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Accretivity of the General Second Order Linear Differential Operator

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Abstract

For the general second order linear differential operator

$$\mathcal{L}_0=\sum_{j,k=1}^n{a_{jk}}\partial_j\partial_k+\sum_{j=1}^n{b_{j}}\partial_j+c$$

with complex-valued distributional coefficients aj,k, bj, and c in an open set Ω ⊆ ℝn (n ≥ 1), we present conditions which ensure that \(-\mathcal{L}_0\) is accretive, i.e., Re \(\langle-\mathcal{L}_0\phi,\phi\rangle\geq0\) for all φ ∈ C ∞0 (Ω).

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References

  1. Adams D. R., Hedberg L. I.: Function Spaces and Potential Theory, Grundlehren der Math. Wissenschaften 314, Springer-Verlag, Berlin—Heibelberg—New York, 1996

  2. Ancona A.: On strong barriers and an inequality of Hardy for domains in Rn. J. London Math. Soc, 34, 274–290 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cialdea A., Maz'ya V. G.: Criterion for the Lp-dissipativity of second order differential operators with complex coefficients. J. Math. Pures Appl., 84, 1067–1100 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Coifman R., Lions P. L., Meyer Y., et al.: Compensated compactness and Hardy spaces. J. Math. Pures Appl., 72, 247–286 (1993)

    MathSciNet  MATH  Google Scholar 

  5. Edmunds D. E., Evans W. D.: Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987

    MATH  Google Scholar 

  6. Prazier M., Nazarov F., Verbitsky I.: Global estimates for kernels of Neumann series and Green's functions. J. London Math. Soc, 90, 903–918 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Giannetti F., Greco L., Moscariello G.: Linear elliptic equations with lower order terms. Diff. Int. Eqs., 26, 623–638 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Guevara C, Phuc N. C: Leray's self-similar solutions to the Navier—Stokes equations with profiles in Marcinkiewicz and Morrey spaces. SIAM J. Math. Anal, 50, 541–556 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hara T.: A refined subsolution estimate of weak subsolutions to second-order linear elliptic equations with a singular vector field. Tokyo J. Math., 38, 75–98 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hartman P.: Ordinary Differential Equations. Second Ed., Classics in Appl. Math., 38, SIAM, Philadelphia PA, 2002

  11. Hille E.: Non-oscillation theorems. Trans. Amer. Math. Soc., 64, 234–252 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jaye B. J., Maz'ya V. G., Verbitsky I. E.: Existence and regularity of positive solutions of elliptic equations of Schrödinger type. J. d'Analyse Math., 118, 577–621, 263-302 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jaye B. J., Maz'ya V. G., Verbitsky I. E.: Quasilinear elliptic equations and weighted Sobolev—Poincaré inequalities with distributional weights. Adv. Math., 232, 513–542, 263-302 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kenig C. E.: Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Regional Conference Ser. Math. vol. 83, Amer. Math. Soc, Providence RI, 1994

  15. Kenig C. E., Pipher J.: The Dirichlet problem for elliptic equations with drift terms. Publ. Math., 45, 199–217 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Koch H., Tataru D.: Well-posedness for the Navier-Stokes equations. Adv. Math., 157, 22–35 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liskevich V. A., Perelmuter M. A., Semenov, Yu. A.: Form-bounded perturbations of generators of sub- Markovian semigroups. Acta Appl. Math., 44, 353–377 (1996)

    MathSciNet  MATH  Google Scholar 

  18. Maz'ya V. G.: The negative spectrum of the higher-dimensional Schrödinger operator. Sov. Math. Dokl., 3, 808–810 (1962)

    MATH  Google Scholar 

  19. Maz'ya V. G.: On the theory of the higher-dimensional Schrödinger operator (Russian). Izv. Akad. Nauk SSSR Ser. Mat., 28, 1145–1172 (1964)

    MathSciNet  MATH  Google Scholar 

  20. Maz'ya V.: Sobolev Spaces, with Applications to Elliptic Partial Differential Equations, 2nd augmented ed., Grundlehren der math. Wissenschaften, vol. 342, Berlin—New York, Springer, 2011

  21. Maz'ya V. G., Verbitsky I. E.: The Schrödinger operator on the energy space: boundedness and compactness criteria. Acta Math., 188, 263–302 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Maz'ya V. G., Verbitsky I. E.: Boundedness and compactness criteria for the one-dimensional Schrödinger operator, In: Function Spaces, Interpolation Theory and Related Topics, Proc. Jaak Peetre Conf, Lund, Sweden, August 17-22, 2000. Eds. M. Cwikel, A. Kufner, G. Sparr, De Gruyter, Berlin, 2002, 369–382

    Google Scholar 

  23. Maz'ya V. G., Verbitsky I. E.: Form boundedness of the general second order differential operator. Commun. Pure Appl. Math., 59, 1286–1329 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Maz'ya V. G., Verbitsky I. E.: The form boundedness criterion for the relativistic Schrödinger operator. Ann. Inst. Fourier, 54, 317–339 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Maz'ya V. G., Verbitsky I. E.: Infinitesimal form boundedness and Trudinger's subordination for the Schrödinger operator. Invent. Math., 162, 81–136 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nazarov A. I., Ural'tseva N. N.: The Harnack inequality and related properties of solutions of elliptic and parabolic equations with divergence-free lower-order coefficientts. St. Petersburg Math. J., 23, 93–115 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Phan T.: Regularity gradient estimates for weak solutions of singular quasi-linear parabolic equations. J. Diff. Eq., 263, 8329–8361 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Quinn S., Verbitsky I. E.: A sublinear version of Schur's lemma and elliptic PDE. Analysis & PDE, 11, 439–466 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Reed M., Simon B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New York-London, 1975

    MATH  Google Scholar 

  30. Rozenblum G. V., Shubin M. A., Solomyak M. Z.: Spectral Theory of Differential Operators. Encyclopaedia of Math. Sci., 64, Partial Differential Equations VII, Ed. M. A. Shubin, Springer-Verlag, Berlin-Heidelberg, 1994

  31. Seregin G., Silvestre L., Sverak V., et al.: On divergence-free drifts. J. Diff. Eq., 252 505–540 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Stein E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Math. Ser., 43, Monographs in Harmonic Analysis, Princeton University Press, Princeton NJ, 1993

  33. Temam R.: Navier—Stokes Equations, Theory and Numerical Analysis, 3rd ed., Studies in Math. Appl. 2, North-Holland, Amsterdam, 1984

  34. Zhikov V. V., Pastukhova S. E.: On operator estimates in homogenization theory. Russian Math. Surveys, 71, 417–511 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Linköping University, SE-581 83, Linköping, Sweden

    V. G. Maz’ya

  2. RUDN University, 6 Miklukho-Maklay St., Moscow, 117198, Russia

    V. G. Maz’ya

  3. Department of Mathematics, University of Missouri, Columbia, Missouri, 65211, USA

    I. E. Verbitsky

Authors
  1. V. G. Maz’ya
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  2. I. E. Verbitsky
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Correspondence to V. G. Maz’ya.

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Dedicated to Carlos Kenig with admiration and deep respect

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Maz’ya, V.G., Verbitsky, I.E. Accretivity of the General Second Order Linear Differential Operator. Acta. Math. Sin.-English Ser. 35, 832–852 (2019). https://doi.org/10.1007/s10114-019-8127-9

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