Spectral asymptotics for quasi-elliptic partial differential equations (Q2754335)

The main result of this paper is the Weyl formula for quasi-elliptic anisotropic operators on compact manifolds.NEWLINENEWLINENEWLINELet \(\Omega, X\) be open subsets of \(\mathbb R^n\) and \(\mathbb R^N\), respectively. Let \((m_1,\cdots,m_n)\) be a given \(n\)-tuple of integer numbers; they set \(m=\max_j\{m_j\}, q=(q_1,\cdots,q_n)=(\frac{m}{m_1},\cdots,\frac{m}{m_n}).\) For \(\xi \in \mathbb R^n\), define \(\|\xi\|_q=\sum_{j=1}^n \|\xi_j\|^{\frac{1}{q_j}}\).NEWLINENEWLINENEWLINE\(S^{\mu,q}(X \times \mathbb R^n)\) denotes the space of all functions \(p(x,\xi) \in C^{\infty}(X \times \mathbb R^n)\) satifying the following condition:NEWLINENEWLINENEWLINEFor every multi-index \(\alpha,\beta\) and for every compact subset \(K \subset X\) there exists \(C_{\alpha,\beta,K}>0\) such that NEWLINE\[NEWLINE \|\partial^{\alpha}_{\xi}\partial^{\beta}_{x}(x,\xi)\|\leq C_{\alpha,\beta,K}(1+\|\xi\|_q)^{\mu-\langle \alpha,q \rangle} NEWLINE\]NEWLINE or every \(x \in K,\) \( \xi \in \mathbb R^n\), where \(\langle \alpha,q \rangle = \sum_{j=1}^n \alpha_j q_j\).NEWLINENEWLINENEWLINEIf \(a \in S^{\mu,q}(\Omega \times \Omega \times \mathbb R^n),\) the pseudodifferential operator \(A=Op(a)\) is defined for \(f \in C_0^{\infty}(\Omega)\) by NEWLINE\[NEWLINE Af(x)=(2\pi)^{-n}\int e^{i(x-y)\xi} a(x,y,\xi) f(y) dy d\xi, NEWLINE\]NEWLINE where the integral is understood as an oscillatory integral.\quad \(L^{\mu,q}(\Omega)\) denotes the space all operators of this form.NEWLINENEWLINENEWLINEIf \(A \in L^{\mu,q}(\Omega)\) is defined by an amplitude \(a(x,\xi)\) independent of \(y\),\quad \(A\) is written \(A=a(x,D).\)NEWLINENEWLINENEWLINELet \(\mu_j, j=0,1,\cdots, \) be a sequence of real numbers, with \(\mu_{j+1} < \mu_j\) for every \(j\) and \(\mu_j \rightarrow -\infty.\) Let \(p_j \in S^{\mu,q}(X \times \mathbb R^n)\). \(p \in S^{\mu,q}(X \times \mathbb R^n)\) has asymptotic expansion \quad \(\sum_{j \geq 0}p_j\) \quad means NEWLINE\[NEWLINE p-\sum_{j < k}p_j \in S^{\mu_k,q}(X \times \mathbb R^n) NEWLINE\]NEWLINE for every \(k=1,2,\cdots,\) and denots \quad \(p \sim \sum_{j \geq 0}p_j.\)NEWLINENEWLINENEWLINEThey say that \(P=p(x,D) \in L^{\mu,q}(\Omega)\) has quasi-homogeneous principal symbol if there exists \(p_{\mu,q}(x,\xi) \in C^{\infty}(\Omega \times (R^n \setminus \{0\}))\) quasi-homogeneous of degree \(\mu\) with respect to \(\xi\), such that \(p(x,\xi)-p_{\mu,q}(x,\xi)=O(\|xi\|_q^{\mu-\epsilon})\) as \(\|xi\|_q \rightarrow \infty\), uniformly for \(x\) in compact subsets of \(\Omega\) and some \(\epsilon > 0.