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

Yıl 2024, , 129 - 141, 16.12.2024

Alberto Cialdea , Flavia Lanzara

https://doi.org/10.33205/cma.1540457

Öz

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 $Eu=0$.

Anahtar Kelimeler

Completeness theorems, Lopatinskii condition, Elliptic equations of higher order, Partial differential equations with constant coefficients

Kaynakça

• 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).