Nodal Sets for “Broken” Quasilinear PDEs
We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part, div ( A S ( x , u ) ∇ u ) = div f → ( x ) , where As (x, u) has “broken” derivatives of order s ≥ 0, such as As (x, u) = a(x) + b(x)(u⁺) s , with (u⁺)⁰ being understoo...
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| Published in: | Indiana University mathematics journal 2019-01, Vol.68 (4), p.1113-1148 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | eng |
| Online Access: | Get full text |
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| Summary: | We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part,
div
(
A
S
(
x
,
u
)
∇
u
)
=
div
f
→
(
x
)
, where As
(x, u) has “broken” derivatives of order s ≥ 0, such as As
(x, u) = a(x) + b(x)(u⁺)
s
, with (u⁺)⁰ being understood as the characteristic function on {u > 0}. The vector
f
→
(
x
)
is assumed to be C
α in case s = 0, and C
1,α (or higher) in case s > 0.
Using geometric methods, we prove almost complete results (in analogy with standard PDEs) concerning the behavior of the nodal sets. More precisely, we show that the nodal sets, where solutions have (linear) nondegeneracy, are locally smooth graphs. Degenerate points are shown to have structures that follow the lines of arguments as that of the nodal sets for harmonic functions, and general PDEs. |
|---|---|
| ISSN: | 0022-2518 1943-5258 1943-5258 |