A connection between a general class of superparabolic functions and supersolutions

Abstract

Abstract. We show to a general class of parabolic equations that every bounded superparabolic function is a weak supersolution and, in particular, has derivatives in a Sobolev sense. To this end, we establish various comparison principles between super-and subparabolic functions, and show that a pointwise limit of uniformly bounded weak supersolutions is a weak supersolution.

Existence for a degenerate Cauchy problem

“…Indeed, pointwise convergence together with the uniform L p -bound implies the strong convergence in L q for any q strictly less than p, see, for example, [28]. Nevertheless, due to higher integrability, see Kinnunen-Lewis [26] and also [32], we can repeat the reasoning for p + ε and, thus, get rid off the restriction q < p.…”

“…The rest of the proof is rather standard, see, for example, BoccardoGallouët [11], and also [28]. For the convenience of the reader, we repeat the proof.…”

Local approximation of superharmonic and superparabolic functions in nonlinear potential theory

“…We may pass from convergence in Ω s 1 ,s 2 Ω T to the full set Ω T by the usual exhaustion argument. An application of Theorem 5.3 in [15] shows that u is a supersolution, as well as the pointwise almost everywhere convergence of the gradients.…”

“…For details, see [13] and [15]. Below, we will also need the fact that the solutions to the obstacle problem have a time derivative in L p (0, T ; W −1,p (Ω)), which is the dual space of…”

Obstacle problem for a class of parabolic equations of generalized p-Laplacian type

“…For this we need the following lemma. The proof follows the guidelines of Theorem 6 in [31], see also [24,6].…”