Vol. 3 No. 2 (2010)
Published:
2010-06-17
Articles
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Articles
On interior regularity of solutions of a class of almost-hypoelliptic equations. Armenian Journal of Mathematics
AbstractIn this paper it is proved that all distributional solutions of the non-degenerate, almost hypoelliptic (hypoelliptic by the one of variables) equation $P(D)u = P(D_{1},D_{2})u = 0$ are infinitely differentiable in the certain strip in $E^{2}$ under a priori assumption that they and its certain derivatives are square integrable with a certain exponential weight.
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Articles
Forced Flow by Powers of the $\textit{m}^{{\rm th}}$ Mean Curvature
AbstractIn this paper, we consider the $m^{{\rm th}}$ mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term. Under the assumption that the initial hypersurface is suitably pinched, we show that the flow may shrink to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if the forcing term is large enough. The flow can also converge to a round sphere for some special forcing term and initial hypersurface. Furthermore, the normalization of the flow is carried out so that long time existence and convergence of the rescaled flow are studied. Our work extends Schulze's flow by powers of the mean curvature and Cabezas-Rivas and Sinestrari's volume-preserving flow by powers of the $m^{{\rm th}}$ mean curvature.
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Articles
Weighted Sobolev theorem in Lebesgue spaces with variable exponent: corrigendum
AbstractWe fill in a gap discovered in the proof of Theorem A, on weighted Sobolev type boundedness for potential operators in variable exponent Lebesgue spaces, in the paper of the authors "Weighted Sobolev theorem in Lebesgue spaces with variable exponent", J. Math. Anal. and Applic., 2007, vol. 335, No 1, 560-583. The proof remains the same in the case where the Matuszewska-Orlich indices $m(w)$ and $M(w)$ of the weight $w$ are both positive or negative, but in the case where they have different signs, the proof needs some additional arguments and requires a slightly different formulation of the result.
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