Local regularity for an anisotropic elliptic equation
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Calculus of Variations
Local regularity for an anisotropic elliptic equation Naian Liao1 · Igor I. Skrypnik2 · Vincenzo Vespri3 Received: 21 November 2019 / Accepted: 30 April 2020 © The Author(s) 2020
Abstract We establish the interior Hölder continuity for locally bounded solutions, and the Harnack inequality for non-negative continuous solutions to a class of anisotropic elliptic equations with bounded and measurable coefficients, whose prototype equation is u x x + Δq,y u = 0
locally in R × R N −1 , for q < 2,
via ideas and tools originating from the regularity theory for degenerate and singular parabolic equations. Mathematics Subject Classification 35J70 · 35J92 · 35B65
1 Introduction 1.1 Notation and the main results Let E be an open set in R N with N ≥ 2. We denote a general point in E by z = (x, y) ∈ R×R N −1 . For a function u defined in E, the symbols Dx u (or u x ) and D yi u (or u yi ) represent the differentiation of u with respect to x and yi variables. Accordingly we also set D y = (D y1 , . . . , D y N −1 ),
D = (Dx , D y ).
Communicated by Y. Giga.
B
Naian Liao [email protected] Igor I. Skrypnik [email protected] Vincenzo Vespri [email protected]
1
Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020 Salzburg, Austria
2
Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Gen. Batiouk Str. 19, Sloviansk 84116, Ukraine
3
Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Via Morgagni 67/A, 50134 Florence, Italy 0123456789().: V,-vol
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N. Liao et al.
For 1 < q < 2, we shall consider the elliptic partial differential equation uxx +
N −1
D yi Ai (z, u, Du) = 0
weakly in E,
(1)
i=1
where the functions Ai (z, u, ξ ) : E × R × R N → R are Carathéodory functions, i.e. they are measurable in (u, ξ ) for all z ∈ E and continuous in z for a.e. (u, ξ ) ∈ R N +1 . Moreover, they are subject to the following structure conditions a.e. in E ⎧ N −1 ⎪ ⎪ ⎨ Ai (z, u, Du) · D yi u ≥ Co |D y u|q (2) i=1 ⎪ ⎪ ⎩|A (z, u, Du)| ≤ C |D u|q−1 i = 1, . . . , N − 1 i
1
y
with given positive constants Co and C1 . The prototype equation is u x x + div y (|D y u|q−2 D y u) = 0. Before stating the main result, let us recall that the anisotropic elliptic partial differential Eq. (1) is a special case of the more general equation N
Dxi Ai (x, u, Du) = 0
weakly in E,
(3)
i=1
where the functions Ai (x, u, ξ ) : E × R × R N → R are Carathéodory functions, and subject to the structure conditions Ai (x, u, Du) · u xi ≥ Co |u xi | pi , (4) |Ai (x, u, Du)| ≤ C1 |u xi | pi −1 , for some constants pi > 1, Co > 0 and C1 > 0. The prototype equation is N ai (x)|u xi | pi −2 u xi x = 0 i
weakly in E.
i=1
Here ai (x), i = 1, . . . , N are measurable functions, satisfying Co ≤ ai (x) ≤ C1 for some positive Co and C1 . Note also we slightly abused the symbols x and D in (3) and (4), which represent a vector in R N and the gradient in x. When p1 = · · · = p N = p > 1, the ge
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