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Completeness theorems related to BVPs satisfying the Lopatinskii condition for higher order elliptic equations

Year 2024, , 129 - 141, 16.12.2024
https://doi.org/10.33205/cma.1540457

Abstract

In this paper, we consider a linear elliptic operator $E$ with real constant coefficients of order $2m$ in two independent variables without lower order terms. For this equation, we consider linear BVPs in which the boundary operators $T_1,\ldots,T_m$ are of order $m$ and satisfy the Lopatinskii-Shapiro condition with respect to $E$. We prove boundary completeness properties for the system $\{(T_1\omega_k,\ldots, T_m\omega_k)\}$, where $\{\omega_k\}$ is a system of polynomial solutions of the equation $Eu=0$.

References

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  • A. Cialdea: Teoremi di completezza connessi con equazioni ellittiche di ordine superiore in due variabili in un campo con contorno angoloso, Rend. Circ. Mat. Palermo (2), 35 (1) (1986), 32–49.
  • A. Cialdea: A general theory of hypersurface potentials, Ann. Mat. Pura Appl., 168 (1995), 37–61.
  • A. Cialdea: Completeness theorems on the boundary in thermoelasticity, In: Analysis as a life, Trends Math. Birkhäuser/Springer, Cham, (2019), 93–115.
  • G. Fichera: Teoremi di completezza sulla frontiera di un dominio per taluni sistemi di funzioni, Ann. Mat. Pura Appl., 27 (1948), 1–28.
  • G. Fichera: Approssimazione uniforme delle funzioni olomorfe mediante funzioni razionali aventi poli semplici prefissati I & II, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 27 (1959), 193–201, 317–323.
  • G. Fichera: Linear elliptic equations of higher order in two independent variables and singular integral equations, with applications to anistropic inhomogeneous elasticity, In: R.E. Langer (ed.), Partial differential equations and continuum mechanics, Univ. Wisconsin Press, Madison, WI, (1960), 55–80.
  • G. Fichera: The problem of the completeness of systems of particular solutions of partial differential equations, In: Numerical mathematics (Sympos., Inst. Appl. Math., Univ. Hamburg, Hamburg, 1979), Internat. Ser. Numer. Math., vol. 49. Birkhäuser, Basel-Boston, Mass., (1979), 25–41.
  • G. Fichera, L. De Vito: Funzioni analitiche di una variabile complessa, Terza edizione, Libreria Eredi Virgilio Veschi, Rome (1967).
  • G. Fichera, P. E. Ricci: The single layer potential approach in the theory of boundary value problems for elliptic equations, In: Function theoretic methods for partial differential equations (Proc. Internat. Sympos., Darmstadt, 1976), Lecture Notes in Math., vol. Vol. 561. Springer, Berlin-New York, (1976), 39–50.
  • F. Lanzara: A representation theorem for solutions of higher order strongly elliptic systems, In: A. Cialdea (ed.), Homage to Gaetano Fichera, Quad. Mat., vol. 7. Dept. Math., Seconda Univ. Napoli, Caserta, (2000), 233–271.
  • F. Lanzara: On BVPs for strongly elliptic systems with higher order boundary conditions, Georgian Math. J., 14 (1) (2007), 145–167.
  • P. E. Ricci: Sui potenziali di semplice strato per le equazioni ellittiche di ordine superiore in due variabili, Rend. Mat. (6), 7 (1974), 1–39.
  • N. P. Vekua: Systems of singular integral equations, P. Noordhoff Ltd., Groningen (1967).
Year 2024, , 129 - 141, 16.12.2024
https://doi.org/10.33205/cma.1540457

