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Classical regularity theory for linear problems

  • Luigi Ambrosio
  • Alessandro Carlotto
  • Annalisa Massaccesi
Chapter
Part of the Publications of the Scuola Normale Superiore book series (PSNS, volume 18)

Abstract

In this chapter we begin studying the local behavior of (weak) solutions\(u \in H_{{\text{loc}}}^1\left( {\Omega ;{\mathbb{R}^m}} \right)\) of a system of equations given by
$$\begin{array}{*{20}{c}} { - \sum\limits_{\alpha ,\beta ,j} {{\partial _{{\chi _\alpha }}}\left( {A_{ij}^{\alpha \beta }{\partial _{{\chi _\beta }}}{u^j}} \right) = {f_i} - \sum\limits_\alpha {{\partial _{{\chi _\alpha }}}F_i^\alpha } } }&{i = 1, \ldots ,m} \end{array}$$
(2.1)
with\(A_{ij}^{\alpha \beta } \in {L^\infty }(\Omega ;\mathbb{R}),\,{f_i} \in L_{{\text{loc}}}^2(\Omega ;\mathbb{R})\) and\(F_i^\alpha \in L_{{\text{loc}}}^2(\Omega ;\mathbb{R})\).

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Copyright information

© Scuola Normale Superiore Pisa 2018

Authors and Affiliations

  • Luigi Ambrosio
    • 1
  • Alessandro Carlotto
    • 2
  • Annalisa Massaccesi
    • 3
  1. 1.Scuola Normale SuperiorePisaItalia
  2. 2.ETHZürichSwitzerland
  3. 3.Università di VeronaVeronaItalia

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