Parabolic Equations on Uniformly Regular Riemannian Manifolds and Degenerate Initial Boundary Value Problems

Abstract

In this work there is established an optimal existence and regularity theory for second order linear parabolic differential equations on a large class of noncompact Riemannian manifolds. Then it is shown that it provides a general unifying approach to problems with strong degeneracies in the interior or at the boundary.

Dedicated to Professor Yoshihiro Shibata on the occasion of his sixtieth birthday

Appendix: Tensor Bundles

Let M be a manifold and V = (V, π, M) a vector bundle of rank n over it. For a nonempty subset S of M we denote by V _| S the restriction π ⁻¹(S) of V to S. If S is a submanifold or a union of connected components of ∂ M, then V _| S  is a vector bundle of rank n over S. As usual, V _p : = V _{p} is the fibre π ⁻¹(p) of V over p. By $\Gamma (S,V )$ we mean the $\mathbb{R}^{S}$ module of all sections of V (no smoothness).

As usual, TM and T ^∗ M are the tangent and cotangent bundles of M. Then $T_{\tau }^{\sigma }M:= TM^{\otimes \sigma }\otimes T^{{\ast}}M^{\otimes \tau }$ is for $\sigma,\tau \in \mathbb{N}$ the $(\sigma,\tau )$-tensor bundle of M, that is, the vector bundle of all tensors on M being contravariant of order $\sigma$ and covariant of order τ. In particular, T ₀ ¹ M = TM and T ₁ ⁰ M = T ^∗ M, as well as $T_{0}^{0}M = M \times \mathbb{R}$.

For $\nu \in \mathbb{N}^{\times }$ we put $\mathbb{J}_{\nu }:=\{ 1,\ldots,m\}^{\nu }$. Then, given local coordinates κ = (x ¹, …, x ^m) and setting

$$\displaystyle{ \frac{\partial } {\partial x^{(i)}}:= \frac{\partial } {\partial x^{i_{1}}} \otimes \cdots \otimes \frac{\partial } {\partial x^{i_{\sigma }}},\quad dx^{(\,j\,)}:= dx^{\,j_{1} } \otimes \cdots \otimes dx^{j_{\tau }}}$$

for $(i) = (i_{1},\ldots,i_{\sigma }) \in \mathbb{J}_{\sigma }$, $(\,j\,) \in \mathbb{J}_{\tau }$, the local representation of a $(\sigma,\tau )$-tensor field $a \in \Gamma (T_{\tau }^{\sigma }M)$ with respect to these coordinates is given by

$$\displaystyle{a = a_{(\,j\,)}^{(\,i\,)} \frac{\partial } {\partial x^{(\,i\,)}} \otimes dx^{(\,j\,)}}$$

with $a_{(j)}^{(i)} \in \mathbb{R}^{U_{\kappa }}$. We use the summation convention for (multi-)indices labeling coordinates or bases. Thus such a repeated index, which appears once as a superscript and once as a subscript, implies summation over its whole range.

Suppose $\sigma _{1},\sigma _{2},\tau _{1},\tau _{2} \in \mathbb{N}$. Then the complete contraction

$$\displaystyle{\Gamma (T_{\tau _{2}+\sigma _{1}}^{\sigma _{2}+\tau _{1} }M) \times \Gamma (T_{\tau _{1}}^{\sigma _{1} }M) \rightarrow \Gamma (T_{\tau _{2}}^{\sigma _{2} }M),\quad (a,b)\mapsto a \cdot b}$$

is defined as follows: Given $(i_{k}) \in \mathbb{J}_{\sigma _{k}}$ and $(j_{k}) \in \mathbb{J}_{\tau _{k}}$ for k = 1, 2, we set

$$\displaystyle{(i_{2};j_{1}):= (i_{2,1},\ldots,i_{2,\sigma _{2}},j_{1,1},\ldots,j_{1,\tau _{1}}) \in \mathbb{J}_{\sigma _{2}+\tau _{1}}}$$

etc., using obvious interpretations if $\min \{\sigma,\tau \}= 0$. Suppose $a \in \Gamma (T_{\tau _{2}+\sigma _{1}}^{\sigma _{2}+\tau _{1}}M)$ and $b \in \Gamma (T_{\tau _{1}}^{\sigma _{1}}M)$ are locally represented on $U_{\kappa }$ by

$$\displaystyle{a = a_{(j_{2};i_{1})}^{(i_{2};j_{1})}\, \frac{\partial } {\partial x^{(i_{2})}} \otimes \frac{\partial } {\partial x^{(j_{1})}} \otimes dx^{(j_{2})} \otimes dx^{(i_{1})},\quad b = b_{ (j_{1})}^{(i_{1})}\, \frac{\partial } {\partial x^{(i_{1})}} \otimes dx^{(j_{1})}.}$$

