## Osaka Journal of Mathematics

### Spectral asymptotics for Dirichlet elliptic operators with non-smooth coefficients

Yoichi Miyazaki

#### Abstract

We consider a $2m$-th-order elliptic operator of divergence form in a domain $\Omega$ of $\mathbb{R}^{n}$, assuming that the coefficients are Hölder continuous of exponent $r \in (0,1]$. For the self-adjoint operator associated with the Dirichlet boundary condition we improve the asymptotic formula of the spectral function $e(\tau^{2m},x,y)$ for $x=y$ to obtain the remainder estimate $O(\tau^{n-\theta}+\dist(x,\partial\Omega)^{-1}\tau^{n-1})$ with any $\theta \in (0,r)$, using the $L^{p}$ theory of elliptic operators of divergence form. We also show that the spectral function is in $C^{m-1,1-\varepsilon}$ with respect to $(x,y)$ for any small $\varepsilon > 0$. These results extend those for the whole space $\mathbb{R}^{n}$ obtained by Miyazaki [19] to the case of a domain.

#### Article information

Source
Osaka J. Math., Volume 46, Number 2 (2009), 441-460.

Dates
First available in Project Euclid: 19 June 2009

https://projecteuclid.org/euclid.ojm/1245415678

Mathematical Reviews number (MathSciNet)
MR2549595

Zentralblatt MATH identifier
1171.35443

Subjects
Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions

#### Citation

Miyazaki, Yoichi. Spectral asymptotics for Dirichlet elliptic operators with non-smooth coefficients. Osaka J. Math. 46 (2009), no. 2, 441--460. https://projecteuclid.org/euclid.ojm/1245415678

#### References

• R. Beals: Asymptotic behavior of the Green's function and spectral function of an elliptic operator, J. Functional Analysis 5 (1970), 484--503.
• M. Bronstein and V. Ivrii: Sharp spectral asymptotics for operators with irregular coefficients I, Pushing the limits, Comm. Partial Differential Equations 28 (2003), 83--102.
• J. Brüning: Zur Abschätzung der Spektralfunktion elliptischer Operatoren, Math. Z. 137 (1974), 75--85.
• L. Hörmander: On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators; in Some Recent Advances in the Basic Sciences 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965--1966), Belfer Graduate School of Science, Yeshiva Univ., New York, 1969, 155--202.
• V. Ivrii: Precise Spectral Asymptotics for Elliptic Operators Acting in Fiberings over Manifolds with Boundary, Lecture Notes in Math. 1100, Springer, Berlin, 1984.
• V. Ivrii: Microlocal Analysis and Precise Spectral Asymptotics, Springer, Berlin, 1998
• V. Ivrii: Sharp spectral asymptotics for operators with irregular coefficients, Internat. Math. Res. Notices (2000), 1155--1166.
• V. Ivrii: Sharp spectral asymptotics for operators with irregular coefficients II, Domains with boundaries and degenerations, Comm. Partial Differential Equations 28 (2003), 103--128.
• K. Maruo and H. Tanabe: On the asymptotic distribution of eigenvalues of operators associated with strongly elliptic sesquilinear forms, Osaka J. Math. 8 (1971), 323--345.
• K. Maruo: Asymptotic distribution of eigenvalues of non-symmetric operators associated with strongly elliptic sesquilinear forms, Osaka J. Math. 9 (1972), 547--560.
• G. Métivier: Valeurs propres de problèmes aux limites elliptiques irrégulières, Bull. Soc. Math. France Suppl. Mém. 51--52 (1977), 125--219.
• G. Métivier: Estimation du reste en théorie spectrale, Conference on linear partial and pseudodifferential operators (Torino, 1982), Rend. Sem. Mat. Univ. Politec. Torino (1983) 157--180.
• Y. Miyazaki: A sharp asymptotic remainder estimates for the eigenvalues of operators associated with strongly elliptic sesquilinear forms, Japan. J. Math. (N.S.) 15 (1989), 65--97.
• Y. Miyazaki: The eigenvalue distribution of elliptic operators with Hölder continuous coefficients, Osaka J. Math. 28 (1991), 935--973.
• Y. Miyazaki: The eigenvalue distribution of elliptic operators with Hölder continuous coefficients II, Osaka J. Math. 30 (1993), 267--301.
• Y. Miyazaki: Asymptotic behavior of spectral functions of elliptic operators with Hölder continuous coefficients, J. Math. Soc. Japan 49 (1997), 539--563.
• Y. Miyazaki: The $L\sp p$ resolvents of elliptic operators with uniformly continuous coefficients, J. Differential Equations 188 (2003), 555--568.
• Y. Miyazaki: Asymptotic behavior of spectral functions for elliptic operators with non-smooth coefficients, J. Funct. Anal. 214 (2004), 132--154.
• Y. Miyazaki: The $L\sp p$ resolvents of second-order elliptic operators of divergence form under the Dirichlet condition, J. Differential Equations 206 (2004), 353--372.
• Y. Miyazaki: The $L\sp p$ theory of divergence form elliptic operators under the Dirichlet condition, J. Differential Equations 215 (2005), 320--356.
• Y. Miyazaki: Higher order elliptic operators of divergence form in $C^{1}$ or Lipschitz domains, J. Differential Equations 230 (2006), 174--195, Corrigendum, J. Differential Equations 244 (2008), 2404--2405.
• Yu. Safarov and D. Vassiliev: The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, Translations of Mathematical Monographs 155, Amer. Math. Soc., Providence, RI, 1997.
• R. Seeley: A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of $\mathbf{R}^{3}$, Adv. in Math. 29 (1978), 244--269.
• H. Tanabe: Functional Analytic Methods for Partial Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics 204, Marcel Dekker, New York, 1997.
• J. Tsujimoto: On the remainder estimates of asymptotic formulas for eigenvalues of operators associated with strongly elliptic sesquilinear forms, J. Math. Soc. Japan 33 (1981), 557--569.
• J. Tsujimoto: On the asymptotic behavior of spectral functions of elliptic operators, Japan. J. Math. (N.S.) 8 (1982), 177--210.
• L. Zielinski: Asymptotic distribution of eigenvalues for some elliptic operators with simple remainder estimates, J. Operator Theory 39 (1998), 249--282.
• L. Zielinski: Asymptotic distribution of eigenvalues for elliptic boundary value problems, Asymptot. Anal. 16 (1998), 181--201.
• L. Zielinski: Asymptotic distribution of eigenvalues for some elliptic operators with intermediate remainder estimate, Asymptot. Anal. 17 (1998), 93--120.
• L. Zielinski: Sharp spectral asymptotics and Weyl formula for elliptic operators with non-smooth coefficients, Math. Phys. Anal. Geom. 2 (1999), 291--321.
• L. Zielinski: Sharp spectral asymptotics and Weyl formula for elliptic operators with non-smooth coefficients II, Colloq. Math. 92 (2002), 1--18.