On the second-order regularity of solutions to the parabolic p-Laplace equation

Abstract

In this paper, we study the second-order Sobolev regularity of solutions to the parabolic p-Laplace equation. For any p-parabolic function u, we show that $$D(\left| Du\right| ^{\frac{p-2+s}{2}}Du)$$ D ( D u p - 2 + s 2 D u ) exists as a function and belongs to $$L^{2}_{\text {loc}}$$ L loc 2 with $$s>-1$$ s > - 1 and $$1<p<\infty $$ 1 < p < ∞ . The range of s is sharp.

Existence and global second-order regularity for anisotropic parabolic equations with variable growth

“…As it is mentioned in [15,Remark 2.3], the global estimate (1.3) with σ = p − 2 is sharp for the solutions to problem (1.4) because in the case u 0 = 0 the norms in (1.3) are bounded below and above by C i f L 2 (QT ) with C i = const. For the weak solutions of the homogeneous equation (1.4), a counterexample confirms that the inclusion D i |∇u| p−2+s 2 D j u ∈ L 2 loc (Q T ) is sharp for s > −1, see [21]. The regularity results of the present work cannot be sharpened.…”

“…In the case of constant p, the second-order regularity of the weak solutions to equations or systems of parabolic equations of the type (1.4) u t = ∆ p u + f, p > 1, were studied by many authors. We refer to [15] for the global estimates (1.3) and to [15,21,22,16,11,20,29,1] for local results, see also references therein. The works [21,22] deal with the homogeneous equation (1.4).…”

“…We refer to [15] for the global estimates (1.3) and to [15,21,22,16,11,20,29,1] for local results, see also references therein. The works [21,22] deal with the homogeneous equation (1.4).…”

“…The class of equations studied in [22] has a more complicated nonlinear structure than (1.4) but contains it as a partial case.…”

The Cauchy problem for the fast p-Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

“…and where M represents the (conserved) mass of B M , b 1 and b 2 are given numerical constants (see (40) and (39) for their definitions). The Barenblatt solution B M has a Dirac delta M δ as initial datum.…”