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Absence of De Giorgi-type theorems for strongly elliptic equations with complex coefficients. (English) Zbl 0562.35030

Translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 115, 156-168 (Russian) (1982; Zbl 0498.35033).

MSC:

35J15 Second-order elliptic equations
35B35 Stability in context of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)

Citations:

Zbl 0498.35033
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Full Text: DOI

References:

[1] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968).
[2] E. De Giorgi, ”Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari,” Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat.,3, 25–43 (1957). · Zbl 0084.31901
[3] E. De Giorgi, ”Un esempio di estremali discontinue per un problems variazionale di tipo ellittico,” Boll. Un. Mat. Ital., Ser. IV, No. 1, 135–137 (1968). · Zbl 0155.17603
[4] V. G. Maz’ya, ”Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients,” Funkts. Anal. Prilozhen.,2, No. 3, 53–57 (1968).
[5] E. Giusti and M. Miranda, ”Un esempio di soluzioni discontinue per un problems di minimo relativo ad un integrale regolare del calcolo delle variazioni,” Boll. Un. Mat. Ital. Ser. IV, No. 2, 219–226 (1968). · Zbl 0155.44501
[6] C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York (1966). · Zbl 0142.38701
[7] A. A. Samarskii, ”On the effect of clamping on the eigenfrequencies of closed domains,” Usp. Mat. Nauk,5, No. 3, 133–134 (1950).
[8] C. A. Swanson, ”Asymptotic variational formulas for eigenvalues,” Can. Math. Bull.,6, No. 1, 15–25 (1963). · Zbl 0111.29803
[9] V. G. Maz’ya, S. A. Nazarov, and B. A. Plamenevskii, Asymptotic Behavior of the Solutions of Elliptic Boundary Value Problems under Singular Perturbations of the Domain [in Russian], Tbilis. Gos. Univ., Inst. Prikl. Mat., Tbilisi (1981).
[10] F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Wiley-Interscience, New York (1955). · Zbl 0067.32101
[11] V. A. Kondrat’ev, ”Boundary-value problems for elliptic equations in domains with conical or angular points,” Tr. Mosk. Mat. Obshch.,16, 209–292 (1967).
[12] V. G. Maz’ya and B. A. Plamenevskii, ”On the coefficients in the asymptotics of the solutions of elliptic boundary-value problems in domains with conical points,” Math. Nachr.,76, 29–60 (1977). · Zbl 0359.35024
[13] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Amer. Math. Soc., Providence, R.I. (1969). · Zbl 0181.13503
[14] I. M. Gel’fand and G. E. Shilov, Generalized Functions. Vol. I: Properties and Operations, Academic Press, New York (1964).
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