JournalsrlmVol. 29, No. 3pp. 557–577

W2,2W^{2,2}-solvability of the Dirichlet problem for a class of elliptic equations with discontinuous coefficients

  • Flavia Giannetti

    Università degli Studi di Napoli Federico II, Italy
  • Gioconda Moscariello

    Università degli Studi di Napoli Federico II, Italy
$W^{2,2}$-solvability of the Dirichlet problem for a class of elliptic equations with discontinuous coefficients cover
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Abstract

We study the Dirichlet problem for the second order elliptic equation

i,j=1Naij(x)2uxixj=f(x)-\sum_{i,j=1}^Na_{ij}(x) \frac{\partial^2u}{\partial x_i\partial x_j}= f(x)

in a bounded regular domain ΩRN,N>2\Omega\subset\mathbb R^N, N > 2. We assume that fL2f\in L^{2} and that the coefficients aija_{ij} are measurable and bounded functions with the first derivatives in the Marcinkiewicz class weak LN-\,L^{N} and having a sufficiently small distance to LL^{\infty}. Under these assumptions we prove the solvability of the problem in W2,2W01,2W^{2,2}\cap W_0^{1,2^*}, where 2=2NN22^*=\frac{2N}{N-2}. An higher integrability result for the gradient of the solution is achieved when fLp,p>2f\in L^p, p > 2.

Cite this article

Flavia Giannetti, Gioconda Moscariello, W2,2W^{2,2}-solvability of the Dirichlet problem for a class of elliptic equations with discontinuous coefficients. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29 (2018), no. 3, pp. 557–577

DOI 10.4171/RLM/820