Dirichlet problem characterization of regularity (Q1340220)

The author studies the regularity of the solutions to the Dirichlet problem \[ Lu\equiv -(a_{ij} u_{x_ i} )_{x_ j}= f \quad\text{in }\Omega, \qquad u=0 \quad \text{in }\partial\Omega \tag{1} \] where \(\mu\) is a given bounded variation measure on \(\Omega\) (bounded open subset of \(\mathbb{R}^ n\)), \(a_{ij}= a_{ji}\in L^ \infty (\Omega)\), \(i,j= 1,\dots, n\) and the ellipticity condition \(\nu^{-1} |\xi |^ 2\leq a_{ij} \xi_ i\xi_ j\leq \nu| \xi|^ 2\), \(\forall\xi \in\mathbb{R}^ n\), \(\nu>0\) is satisfied. By means of the definition of very weak solutions to (1) and Green functions he finds spaces to which the measure \(\mu\) must belong in order to get the solution \(u\) is in some \(L^ p\) classes, in \(L^ \infty\), in \(C^ 0\), and in \(C^{0,\alpha}\). The following propositions are proved: Let \(f\in L^ 1 (\Omega)\), \(f\geq 0\) and \(u\in L^ 1 (\Omega)\) be the very weak solution to (1). Then (a) \(u\in L^ p_{\text{loc}} (\Omega)\) iff \(f\in M^ p_{\text{loc}} (\Omega)\), \(1<p <\infty\) (Schechter spaces), (b) \(u\in L^ \infty_{\text{loc}} (\Omega)\) iff \(f\in \widetilde {S}_{\text{loc}} (\Omega)\) (Stummel-Kato spaces), (c) \(u\in C^ 0_{\text{loc}} (\Omega)\) iff \(f\in S_{\text{loc}} (\Omega)\) (Stummel-Kato spaces), (d) \(u\in C^{0,\alpha}_{\text{loc}} (\overline {\Omega})\) iff \(\widetilde {f}\in L_{\text{loc}}^{1,n- 2+ 2\alpha} (\Omega)\) (Morrey spaces). Here he uses the functions on the right hand side of (1) for simplicity in the statements and proofs only.