AbstractOn a manifold X with boundary and bounded geometry we consider a strongly elliptic second order operator A together with a degenerate boundary operator T of the form $$T=\varphi _0\gamma _0 + \varphi _1\gamma _1$$
T
=
φ
0
γ
0
+
φ
1
γ
1
. Here $$\gamma _0$$
γ
0
and $$\gamma _1$$
γ
1
denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that $$\varphi _0, \varphi _1\ge 0$$
φ
0
,
φ
1
≥
0
, and $$\varphi _0+\varphi _1\ge c$$
φ
0
+
φ
1
≥
c
, for some $$c>0$$
c
>
0
, where either $$\varphi _0,\varphi _1\in C^{\infty }_b(\partial X)$$
φ
0
,
φ
1
∈
C
b
∞
(
∂
X
)
or $$\varphi _0=1 $$
φ
0
=
1
and $$\varphi _1=\varphi ^2$$
φ
1
=
φ
2
for some $$\varphi \in C^{2+\tau }(\partial X)$$
φ
∈
C
2
+
τ
(
∂
X
)
, $$\tau >0$$
τ
>
0
. We also assume that the highest order coefficients of A belong to $$C^\tau (X)$$
C
τ
(
X
)
and the lower order coefficients are in $$L_\infty (X)$$
L
∞
(
X
)
. We show that the $$L_p(X)$$
L
p
(
X
)
-realization of A with respect to the boundary operator T has a bounded $$H^\infty $$
H
∞
-calculus. We then obtain the unique solvability of the associated boundary value problem in adapted spaces. As an application, we show the short time existence of solutions to the porous medium equation.
A nonlinear oblique derivative boundary value problem for the heat equation: Analogy with the porous medium equation
