Classical regularity theory for linear problems

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


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
$$\begin{array}{*{20}{c}} { - \sum\limits_{\alpha ,\beta ,j} {{\partial _{{\chi _\alpha }}}\left( {A_{ij}^{\alpha \beta }{\partial _{{\chi _\beta }}}{u^j}} \right) = {f_i} - \sum\limits_\alpha {{\partial _{{\chi _\alpha }}}F_i^\alpha } } }&{i = 1, \ldots ,m} \end{array}$$
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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