2 Added discussion of 1D behavior
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It appears that PlotRange must be specified to assure that the Full range of data is displayed in the second DensityPlot. Also, specifying PlotPoints improves the appearance of the first DensityPlot.

DensityPlot[Max[0, 0.1 - Abs[1. - Sqrt[x^2 + y^2]]], {x, -1.1, 1.1}, {y, -1.1, 1.1}, 
ColorFunction -> (Opacity[#, Blue] &), PlotPoints -> 100, Frame -> False, PlotRange -> Full]

enter image description here

DensityPlot[Max[0, 0.1 - Abs[1. - Sqrt[x^2 + y^2]]], {x, -1.2, 1.2}, {y, -1.2, 1.2}, 
ColorFunction -> (Opacity[#, Blue] &), PlotPoints -> 100, Frame -> False, PlotRange -> Full]

enter image description here

Update - Corresponding 1-D behavior

Plot exhibits similar behavior

Plot[Max[0, 0.1 - Abs[1. - Sqrt[x^2]]], {x, -1.2, 1.2}]

enter image description here

omits part of the curve unless PlotPoints is set to 100 or more. And,

Plot[Max[0, 0.1 - Abs[1. - Sqrt[x^2]]], {x, -1.1, 1.1}]

enter image description here

has far too small a PlotRange, unless it is set explicitly to All or Full. This same behavior persists for ranges as large as about {x, -1.14, 1.14}, and as small as about {x, -1.09, 1.09}. It is not uncommon that Mathematica plotting routines have difficulty with functions that are zero except in narrow ranges.

It appears that PlotRange must be specified to assure that the Full range of data is displayed in the second DensityPlot. Also, specifying PlotPoints improves the appearance of the first DensityPlot.

DensityPlot[Max[0, 0.1 - Abs[1. - Sqrt[x^2 + y^2]]], {x, -1.1, 1.1}, {y, -1.1, 1.1}, 
ColorFunction -> (Opacity[#, Blue] &), PlotPoints -> 100, Frame -> False, PlotRange -> Full]

enter image description here

DensityPlot[Max[0, 0.1 - Abs[1. - Sqrt[x^2 + y^2]]], {x, -1.2, 1.2}, {y, -1.2, 1.2}, 
ColorFunction -> (Opacity[#, Blue] &), PlotPoints -> 100, Frame -> False, PlotRange -> Full]

enter image description here

It appears that PlotRange must be specified to assure that the Full range of data is displayed in the second DensityPlot. Also, specifying PlotPoints improves the appearance of the first DensityPlot.

DensityPlot[Max[0, 0.1 - Abs[1. - Sqrt[x^2 + y^2]]], {x, -1.1, 1.1}, {y, -1.1, 1.1}, 
ColorFunction -> (Opacity[#, Blue] &), PlotPoints -> 100, Frame -> False, PlotRange -> Full]

enter image description here

DensityPlot[Max[0, 0.1 - Abs[1. - Sqrt[x^2 + y^2]]], {x, -1.2, 1.2}, {y, -1.2, 1.2}, 
ColorFunction -> (Opacity[#, Blue] &), PlotPoints -> 100, Frame -> False, PlotRange -> Full]

enter image description here

Update - Corresponding 1-D behavior

Plot exhibits similar behavior

Plot[Max[0, 0.1 - Abs[1. - Sqrt[x^2]]], {x, -1.2, 1.2}]

enter image description here

omits part of the curve unless PlotPoints is set to 100 or more. And,

Plot[Max[0, 0.1 - Abs[1. - Sqrt[x^2]]], {x, -1.1, 1.1}]

enter image description here

has far too small a PlotRange, unless it is set explicitly to All or Full. This same behavior persists for ranges as large as about {x, -1.14, 1.14}, and as small as about {x, -1.09, 1.09}. It is not uncommon that Mathematica plotting routines have difficulty with functions that are zero except in narrow ranges.

1
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It appears that PlotRange must be specified to assure that the Full range of data is displayed in the second DensityPlot. Also, specifying PlotPoints improves the appearance of the first DensityPlot.

DensityPlot[Max[0, 0.1 - Abs[1. - Sqrt[x^2 + y^2]]], {x, -1.1, 1.1}, {y, -1.1, 1.1}, 
ColorFunction -> (Opacity[#, Blue] &), PlotPoints -> 100, Frame -> False, PlotRange -> Full]

enter image description here

DensityPlot[Max[0, 0.1 - Abs[1. - Sqrt[x^2 + y^2]]], {x, -1.2, 1.2}, {y, -1.2, 1.2}, 
ColorFunction -> (Opacity[#, Blue] &), PlotPoints -> 100, Frame -> False, PlotRange -> Full]

enter image description here