial H?lder continuity of the horizontal gradient for non-linear systems with p-growth for p≥2 via the -harmonic approximation technique.     

Elliptic systems with nonlinearity q greater or equal than two with controlled growth. Global Hölder continuity of solutions to the Dirichlet problem

Hölder regularity up to the boundary of the solutions to the Dirichlet problem for second order elliptic systems with nonlinearity q>2 and with controlled growth is proved when n?q+2.     

Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary

We present some new regularity criteria for “suitable weak solutions” of the Navier-Stokes equations near the boundary in dimension three. We prove that suitable weak solutions are Hölder continuous up to the boundary provided that the scaled mixed norm with 3/p+2/q?2, 2<q?∞, (p,q)≠(3/2,∞) is small near the boundary. Our methods yield new results in the interior case as well. Partial regularity of weak solutions is also analyzed under some additional integral conditions.     

On nonlinear higher order elliptic systems positive in the first derivatives

Elliptic systems of a divergent type with a natural energy spaceW _p ^m W _q ¹ are considered. Under certain restrictions imposed on the modulus of ellipticity of the lower part of the system, the Hölder property of the generalized solutions to this system and the Liouville theorem are established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 942–947, July, 1993.     

Global regularity for solutions to Dirichlet problem for discontinuous elliptic systems with nonlinearity q > 1 and with natural growth

In this paper we deal with the Hölder regularity up to the boundary of the solutions to a nonhomogeneous Dirichlet problem for second-order discontinuous elliptic systems with nonlinearity q > 1 and with natural growth. The aim of the paper is to clarify that the solutions of the above problem are always global Hölder continuous in the case of the dimension n = q without any kind of regularity assumptions on the coefficients. As a consequence of this sharp result, the singular sets $\Omega_0 \subset \OmegaIn this paper we deal with the H?lder regularity up to the boundary of the solutions to a nonhomogeneous Dirichlet problem for second-order discontinuous elliptic systems with nonlinearity q > 1 and with natural growth. The aim of the paper is to clarify that the solutions of the above problem are always global H?lder continuous in the case of the dimension n = q without any kind of regularity assumptions on the coefficients. As a consequence of this sharp result, the singular sets , are always empty for n = q. Moreover we show that also for 1 < q < 2, but q close enough to 2, the solutions are global H?lder continuous for n = 2.      

Hölder Regularity for Nonhomogeneous Elliptic Systems with Nonlinearity Greater than Two

Regularity results for elliptic systems of second order quasilinear PDEs with nonlinear growth of order q > 2 are proved, extending results of [7] and [10]. In particular Hölder regularity of the solutions is obtained if the dimension n is less than or equal to q + 2.     

Hölder continuity and Harnack inequality for De Giorgi classes related to Hörmander vector fields

Summary In this paper De Giorgi classes (see [DG.], [L.])related to Hörmander vector fields (see [H.]₁)are considered. Hölder continuity and Harnack inequality (with respect to the intrinsic balls) are proved. These properties are valid, in particular, for Q-minima (see [GG.])and for solutions of certain non-linear operators related to Hörmander vector fields.     

Partial applicability of Moser's method to nonlinear elliptic systems

We consider nonlinear elliptic systems of divergent-type second-order partial differential equations with solutionsu W _p ¹ . It is proved thatDu L _q with someq (p; +) and it is explicitly shown howq depends on the ellipticity modulus of the system. Some conditions on the ellipticity modulus are obtained under which the solutions satisfy the Hölder conditions and the Liouville theorem holds.Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 547–557, October, 1995.     

Holder Continuity of the Dirichlet Solution for a General Domain

Let D be a bounded domain in Rⁿ. For a function f on the boundaryD, the Dirichlet solution of f over D is denoted by H_Df, providedthat such a solution exists. Conditions on D for H_D to transforma Hölder continuous function on D to a Hölder continuousfunction on D with the same Hölder exponent are studied.In particular, it is demonstrated here that there is no boundeddomain that preserves the Hölder continuity with exponent1. It is also also proved that a bounded regular domain D preservesthe Hölder continuity with some exponent , 0<<1,if and only if D satisfies the capacity density condition, whichis equivalent to the uniform perfectness of D if n = 2. 2000Mathematics Subject Classification 31A05, 31A20, 31B05, 31B25.     

