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 Mat. Sb., 1996, Volume 187, Number 1, Pages 17–40 (Mi msb98)

A priori estimates and smoothness of solutions of a system of quasi-linear equations that is elliptic in the Douglis–Nirenberg sense

G. V. Grishina

N. E. Bauman Moscow State Technical University

Abstract: We study a Douglis–Nirenberg elliptic system of quasi-linear equations. We solve the problem of the limiting admissible rate of growth of the non-linear terms of the system with respect to their arguments consistent with the possibility of obtaining estimates of the derivatives of a solution in terms of its maximum absolute value. The restrictions on the smoothness of the non-linear terms are minimal and the results are sharp. We construct an example that shows the optimality of the upper bound for the exponent of growth. A priori $L_p$-estimates are obtained both inside the domain for solutions belonging to certain Sobolev spaces. We obtain estimates of the Hölder norms of the derivatives of a solutions. We prove a theorem on a removable isolated singularity of bounded solutions of general elliptic systems of quasi-linear equation. All results are new, even for a single second-order equation.

DOI: https://doi.org/10.4213/sm98

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English version:
Sbornik: Mathematics, 1996, 187:1, 15–38

Bibliographic databases:

UDC: 517.956.2
MSC: Primary 35J30; Secondary 35B05

Citation: G. V. Grishina, “A priori estimates and smoothness of solutions of a system of quasi-linear equations that is elliptic in the Douglis–Nirenberg sense”, Mat. Sb., 187:1 (1996), 17–40; Sb. Math., 187:1 (1996), 15–38

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb98
• https://doi.org/10.4213/sm98
• http://mi.mathnet.ru/eng/msb/v187/i1/p17

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