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A priori majorization of the solutions of the boundary value problems

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Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2,volume 2)

Abstract

In the scope of the study of boundary value problems for nonlinear elliptic equations, it is essential to deepen the study of the solutions of linear equations for the purpose of establishing certain majorization formulas for them, which are refined as much more as possible. Along this line of thought, the first problem which presents itself is that of majorizing, by functions of the data of the problem, the maximum moduli and the HÖLDER coefficients of the solution and its derivatives and of establishing which regularity properties hold for this solution as consequences of appropriate regularity properties on the data of the problem. The importance of this research was recognized by G. Giraud since his first works [2, 4] in which, departing from theorems proved by himself on existence and on the representation of the solution by means of potentials, he succeeded in bringing some contribution to the study of the question.

Keywords

  • Weak Solution
  • Elliptic Equation
  • DIRICHLET Problem
  • Existence Theorem
  • Elliptic Operator

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Reference

  1. J. Marcinkiewicz: Sur les multiplicateurs des séries de Fourier. Studia Math. 8 (1939) 78–91.

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  2. G. Stampacchia: The spaces I(p,), N(p,) and interpolation. Ann. Sc. N. Sup. Pisa 19 (1965) 443–462.

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© 1970 Springer-Verlag Berlin · Heidelberg

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Miranda, C. (1970). A priori majorization of the solutions of the boundary value problems. In: Partial Differential Equations of Elliptic Type. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-87773-5_5

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  • DOI: https://doi.org/10.1007/978-3-642-87773-5_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-87775-9

  • Online ISBN: 978-3-642-87773-5

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