Then the local representation of a ⋅ b on $U_{\kappa }$ is given by

$$\displaystyle{a_{(j_{2};i_{1})}^{(i_{2};j_{1})}\,b_{ (j_{1})}^{(i_{1})}\, \frac{\partial } {\partial x^{(i_{2})}} \otimes dx^{(j_{2})}.}$$

Let g be a Riemannian metric on TM. We write $g_{\flat }: TM \rightarrow T^{{\ast}}M$ for the (fiber-wise defined) Riesz isomorphism. Thus $\langle g_{\flat }X,Y \rangle = g(X,Y )$ for $X,Y \in \Gamma (TM)$, where $\langle \cdot,\cdot \rangle: \Gamma (T^{{\ast}}M) \times \Gamma (TM) \rightarrow \mathbb{R}^{M}$ is the natural (fiber-wise defined) duality pairing. The inverse of g _♭ is denoted by $g^{\sharp }$. Then g ^∗, the adjoint Riemannian metric on T ^∗ M, is defined by $g^{{\ast}}(\alpha,\beta ):= g(g^{\sharp }\alpha,g^{\sharp }\beta )$ for $\alpha,\beta \in \Gamma (T^{{\ast}}M)$. In local coordinates

$$\displaystyle{ g = g_{ij}\,dx^{i} \otimes dx^{j},\quad g^{{\ast}} = g^{ij}\, \frac{\partial } {\partial x^{i}} \otimes \frac{\partial } {\partial x^{j}}, }$$

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[g ^ij] being the inverse of the (m × m)-matrix [g _ij ].

The metric g induces a vector bundle metric on $T_{\tau }^{\sigma }M$ which we denote by $g_{\sigma }^{\tau }$. In local coordinates

$$\displaystyle{ g_{\sigma }^{\tau }(a,b) = g_{(i)(j)}g^{(k)(\ell)}a_{ (k)}^{(i)}b_{ (\ell)}^{(j)},\qquad a,b \in \Gamma (T_{\tau }^{\sigma }M), }$$

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where

$$\displaystyle{ g_{(i)(j)}:= g_{i_{1}j_{1}}\cdots g_{i_{\sigma }j_{\sigma }},\quad g^{(k)(\ell)}:= g^{k_{1}\ell_{1} }\cdots g^{k_{\tau }\ell_{\tau }} }$$

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for $(i),(j) \in \mathbb{J}_{\sigma }$ and $(k),(\ell) \in \mathbb{J}_{\tau }$. Note g ₁ ⁰ = g and g ₀ ¹ = g ^∗ and g ₀ ⁰(a, b) = ab for $a,b \in \Gamma (M \times \mathbb{R}) = \mathbb{R}^{M}$. Moreover,

$$\displaystyle{ \vert \cdot \vert _{g_{\sigma }^{\tau }}: \Gamma (T_{\tau }^{\sigma }M) \rightarrow (\mathbb{R}^{+})^{M},\quad a\mapsto \sqrt{g_{\sigma }^{\tau }(a, a)} }$$

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is the vector bundle norm on $T_{\tau }^{\sigma }M$ induced by g. It follows that the complete contraction satisfies

$$\displaystyle{ \vert a \cdot b\vert _{g_{\sigma _{ 2}}^{\tau _{2}}} \leq \vert a\vert _{g_{\sigma _{ 2}+\tau _{1}}^{\tau _{2}+\sigma _{1}}}\,\vert b\vert _{g_{\sigma _{ 1}}^{\tau _{1}}},\qquad a \in \Gamma (T_{\tau _{2}+\sigma _{1}}^{\sigma _{2}+\tau _{1} }M),\quad b \in \Gamma (T_{\tau _{1}}^{\sigma _{1} }M). }$$

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We define a vector bundle isomorphism $T_{\tau +1}^{\sigma }M \rightarrow T_{\tau }^{\sigma +1}M$, $a\mapsto a^{\sharp }$ by

$$\displaystyle{ a^{\sharp }(\alpha _{ 1},\ldots,\alpha _{\sigma },\alpha,X_{1},\ldots,X_{\tau }):= a(\alpha _{1},\ldots,\alpha _{\sigma },X_{1},\ldots,X_{\tau },g^{\sharp }\alpha ) }$$

