𝐿²-regularity theory of linear strongly elliptic Dirichlet systems of order 2𝑚 with minimal regularity in the coefficients
Authors: Ebenfeld Stefan 
Published: December 2016

In this article, we consider the following Dirichlet system of order

2
m

2m

:
\[

L

(
x
,
∇
)

u
=
f

(
x
)

i
n
Ω

L\left ( x, \nabla \right )u = f\left ( x \right ) \qquad in \Omega

\]
,
\[

∇
k

u
=
0

o
n
∂
Ω

(
k
=
0
,
.
.
.
,
m
−
1
)

{\nabla ^k}u = 0 \qquad on \partial \Omega \left ( k = 0,...,m - 1 \right )

\]
. Here,

Ω
\Omega

is a smooth bounded domain in

R

n

{\mathbb {R}^n}

and the differential operator

L

(
x
,
∇
)

L\left ( x, \nabla \right )

given by (1) satisfies the Legendre-Hadamard condition (4). From the general elliptic theory we know that for sufficiently smooth coefficients
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript alpha beta Superscript left-parenthesis m right-parenthesis Baseline comma upper B Subscript alpha beta Superscript left-parenthesis k m right-parenthesis Baseline comma upper C Subscript alpha Superscript left-parenthesis k right-parenthesis">

A

α
β

(
m
)

,

B

α
β

(

k
m

)

,

C
α

(
k
)

A_{\alpha \beta }^{\left ( m \right )}, B_{\alpha \beta }^{\left ( {km} \right )}, C_\alpha ^{\left ( k \right )}

and for

f
∈

H

−
m
+
s

(
Ω
,

R

N

)

f \in \\ {H^{ - m + s}}\left ( \Omega , {\mathbb {R}^N} \right )

, every weak solution

u
∈

H

0

m

(
Ω
,

R

N

)

u \in H_{0}^{m}\left ( \Omega , {\mathbb {R}^N} \right )

is actually in

H

m
+
s

(
Ω
,

R

N

)

{H^{m + s}}\left ( \Omega , {\mathbb {R}^N} \right )

and satisfies an a priori estimate of the following form:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar u double-vertical-bar Subscript upper M Sub Superscript m plus s Subscript left-parenthesis normal upper Omega comma double-struck upper R Sub Superscript upper N Subscript right-parenthesis Baseline less-than-or-equal-to ModifyingAbove upper C With caret double-vertical-bar f double-vertical-bar Subscript upper H Sub Superscript negative m plus s Subscript left-parenthesis normal upper Omega comma double-struck upper R Sub Superscript upper N Subscript right-parenthesis Baseline plus ModifyingAbove upper K With caret double-vertical-bar u double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega comma double-struck upper R Sub Superscript upper N Subscript right-parenthesis">

‖
u
‖

M

m
+
s

(
Ω
,

R

N

)

≤

C
^

‖
f
‖

H

−
m
+
s

(
Ω
,

R

N

)

+

K
^

‖
u
‖

L
2

(
Ω
,

R

N

)

{\left \| u \right \|_{{M^{m + s}}\left ( \Omega , {\mathbb {R}^N} \right )}} \le \hat C{\left \| f \right \|_{{H^{ - m + s}}\left ( \Omega ,{\mathbb {R}^N} \right )}} + \hat K{\left \| u \right \|_{{L^2}\left ( \Omega ,{\mathbb {R}^N} \right )}}

\]
. The latter a priori estimate is of particular interest in applications to nonlinear PDEs (see, e.g., [6] and [10]). There the coefficients of

L

(
x
,
∇
)

L\left ( x, \nabla \right )

result from a linearization procedure and consequently they cannot be chosen as smooth as one likes. Therefore, e.g. in [10] (Kato), the author cannot use the famous results stated in [4] (Agmon-Douglis-Nirenberg) but refers to [14] (Milani) instead.