A Boundary Estimate for Degenerate Parabolic Diffusion Equations
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A Boundary Estimate for Degenerate Parabolic Diffusion Equations Ugo Gianazza1 · Naian Liao2 Received: 23 July 2018 / Accepted: 22 July 2019 / © Springer Nature B.V. 2019
Abstract We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to degenerate parabolic equations of p-laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic p-capacity. Keywords Degenerate parabolic p-laplacian · Boundary estimates · Continuity · Elliptic p-capacity · Wiener-type integral Mathematics Subject Classification (2010) Primary 35K65 · 35B65; Secondary 35B45 · 35K20
1 Introduction Let E be an open set in RN and for T > 0 let ET denote the cylindrical domain E × (0, T ]. Moreover let ST = ∂E × (0, T ], ∂p ET = ST ∪ (E¯ × {0}) denote the lateral, and the parabolic boundary respectively. We shall consider quasi-linear, parabolic partial differential equations of the form ut − div A(x, t, u, Du) = 0 RN+1
→ where the function A : ET × to the structure conditions A(x, t, u, ξ ) · ξ ≥ Co |ξ |p |A(x, t, u, ξ )| ≤ C1 |ξ |p−1
RN
weakly in ET ,
(1.1)
is only assumed to be measurable and subject
a.e. (x, t) ∈ ET , ∀ u ∈ R, ∀ξ ∈ RN ,
where Co and C1 are given positive constants, and p > 2. Naian Liao
[email protected] Ugo Gianazza [email protected] 1
Dipartimento di Matematica “F. Casorati”, Universit`a di Pavia, via Ferrata 1, 27100 Pavia, Italy
2
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China
(1.2)
U. Gianazza, N. Liao
We refer to the parameters {p, N, Co , C1 } as our structural data, and we write γ = γ (p, N, Co , C1 ) if γ can be quantitatively determined a priori only in terms of the above quantities. A function p 1,p u ∈ C 0, T ; L2loc (E) ∩ Lloc 0, T ; Wloc (E) (1.3) is a local, weak sub(super)-solution to Eqs. 1.1–1.2 if for every compact set K ⊂ E and every sub-interval [t1 , t2 ] ⊂ (0, T ] t2 t 2 − uϕt + A(x, t, u, Du) · Dϕ dxdt ≤ (≥)0 uϕdx + (1.4) K
t1
t1
K
for all non-negative test functions p 1,p 1,2 0, T ; L2 (K) ∩ Lloc 0, T ; Wo (K) . ϕ ∈ Wloc This guarantees that all the integrals in Eq. 1.4 are convergent. For any k ∈ R, let (v − k)− = max{−(v − k), 0},
(v − k)+ = max{v − k, 0}.
We require (1.1)–(1.2) to be parabolic, namely that whenever u is a weak solution, for all k ∈ R, the functions (u − k)± are weak sub-solutions, with A(x, t, u, Du) replaced by ±A(x, t, k ± (u − k)± , ±D(u − k)± ). As discussed in condition (A6 ) of [3, Chapter II] or Lemma 1.1 of [4, Chapter 3], such a condition is satisfied, if for all (x, t, u) ∈ ET × R we have A(x, t, u, η) · η ≥ 0 ∀ η ∈ RN , which is guaranteed by Eq. 1.2. For y ∈ RN and ρ > 0, Kρ (y) denotes the cube of edge 2ρ, centered at y with faces parallel to the coordinate planes. When y is the origin of RN , we simply write Kρ . We are interested in the boundary behaviour of solutions to the Cauchy-Dirichlet problem ⎧ ut − div A(x, t, u, Du) = 0 weakly in ET ⎪ ⎪ ⎨ u(·, t) = g(·, t)
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