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for $X_{1},\ldots,X_{\tau } \in \Gamma (TM)$ and $\alpha,\alpha _{1},\ldots,\alpha _{\sigma } \in \Gamma (T^{{\ast}}M)$. If a _(j; k) ⁽ⁱ⁾ with $(i) \in \mathbb{J}_{\sigma }$, $(j) \in \mathbb{J}_{\tau }$, and $k \in \mathbb{J}_{1}$ is the coefficient of a in a local coordinate representation, then

$$\displaystyle{ (a^{\sharp })_{ (j)}^{(i;k)} = g^{k\ell}a_{ (j;\ell)}^{(i)}. }$$

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This implies

$$\displaystyle{ \vert a^{\sharp }\vert _{ g_{\sigma +1}^{\tau }} = \vert a\vert _{g_{\sigma }^{\tau +1}}. }$$

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The Levi-Civita connection on TM is denoted by $\nabla = \nabla _{g}$. We use the same symbol for its natural extension to a metric connection on $T_{\tau }^{\sigma }M$. Then the corresponding covariant derivative is the linear map

$$\displaystyle{\nabla: C^{\infty }(T_{\tau }^{\sigma }M) \rightarrow C^{\infty }(T_{\tau +1}^{\sigma }M),\quad a\mapsto \nabla a,}$$

defined by $\langle \nabla a,b \otimes X\rangle:=\langle \nabla _{X}a,b\rangle$ for $b \in C^{\infty }(T_{\sigma }^{\tau }M)$ and $X \in C^{\infty }(TM)$. It is a well-defined continuous linear map from $C^{1}(T_{\tau }^{\sigma }M)$ into $C(T_{\tau +1}^{\sigma }M)$, as follows from its local representation. For $k \in \mathbb{N}$ we define

$$\displaystyle{\nabla ^{k}: C^{k}(T_{\tau }^{\sigma }M) \rightarrow C(T_{\tau +k}^{\sigma }M),\quad a\mapsto \nabla ^{k}a}$$

by ∇⁰ a: = a and ∇^k+1: = ∇∘∇^k.

In local coordinates κ = (x ¹, …, x ^m) the volume measure dv = dv _g of (M, g) is represented by $\kappa _{{\ast}}\,dv =\kappa _{{\ast}}\sqrt{g}\,dx$, where $\sqrt{g}:={\bigl (\det [g_{ij}]\bigr )}^{1/2}$ and dx is the Lebesgue measure on $\mathbb{R}^{m}$.

The contraction $\mathsf{C}: T_{\tau +1}^{\sigma +1}M \rightarrow T_{\tau }^{\sigma }M$, a ↦ C a is given in local coordinates by (C a)_(j) ⁽ⁱ⁾: = a _(j; k) ^(i; k). It follows

$$\displaystyle{ \vert \mathsf{C}a\vert _{g_{\sigma }^{\tau }} = \vert a\vert _{g_{\sigma +1}^{\tau +1}}. }$$

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Recall that the divergence of tensor fields is the map

$$\displaystyle{ \mathop{\mathrm{div}}\nolimits =\mathop{ \mathrm{div}}\nolimits _{g}: C^{1}(T_{\tau }^{\sigma +1}M) \rightarrow C(T_{\tau }^{\sigma }M),\quad a\mapsto \mathop{\mathrm{div}}\nolimits a:= \mathsf{C}(\nabla a). }$$

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If X is a C ¹ vector field on M, then $\mathop{\mathrm{div}}\nolimits X$ has the well-known local representation

$$\displaystyle{ \frac{1} {\sqrt{g}}\, \frac{\partial } {\partial x^{i}}{\bigl (\sqrt{g}\,X^{i}\bigr )},\qquad X = X^{i}\, \frac{\partial } {\partial x^{i}}. }$$

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The gradient, $\mathop{\mathrm{grad}}\nolimits u =\mathop{ \mathrm{grad}}\nolimits _{g}u$, of a C ¹ function u is the continuous vector field $g^{\sharp }du$.

Suppose a ∈ C ¹(T ₁ ¹ M). Then, in terms of covariant derivatives,

$$\displaystyle{ \mathop{\mathrm{div}}\nolimits (a\mathop{\mathrm{grad}}\nolimits u) = a^{\sharp } \cdot \nabla ^{2}u +\mathop{ \mathrm{div}}\nolimits (a^{\sharp }) \cdot \nabla u. }$$

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