Partial Regularity of the Gradient for Subsolutions

Abstract For any subharmonic function u , we prove that $$|\partial _j u|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> ( $$j=1, \ldots , n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> ) is upper semi-continuous, provided that the super-level sets of u can be touched from the exterior by uniform $$C^{1,\text {Dini}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mtext>Dini</mml:mtext> </mml:mrow> </mml:msup> </mml:math> domains at every point. This idea extends to a class of general operators, as well as to the boundary behaviour of the gradient of solutions of the Dirichlet problem in a domain whose boundary satisfies this geometric condition.