49 research outputs found
Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth
We establish global regularity for weak solutions to quasilinear divergence
form elliptic and parabolic equations over Lipschitz domains with controlled
growth conditions on low order terms. The leading coefficients belong to the
class of BMO functions with small mean oscillations with respect to .Comment: 24 pages, to be submitte
Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces
We prove optimal integrability results for solutions of the p(x)-Laplace equation in the scale of (weak) Lebesgue spaces.
To obtain this, we show that variable exponent Riesz and Wolff potentials map L1 to variable exponent weak Lebesgue spaces
A sharp quantitative isoperimetric inequality in higher codimension
We establish the validity of a quantitative isoperimetric inequality in higher codimension. To be precise we show for any closed (n-1)-dimensional manifold Î in R^{n+k} that the quantitative isoperimetric inequality
D(Î)â„ C_1 d^2(Î)
holds true. Here D(Î) stands for the isoperimetric deficit of Î, i.e., the deviation in measure of Î being a round sphere. Further, d(Î ) denotes a natural generalization to higher codimension of the Fraenkel asymmetry index of Î
MoserâNash kernel estimates for degenerate parabolic equations
In this paper we deal with the Cauchy problem associated to a class of nonlinear degenerate parabolic equations, whose prototype is the parabolic p-Laplacian (2>p>â). In his seminal paper, after stating the Harnack estimates, Moser proved almost optimal estimates for the parabolic kernel by using the so called âHarnack chainâ method. In the linear case sharp estimates come by using Nash's approach. Fabes and Stroock proved that Gaussian estimates are equivalent to a parabolic Harnack inequality. In this paper, by using the DiBenedettoâDeGiorgi approach we prove optimal kernel estimates for degenerate quasilinear parabolic equations. To obtain this result we need to prove the finite speed of propagation of the support and to establish optimal estimates. Lastly we use these results to prove existence and sharp pointwise estimates from above and from below for the fundamental solutions