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Fritz John pp 258-266 | Cite as

Derivatives of Continuous Weak Solutions of Linear Elliptic Equations

  • Fritz John
Chapter
Part of the Contemporary Mathematicians book series (CM)

Abstract

Let L be a linear elliptic differential operator in n-space of order m. Let M denote the adjoint operator to L. An integrable function u is called a weak solution of the equation L[u] = f in a domain D, if for every w of class C , which vanishes outside a compact subset of D, the relation
$$ \int_D {(uM[w]) - fw)d{x_1}...d{x_n}} $$
holds; u will be called a strict solution, if u is of class C m and satisfies L[u] =f in the ordinary sense. One of the remarkable facts concerning elliptic equations is that under suitable regularity assumptions on f and the coefficients of L a weak solution can be differentiated any number of times and is a strict solution. In a recent paper F. E. Browder1 gives the theorem: a weak solution which is square integrable on every compact subset of D is almost everywhere equal to a strict solution, provided f is in C 1 and the coefficients of the j-th derivatives in the operator L are in C m+i . Browder’s proof of this generalization of “Weyl’s lemma” makes use of the fundamental solution of elliptic equations with analytic coefficients.2 Results contained in a paper by K. O. Friedrichs,3 which appears in the same issue, imply the theorem that a weak solution is a strict solution, provided f is in C (n+1)/2 and the coefficients of L are in C m/2.

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Copyright information

© Springer Science+Business Media New York 1985

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  • Fritz John

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