\)NEWLINENEWLINENEWLINEAn operator \(P=p(x,D) \in L^{\mu,q}(\Omega)\) is called quasi-elliptic if its symbol \(p(x,\xi)\) satisfies the following condition: for every compact subset \(K \subset \Omega\) there exist constants \(C,R > 0\) such that NEWLINE\[NEWLINE |p(x,\xi)|\geq C\|\xi\|_q^{\mu} \text{for} \|\xi\|_q >R. NEWLINE\]NEWLINE The quasi-ellipticity is the same as for the principal symbol NEWLINE\[NEWLINE p_{\mu,q}(x,\xi) \neq 0 \text{for} x \in \Omega, \xi \in \mathbb R^n \setminus \{0\}. NEWLINE\]NEWLINELet \(M\) be a compact Riemanian \(q-\)manifold. The quasi-elliptic operator \(A \in L^{\mu,q}(M), \) \(\mu >0,\) can be regarded as a closed unbounded operator on \(L^2(M)\) with dense domain \(H^{\mu,q}(M).\) Its resolvent is compact and if \(A\) is formally self-adjoint, it has a spectrum made of real eigenvalues \(\{\lambda_j \}_{j \geq 0}\) of finite multiplicity, clustering at infinity.NEWLINENEWLINENEWLINEThey assume \(A\) satisfies the following properties (*): NEWLINE\[NEWLINE A^{\ast}=A \in L^{\mu,q}(M), \mu > 0, \text{formally self-adjoint with quasi-homogeneous principal symbol} NEWLINE\]NEWLINE NEWLINE\[NEWLINE a_{\mu}(x,\xi) > 0 \text{in} T^{\ast}_q M \setminus 0.NEWLINE\]NEWLINE Let \(\{U(t)\}_{t>0}\) be a one-parameter family of continuous maps, \(U: C^{\infty}(M) \rightarrow C^{\infty}([0,\infty) \times M),\) which solves the initial value problem NEWLINE\[NEWLINE \begin{cases} (\partial_t + A)U(t)f = K(t)f \cr U(0)f=f \end{cases} NEWLINE\]NEWLINE for every \(f \in C^{\infty}(M)\) and for some operator \(K(t)\) with kernel in \(C^{\infty}([0,\infty)\times M \times M).\)NEWLINENEWLINENEWLINEThe method used here is to determine the singularities of the heat kernel \(H(t,x,y)\) of \(e^{-tA}\) for small \(t\), through an estimate of the heat parametrics \(U(t)\). Using the formula \(\text{Tr }U(t)=\int e^{-t\lambda} dN(\lambda) + O(t)\) as \(t \rightarrow 0^+\) and Karamata's Tauberian theorem they obtain the main result.NEWLINENEWLINENEWLINELet \(N(\lambda):= \sum_{\lambda_j < \lambda} 1 \) be the counting function associated with \(A\).NEWLINENEWLINENEWLINEThen the trace of \(U(t)\) NEWLINE\[NEWLINE\text{Tr }U(t)=\int_M U(t,x,x) dx=\int e^{-t \lambda} dN(\lambda)+O(t) \text{as} t \rightarrow 0^+. NEWLINE\]NEWLINE The main result of this paper is: NEWLINENEWLINENEWLINETheorem: The following estimate holds NEWLINE\[NEWLINE\text{Tr }U(t)=\Gamma \left(1+\frac{|q|}{\mu}\right)C t^{-\frac{|q|}{\mu}}+O\left(t^{-\frac{|q|}{\mu}+\frac{min(\epsilon,\theta)}{ \mu}}\right) \text{as} t \rightarrow 0^+. NEWLINE\]NEWLINE where \(C\) is defined \(C=(2\pi)^{-n} \int_{a_{\mu}(x,\xi) \leq 1} dx d\xi\) where \(dx d\xi\) is the canonical volume density in \(T^{\ast}_q M.\) NEWLINENEWLINENEWLINETheorem: \(N(\lambda)\) can be estimated as follows: NEWLINE\[NEWLINE N(\lambda)=C \lambda^{\frac{|q|}{\mu}}+o( \lambda^{\frac{|q|}{\mu}}) \text{as} \lambda \rightarrow \infty, NEWLINE\]NEWLINE They also illustrate the result.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00008].