[Paper Review] Flows and invariance for elliptic operators

Tom Ter Elst, Derek W. Robinson+1 more|ArXiv.org|Mar 31, 2009

Spectral Theory in Mathematical Physics12 references3 citations

TL;DR

This paper establishes a characterization of $ L^2(\Omega) $-invariance under the submarkovian semigroup generated by a degenerate elliptic operator $ H $ in divergence form with $ W^{1,\infty} $ coefficients. It proves that $ S_t L^2(\Omega) \subseteq L^2(\Omega) $ for all $ t > 0 $ if and only if the semigroup is invariant under the flows generated by the vector fields $ \sum_{l=1}^d c_{kl} \partial_l $, under mild regularity conditions on $ \Omega $ or the core property of $ C_c^\infty(\mathbb{R}^d) $.

ABSTRACT

Let $S$ be the submarkovian semigroup on $L_2({\bf R}^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with $W^{1,\infty}$ coefficients $c_{kl}$. Further let $Ω$ be an open subset of ${\bf R}^d$. Under mild conditions we prove that $S$ leaves $L_2(Ω)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $\sum_{l=1}^d c_{kl} \partial_l$ for all $k$.

Motivation & Objective

• To characterize the invariance of $ L^2(\Omega) $ under the submarkovian semigroup $ S $ generated by a degenerate elliptic operator $ H $ with $ W^{1,\infty} $ coefficients.
• To establish a connection between semigroup invariance and the invariance of $ L^2(\Omega) $ under flows generated by the vector fields $ \sum_{l=1}^d c_{kl} \partial_l $.
• To determine sufficient conditions under which $ C_c^\infty(\mathbb{R}^d) $ is a core for $ H $, ensuring the equivalence of semigroup and flow invariance.
• To extend the characterization to flows generated by $ C_c^\infty $-smooth functions $ \psi $, showing equivalence with the original family of flows.

Proposed method

• Define the self-adjoint operator $ H $ as the Friedrichs extension of the symmetric operator $ H_0 = -\sum_{k,l} \partial_k c_{kl} \partial_l $ with $ c_{kl} \in W^{1,\infty}(\mathbb{R}^d) $, ensuring $ H $ generates a submarkovian semigroup $ S $.
• Introduce the vector fields $ Y_k = \sum_{l=1}^d c_{kl} \partial_l $, whose $ L^2 $-closures generate one-parameter unitary groups $ T^{(k)} $, referred to as flows.
• Prove that $ S $-invariance of $ L^2(\Omega) $ implies $ T^{(k)} $-invariance for all $ k $, using spectral theory and the skew-adjointness of $ Y_k $.
• Establish equivalence between $ T^{(k)} $-invariance and invariance under flows $ T^\psi $ generated by $ \sum_{k,l} (\partial_k \psi) c_{kl} \partial_l $ for all $ \psi \in C_c^\infty(\mathbb{R}^d) $.
• Use partition-of-unity techniques and local approximations via cutoff functions to show that $ C_c^\infty(\mathbb{R}^d) $ is a core for $ H $ under $ W^{2,\infty} $-regularity or local invertibility and smoothness of coefficients.
• Apply the locality of Dirichlet forms and the closedness of quadratic forms to transfer invariance properties between local and global operators.

Experimental results

Research questions

• RQ1 Under what conditions is the semigroup $ S $ generated by a degenerate elliptic operator $ H $ invariant on $ L^2(\Omega) $?
• RQ2 Is the invariance of $ L^2(\Omega) $ under $ S $ equivalent to invariance under the flows generated by the vector fields $ \sum_{l=1}^d c_{kl} \partial_l $?
• RQ3 What regularity conditions on $ \Omega $ or the coefficients $ c_{kl} $ ensure that $ C_c^\infty(\mathbb{R}^d) $ is a core for $ H $?
• RQ4 How are the flows generated by $ C_c^\infty $-functions $ \psi $ related to the flows generated by the individual vector fields $ Y_k $?
• RQ5 Can the invariance of $ L^2(\Omega) $ under $ S $ be reduced to invariance under a family of flows associated with smooth test functions?

Key findings

• The semigroup $ S $ leaves $ L^2(\Omega) $ invariant if and only if the flows $ T^{(k)} $ generated by the vector fields $ \sum_{l=1}^d c_{kl} \partial_l $ leave $ L^2(\Omega) $ invariant, provided $ \Omega $ is open with locally Lipschitz boundary or $ C_c^\infty(\mathbb{R}^d) $ is a core for $ H $.
• Invariance under the flows $ T^{(k)} $ is equivalent to invariance under the flows $ T^\psi $ generated by $ \sum_{k,l} (\partial_k \psi) c_{kl} \partial_l $ for all $ \psi \in C_c^\infty(\mathbb{R}^d) $.
• The condition that $ C_c^\infty(\mathbb{R}^d) $ is a core for $ H $ holds if $ c_{kl} \in W^{2,\infty}(\mathbb{R}^d) $, or more generally if $ c_{kl} $ are locally $ W^{2,\infty} $ and the coefficient matrix is locally invertible.
• A counterexample shows that $ W^{1,\infty} $-regularity of $ c_{kl} $ alone does not guarantee that $ C_c^\infty(\mathbb{R}^d) $ is a core for $ H $, as demonstrated by a one-dimensional operator with $ c(x) = |x|^{2\delta}(1+x^2)^{-\delta} $ for $ \delta \in [1/2, 3/4) $.
• The core property can be established via partition-of-unity arguments using cutoff functions and local extensions of $ H $ to $ H_1 $ and $ H_2 $, ensuring that $ C_c^\infty(\mathbb{R}^d) $ is dense in $ D(H) $.
• The proof relies on the locality of the Dirichlet form and the fact that $ H $-invariance on a set $ U $ implies $ H $-invariance on $ U $ for the associated local operator, enabling global approximation.