Regularizing effect of the interplay between coefficients in Dirichlet problems with convection or drift terms
Abstract: There are very important results by Enrique Zuazua on the subject of the convection-diffusion equation. In some sense this paper deals with a linear elliptic counterpart of the above equation if $d$ is not constant. We prove regularizing results on the solutions, under assumptions of interplay between the datum and the coefficient of the zero order term or between the modulus of the drift and the coefficient of the zero order term.
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Cited by 3 publications
References 13 publications
We study of the regularizing effect of the interaction between
the coefficient of the zero-order term and the lower-order term in quasilinear Dirichlet problems whose model is
∫
Ω
M
(
x
,
u
)
∇
u
⋅
∇
φ
+
∫
Ω
a
(
x
)
u
φ
=
∫
Ω
b
(
x
)
|
∇
u
|
q
φ
+
∫
Ω
f
(
x
)
φ
for all
φ
∈
W
0
1
,
2
(
Ω
)
∩
L
∞
(
Ω
)
,
\int_{\Omega}M(x,u)\nabla u\cdot\nabla\varphi+\int_{\Omega}a(x)u\varphi=\int_{%
\Omega}b(x)|\nabla u|^{q}\varphi+\int_{\Omega}f(x)\varphi\quad\text{for all }%
\varphi\in W_{0}^{1,2}(\Omega)\cap L^{\infty}(\Omega),
where Ω is a bounded open set of
ℝ
N
{\mathbb{R}^{N}}
,
M
(
x
,
s
)
{M(x,s)}
is a Carathéodory matrix on
Ω
×
ℝ
{\Omega\times\mathbb{R}}
which is elliptic (that is,
M
(
x
,
s
)
ξ
⋅
ξ
≥
α
|
ξ
|
2
>
0
{M(x,s)\xi\cdot\xi\geq\alpha|\xi|^{2}>0}
for every
(
x
,
s
,
ξ
)
∈
Ω
×
ℝ
×
(
ℝ
N
∖
{
0
}
)
{(x,s,\xi)\in\Omega\times\mathbb{R}\times(\mathbb{R}^{N}\setminus\{0\})}
) and bounded
(that is,
|
M
(
x
,
s
)
|
≤
β
{|M(x,s)|\leq\beta}
for every
(
x
,
s
)
∈
Ω
×
ℝ
{(x,s)\in\Omega\times\mathbb{R}}
),
b
(
x
)
∈
L
2
2
-
q
(
Ω
)
{b(x)\in L^{\frac{2}{2-q}}(\Omega)}
,
1
<
q
<
2
{1<q<2}
and
0
≤
a
(
x
)
∈
L
1
(
Ω
)
{0\leq a(x)\in L^{1}(\Omega)}
.
We prove the existence of a weak solution u belonging to
W
0
1
,
2
(
Ω
)
{W_{0}^{1,2}(\Omega)}
and to
L
∞
(
Ω
)
{L^{\infty}(\Omega)}
when either
b
∈
L
2
m
2
-
q
(
Ω
)
for some
m
>
N
2
and
\displaystyle b\in L^{\frac{2m}{2-q}}(\Omega)\text{ for some }m>\frac{N}{2}%
\text{ and}
(0.1)
∃
Q
>
0
such that
|
f
(
x
)
|
≤
Q
a
(
x
)
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