<p>In this paper we prove “regularizing effects” for the boundary value problems (with convection terms) of the type <Equation ID="Equ35"> <EquationSource Format="TEX">\( {\left\{ \begin{array}{ll} -\textrm{div}(M(x){D} u)+\textrm{div}(u\,E(x))+a(x)\,u= f(x), &amp; \text{ in } \Omega ,\\ u (x) = 0, &amp; \text{ on } \partial \Omega , \end{array}\right. } \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mo>-</mo> <mtext>div</mtext> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>D</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mtext>div</mtext> <mo stretchy="false">(</mo> <mi>u</mi> <mspace width="0.166667em" /> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="0.166667em" /> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>on</mtext> <mspace width="0.333333em" /> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation></p>