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 $x$.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 Γ in R^{n+k} that the quantitative isoperimetric inequality D(Γ)≥ C_1 d^2(Γ) holds true. Here D(Γ) stands for the isoperimetric deficit of Γ, i.e., the deviation in measure of Γ being a round sphere. Further, d(Γ ) denotes a natural generalization to higher codimension of the Fraenkel asymmetry index of Γ

Moser–Nash 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>∞). In his seminal paper, after stating the Harnack estimates, Moser proved almost optimal estimates for the parabolic kernel by using the so called ‘Harnack chain’ 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 DiBenedetto–DeGiorgi 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