Abstract

References

  • S. Agmon: Multiple layer potentials and the Dirichlet problem for higher order elliptic equations in the plane I, Comm. Pure Appl. Math., 10 (1957), 179–239.
  • A. Cialdea: Teoremi di completezza connessi con equazioni ellittiche di ordine superiore in due variabili in un campo con contorno angoloso, Rend. Circ. Mat. Palermo (2), 35 (1) (1986), 32–49.
  • A. Cialdea: A general theory of hypersurface potentials, Ann. Mat. Pura Appl., 168 (1995), 37–61.
  • A. Cialdea: Completeness theorems on the boundary in thermoelasticity, In: Analysis as a life, Trends Math. Birkhäuser/Springer, Cham, (2019), 93–115.
  • G. Fichera: Teoremi di completezza sulla frontiera di un dominio per taluni sistemi di funzioni, Ann. Mat. Pura Appl., 27 (1948), 1–28.
  • G. Fichera: Approssimazione uniforme delle funzioni olomorfe mediante funzioni razionali aventi poli semplici prefissati I & II, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 27 (1959), 193–201, 317–323.
  • G. Fichera: Linear elliptic equations of higher order in two independent variables and singular integral equations, with applications to anistropic inhomogeneous elasticity, In: R.E. Langer (ed.), Partial differential equations and continuum mechanics, Univ. Wisconsin Press, Madison, WI, (1960), 55–80.
  • G. Fichera: The problem of the completeness of systems of particular solutions of partial differential equations, In: Numerical mathematics (Sympos., Inst. Appl. Math., Univ. Hamburg, Hamburg, 1979), Internat. Ser. Numer. Math., vol. 49. Birkhäuser, Basel-Boston, Mass., (1979), 25–41.
  • G. Fichera, L. De Vito: Funzioni analitiche di una variabile complessa, Terza edizione, Libreria Eredi Virgilio Veschi, Rome (1967).
  • G. Fichera, P. E. Ricci: The single layer potential approach in the theory of boundary value problems for elliptic equations, In: Function theoretic methods for partial differential equations (Proc. Internat. Sympos., Darmstadt, 1976), Lecture Notes in Math., vol. Vol. 561. Springer, Berlin-New York, (1976), 39–50.
  • F. Lanzara: A representation theorem for solutions of higher order strongly elliptic systems, In: A. Cialdea (ed.), Homage to Gaetano Fichera, Quad. Mat., vol. 7. Dept. Math., Seconda Univ. Napoli, Caserta, (2000), 233–271.
  • F. Lanzara: On BVPs for strongly elliptic systems with higher order boundary conditions, Georgian Math. J., 14 (1) (2007), 145–167.
  • P. E. Ricci: Sui potenziali di semplice strato per le equazioni ellittiche di ordine superiore in due variabili, Rend. Mat. (6), 7 (1974), 1–39.
  • N. P. Vekua: Systems of singular integral equations, P. Noordhoff Ltd., Groningen (1967).
There are 14 citations in total.

Details

Primary Language English
Subjects Approximation Theory and Asymptotic Methods
Journal Section Articles
Authors

Alberto Cialdea 0000-0002-0009-5957

Flavia Lanzara 0000-0002-2052-4202

Early Pub Date December 16, 2024
Publication Date December 16, 2024
Submission Date August 29, 2024
Acceptance Date November 13, 2024
Published in Issue Year 2024

Cite

APA Cialdea, A., & Lanzara, F. (2024). Completeness theorems related to BVPs satisfying the Lopatinskii condition for higher order elliptic equations. Constructive Mathematical Analysis, 7(Special Issue: AT&A), 129-141. https://doi.org/10.33205/cma.1540457
AMA Cialdea A, Lanzara F. Completeness theorems related to BVPs satisfying the Lopatinskii condition for higher order elliptic equations. CMA. December 2024;7(Special Issue: AT&A):129-141. doi:10.33205/cma.1540457
Chicago Cialdea, Alberto, and Flavia Lanzara. “Completeness Theorems Related to BVPs Satisfying the Lopatinskii Condition for Higher Order Elliptic Equations”. Constructive Mathematical Analysis 7, no. Special Issue: AT&A (December 2024): 129-41. https://doi.org/10.33205/cma.1540457.
EndNote Cialdea A, Lanzara F (December 1, 2024) Completeness theorems related to BVPs satisfying the Lopatinskii condition for higher order elliptic equations. Constructive Mathematical Analysis 7 Special Issue: AT&A 129–141.
IEEE A. Cialdea and F. Lanzara, “Completeness theorems related to BVPs satisfying the Lopatinskii condition for higher order elliptic equations”, CMA, vol. 7, no. Special Issue: AT&A, pp. 129–141, 2024, doi: 10.33205/cma.1540457.
ISNAD Cialdea, Alberto - Lanzara, Flavia. “Completeness Theorems Related to BVPs Satisfying the Lopatinskii Condition for Higher Order Elliptic Equations”. Constructive Mathematical Analysis 7/Special Issue: AT&A (December 2024), 129-141. https://doi.org/10.33205/cma.1540457.
JAMA Cialdea A, Lanzara F. Completeness theorems related to BVPs satisfying the Lopatinskii condition for higher order elliptic equations. CMA. 2024;7:129–141.
MLA Cialdea, Alberto and Flavia Lanzara. “Completeness Theorems Related to BVPs Satisfying the Lopatinskii Condition for Higher Order Elliptic Equations”. Constructive Mathematical Analysis, vol. 7, no. Special Issue: AT&A, 2024, pp. 129-41, doi:10.33205/cma.1540457.
Vancouver Cialdea A, Lanzara F. Completeness theorems related to BVPs satisfying the Lopatinskii condition for higher order elliptic equations. CMA. 2024;7(Special Issue: AT&A):129-41.