Filippo Chiarenza

^{2,}-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients

We prove a well-posedness result in the class W 2,p ∩ W 0 1,p for the Dirichlet problem Lu = f a.e. in Ω, u = 0 on ∂Ω. We assume the coefficients of the elliptic nondivergence form equation that we study are in VMO ∩ L∞

A Harnack inequality for degenerate parabolic equations

On etablit un principe de Harnack pour les solutions positives faibles dequations lineaires paraboliques degenerees

Degenerate parabolic equations and Harnack inequality

SuntoViene risolto il problema di Cauchy Dirichlet relativo alloperatore parabolico degenere ∂u/∂t−∂/∂xi(aij(x, t) ∂u/∂xj), in opportune ipotesi di integrabilità per gli autovalori di aij(x, t). Vengono inoltre forniti controesempi circa limpossibilità di risultati di regolarità per le soluzioni deboli mostrando in tal modo che operatori parabolici degeneri hanno un comportamento radicalmente differente da quello dei corrispondenti operatori ellittici degeneri.

Boundedness for the solutions of a degenerate parabolic equation

Global and local boundedness results for the solutions of a certain class of A2-degenerate parabolic equations are proved.

A Remark on a Paper by C. Fefferman

We give a simplified proof of an imbedding theorem by C. Fefferman [3]. The purpose of this paper is to provide a simplified proof of a deep result by C. Fefferman (see [3,1]) concerning the imbedding (1.1) / u{x)pV{x)dx<c f Vu(x)pdx, Vue OR). Jr Jr In fact (1.1) was proved in [3], for p = 2, assuming VeLr~2r(Rn) l<r<n/2. Here Lr~ r(R) = Lrn~r is the classical Morrey space of the LXoc(R) functions such that sup xetL p JB(x,p) p>0 -hrrl V{y)rdy^Vrrn_2r<+(X> y b where we set B(x, p) = {y 6 R : x y < p}. Our proof rests on the following nice feature of the space Lr~2r : given V e Lrn~r there exists an Ax weight in the same class majorizing V. Such a property is not shared by L ~ which is well known to be necessary but not sufficient for ( 1.1 ) to hold. Our result is the following: Theorem. Let 1 < p < n , 1 < r < n/p , V e Lrn~pr. Then (1.2) f u(x)pV(x)dx<cVr ( Vu{x)pdx, Vh6C0°°(R). Jr Jr» Here c depends on n and p only. Proof. To prove (1.2) we assume for a moment that V e Ax (i.e. MV(x) < cV(x) a.e. in R, where MV is the usual Hardy-Littlewood maximal function). This assumption will be removed later (see Lemma 1). Received by the editors March 10, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46E35; Secondary 42B25. This work was partially financially supported by a national project of the Italian Ministero della Pubblica Istruzione.

Pointwise estimates for degenerate parabolic equations

In this paper we establish an invariant Harnack type inequality for positive solutions of linear degenerate parabolic equations. Consequently we prove Holder continuity of weak solutions.The cylinders on which this invariant Harnack inequality holds are not anymore the usual parabolic cylinders. On the contrary their shape is different from point to point depending on the degeneracy of the operator

A note on a weighted Sobolev inequality

We give a simple proof of a weighted imebedding theorem whose proof was originally given in (3).

Regularity for Solutions of Quasilinear Elliptic Equations Under Minimal Assumptions

We review some recent results in the regularity theory for elliptic second order P.D.E. obtained under assumptions which can be shown in some instances to be necessary.

De giorgi-moser theorem for a class of degenerate non-uniformly elliptic equations

On demontre que des solutions faibles dune classe dequations lineaires non uniformement elliptiques de forme divergence sont continues de Holder