On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems

Summary The Hölder continuity of bounded weak solutions of quasilinear parabolic systems with main part in diagonal form is proved via a parabolic hole-filling technique.This research was supported by the Sonderforschungsbereich 72 of the Deutsche Forschungsgemeinschaft     

Remarks on Hölder continuity for solutions of the p-Laplace evolution equations

We study the interior Hölder regularity problem for the gradient of solutions of the p-Laplace evolution equations with the external forces. Misawa gave some conditions for the Hölder continuity of the gradient of solutions. We show Hölder estimates of the solutions with weaker condition as for Misawa.     

Heat equation with degeneration in Holder and Slobodetskii spaces

We study initial-boundary value problems for the heat equation in which heat conductivity ²(x) may depend on the space variablex ⁺; the nonnegative function(x) is allowed to tend to infinity (respectively, zero) asx + (respectively,x +0). We prove that these problems are well posed and examine the smoothness of solutions. It is shown that criteria for smoothness of the solutions can be stated in terms of certain functionals, namely, the Hölder constant (for Hölder spaces) and the generalized Hölder constant (for Slobodetskii spaces).Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 189–203, August, 1995.     

Estimates of the Hölder constant for functions satisfying a uniformly elliptic or a uniformly parabolic quasilinear inequality with unbounded coefficients

One obtains inner and boundary estimates of the Hölder constants for functions u(·) satisfying a uniformly elliptic or uniformly parabolic quasilinear inequality of nondivergence form with unbounded coefficients. It is shown that the Hölder exponents in them depend only on the dimension W and on the constants and occurring in the ellipticity conditions. In the boundary estimates they depend also on the constant ₀, occurring in the condition (A) on the boundary and on the Hölder exponent for the boundary values of u(·).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 72–94, 1985.     

An inverse problem for a semilinear parabolic equation

Summary In this paper we are concerned with the study of the stability of an unknown non-linear term in a parabolic equation in dependence on over specified Cauchy-Dirichlet data prescribed on the parabolic boundary of the open set under consideration. Since, in general, the dependence of the nonlinear term upon the data is not stable with respect to L -metrics, we show how a Hölder continuity may be restored under mild restrictions for the set of admissible solutions.Lavoro eseguito nell'amMto del G.N.A.F.A. del C.N.R.     

Continuity properties of solutions to some degenerate elliptic equations

We consider a nonlinear (possibly) degenerate elliptic operator where the field a and the function b are (unnecessarily strictly) monotonic and a satisfies a very mild ellipticity assumption. For a given boundary datum ? we prove the existence of the maximum and the minimum of the solutions and formulate a Haar-Radò type result, namely a continuity property for these solutions that may follow from the continuity of ?. In the homogeneous case we formulate some generalizations of the Bounded Slope Condition and use them to obtain the Lipschitz or local Lipschitz regularity of solutions to Lu=0. We prove the global Hölder regularity of the solutions in the case where ? is Lipschitz.     

Partial Hölder continuity for solutions of subquadratic elliptic systems in low dimensions

We consider weak solutions of second order nonlinear elliptic systems in divergence form under standard subquadratic growth conditions with boundary data of class C¹. In dimensions n∈{2,3} we prove that u is locally Hölder continuous for every exponent outside a singular set of Hausdorff dimension less than n−p. This result holds up to the boundary both for non-degenerate and degenerate systems. In the proof we apply the direct method and classical Morrey-type estimates introduced by Campanato.     

The boundary regularity of non-linear parabolic systems I

This is the first part of a work aimed at establishing that for solutions to Cauchy–Dirichlet problems involving general non-linear systems of parabolic type, almost every parabolic boundary point is a Hölder continuity point for the spatial gradient of solutions. Here we develop the basic necessary and sufficient condition for establishing the regular nature of a boundary point.     

Sharp hölder estimates for\overline \partial on ellipsoids

We solve the -equation on real and on complex ellipsoids in ^N. It is proved that the solution satisfies sharp Hölder estimates. That is, the Hölder exponent equals the reciprocal of the maximal order of contact of the boundary of the ellipsoid with complex-analytic curves.Supported by NSF grant DMS 8401273.Supported by the Netherlands' organization for the advancement of pure research ZWO.