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Nonlinear parabolic thin sets and parabolic Wolff inequalities

Marcelo F. de Almeida, Edilson P. dos Santos Filho

Abstract

We prove a parabolic analogue of Wolff's inequality adapted to the intrinsic scaling $δ_c(x,t)=(cx,c^2t)$ and formulated in terms of time-backward parabolic dyadic rectangles. As a consequence, we obtain equivalent characterizations of parabolic $(α,q)$-thinness in this geometric setting and establish the associated Kellogg and Choquet properties. We further use the notion of $(α,2)$-thinness defined in terms of fractional heat balls and prove that the sets of irregular boundary points $z_0\in\partialΩ$ for the heat operator $\partial_t-Δ$ and for the degenerate operator $\mathscr{L}a=\partial_t(|y|^a\cdot)-\operatorname{div}(|y|^a\nabla\cdot)$ in $Ω\subset\mathbb{R}^{d+1}$ are negligible with respect to the thermal capacity $\mathrm{cap}^{\mathcal T}$ and the parabolic Bessel capacity $C_{α,2}$, respectively.

Nonlinear parabolic thin sets and parabolic Wolff inequalities

Abstract

We prove a parabolic analogue of Wolff's inequality adapted to the intrinsic scaling and formulated in terms of time-backward parabolic dyadic rectangles. As a consequence, we obtain equivalent characterizations of parabolic -thinness in this geometric setting and establish the associated Kellogg and Choquet properties. We further use the notion of -thinness defined in terms of fractional heat balls and prove that the sets of irregular boundary points for the heat operator and for the degenerate operator in are negligible with respect to the thermal capacity and the parabolic Bessel capacity , respectively.
Paper Structure (20 sections, 36 theorems, 193 equations, 3 figures)

This paper contains 20 sections, 36 theorems, 193 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Parabolic potential of measures
  4. Fractional heat balls
  5. Parabolic capacity
  6. Parabolic dyadic and continuous Wolff's inequality
  7. Dyadic Wolff's inequality
  8. Continuous Wolff's inequality
  9. Regularized dyadic Wolff's potential
  10. Parabolic dyadic capacity
  11. Extremal measures
  12. C-capacitary measure associated to Wolff's potential
  13. Characterizations via Wolff's potential
  14. Capacitary measure for arbitrary sets
  15. Thermal thinness, Kellogg and Choquet properties
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $d\geq 3$ and $\Omega\subset\mathbb{R}^d$ be a Borel set. Then $x \in \partial\Omega$ is a regular point for dirichlet if and only if where $\Omega^c=\mathbb{R}^d\backslash \Omega$ and $C_2(\cdot):=c_{1,2}(\cdot)$ denotes the Bessel capacity of set $E$ defined by and $G_{\alpha}$ is the so-called Bessel kernel.

Figures (3)

  • Figure 1: Fractional heat ball centered at $z=0$ (red) and time-backward balls $Q_{c}(z_k)$
  • Figure 2: The set $\check{\varTheta}_{r}^{\alpha}(0)$ (blue) is simply the reflected counterpart of the heat ball (green).
  • Figure 3: Time-forward and time-backward parabolic rectangle in $\mathbb{R}^{d+1}_+$.

Theorems & Definitions (63)

  • Theorem 1.1: Wiener's Criterion
  • Theorem 1.2: LanconelliEG
  • Definition 1.3: Nonlinear parabolic thin sets
  • Theorem 1.4
  • Theorem 1.5: Parabolic Wolff's inequality
  • Theorem 1.6: Parabolic Choquet property
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • ...and 53 more